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AVALANCHES IN DRIVEN SYSTEMS

Ph.D. Degree in Physics

Submitted bySANJIB SABHAPANDIT

Under the Guidance ofDEEPAK DHAR

Tata Institute of Fundamental ResearchMumbai 400 005, India

2002

AVALANCHES IN DRIVEN SYSTEMS

SANJIB SABHAPANDIT

Tata Institute of Fundamental ResearchMumbai 400 005, India

A thesis submitted to the

University of Mumbai

for thePh.D. Degree in Physics

2002

Acknowledgments

I am deeply indebted to my thesis advisers Deepak Dhar for his constant invaluable guid-ance and encouragement throughout the last five years. His patience and persistence, in-sights into various problems, critical thinking and insistence on clarity have been mostuseful and inspiring. My words will never be adequate to express my gratitude towardshim.

Mustansir has always been available for any kind of discussion. I learnt many things fromhis two courses on statistical mechanics.

Satya has been a very cheerful friend. Discussions with him about physics and non-physicstopics have always been very enjoyable.

I have had a nice time with many friends in the Institute. Lots of my free time in TIFRwas spent in the enjoyable company of Rajesh, Arun, Dibyendu, Anwesh, Anu, Bhaswati,Ritu, Palit, Patta, Ajay Nandgaonkar. Azizur, Praveen, Vinod, Gulab, Gagan have beengood friends since my joining in TIFR. I would like to thank all my other friends, withwhom I had numerous weekend (!) parties.

The theory students room has been a very lively place. I would like to thank Goutam, Ab-hishek, Keshav, Saumen, Justin, Rajesh, Ghosal, Arun, Dibyendu, Nemani, Patta, Kavita,Dariush, Arti, Sumedha, Ramanan, Ajay, Apoorva, Ashik, Ashotosh and Punyabrata forall the enjoyable discussions.

A special word of thanks to Bathija, Girish, Pawar and Shinde for helping out on officialmatters and for their friendliness.

I acknowledge the “Kanwal Rekhi Scolarship for Career Devlopment” awarded to me bythe TIFR Endowment Fund in the academic year 2001-2002.

I would like to thank all the people involved in the development of GNU/Linux and TEX(LATEX),without whom this work would have been very difficult.

Finally, I would like to express my heartfelt gratitude to my parents, and brothers for theirlove and support.

STATUTORY DECLARATIONS

Name of the Candidate : Sanjib Sabhapandit

Title of the Thesis : Avalanches in Driven Systems

Degree : Doctor of Philosophy (Ph. D.)

Subject : Physics

Name of the Guide : Prof. Deepak Dhar

Registration number and date : TIFR/200, 30��

March 1999

Place of Research : Tata Institute of Fundamental Research,

Mumbai 400005

STATEMENT BY THE CANDIDATE

As required by the University Ordinances 770 and 771, I wish to state that the work

embodied in this thesis titled “Avalanches in Driven Systems” forms my own contribu-

tion to the research work carried out under the guidance of Prof. Deepak Dhar at the

Tata Institute of Fundamental Research and in collaboration with Prof. Prabodh Shukla of

North Eastern Hill University. This work has not been submitted for any other degree of

this or any other University. Whenever references have been made to previous works of

others, it has been clearly indicated as such and included in the Bibliography.

Signature of Candidate

Name Sanjib Sabhapandit

Certified by

Signature of Guide

Name Prof. Deepak Dhar

Contents

List of Figures iii

Synopsis v

List of Publications xv

1 Introduction 1

1.1 Hysteresis in ferromagnets . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Barkhausen effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Hysteresis in random field Ising model . . . . . . . . . . . . . . . . . . . 7

1.4 Equilibrium properties of random field Ising model . . . . . . . . . . . . 12

1.5 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Earlier exact results on hysteresis in random field Ising model 16

2.1 Return point memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Abelian property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Hysteresis in the infinite-range interaction model . . . . . . . . . . . . . 19

2.4 Hysteresis on the Bethe lattice . . . . . . . . . . . . . . . . . . . . . . . 20

3 Distribution of avalanche sizes on the Bethe lattice 26

3.1 Generating function for avalanche distribution . . . . . . . . . . . . . . . 26

3.2 Explicit calculation for the rectangular distribution . . . . . . . . . . . . 30

3.3 General distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Minor hysteresis loops on the Bethe lattice 38

4.1 Magnetization on minor loops . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 Merging of different stable configurations . . . . . . . . . . . . . . . . . 45

i

Contents ii

5 Hysteresis on regular lattices in the low disorder limit 48

5.1 Hysteresis on three coordinated lattices . . . . . . . . . . . . . . . . . . 49

5.2 Bootstrap instability in RFIM on square lattice . . . . . . . . . . . . . . . 52

5.3 Bootstrap instability in RFIM on cubic lattice . . . . . . . . . . . . . . . 56

6 Discussion 59

Appendix A 62

A.1 Avalanche distribution on a linear chain . . . . . . . . . . . . . . . . . . 62

A.2 Avalanche distribution on a three coordinated Bethe lattice . . . . . . . . 64

Reprints 66

Bibliography 84

List of Figures

0.1 A Cayley tree of coordination number �� and height �� . . . . . . . viii

0.2 Behavior of RFIM in the magnetic field - disorder plane for a linear chain. ix

0.3 Minor hysteresis loops . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

1.1 Rayleigh hysteresis loop . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Magnetization curve of single crystal of silicon iron . . . . . . . . . . . . 4

1.3 Barkhausen effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Barkhausen jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 An example of experimental Barkhausen signal . . . . . . . . . . . . . . 8

1.6 Time series of the avalanches in random field Ising model . . . . . . . . . 8

1.7 Experimental data for the distribution of Barkhausen avalanches . . . . . 9

1.8 Evolution of spin configuration. . . . . . . . . . . . . . . . . . . . . . . 11

1.9 Domain of reverse spins. . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1 Return point memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Partial ordering and return point memory . . . . . . . . . . . . . . . . . 18

2.3 Magnetization curve for the random field Ising model with infinite-range

interaction at various values of disorder . . . . . . . . . . . . . . . . . . 21

2.4 A Cayley tree of coordination number 3 and 4 generations. . . . . . . . . 22

2.5 Variation of � �� with � for the Bethe lattice with �� . . . . . . . . . 25

2.6 Magnetization in the increasing field for the Bethe lattice with �� . . . . 25

3.1 A sub-tree �� formed by � and its descendents. . . . . . . . . . . . . . . 27

3.2 Behavior of RFIM in the magnetic field - disorder plane for a linear chain. 31

3.3 Hysteresis loops for the linear chain for the rectangular distribution of

quenched fields with different widths. . . . . . . . . . . . . . . . . . . . 32

3.4 Behavior of RFIM in the magnetic field - disorder plane for Bethe lattice

of coordination number 3. . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.5 A schematic plot of a unimodal random field distribution. . . . . . . . . . 35

iii

List of Figures iv

4.1 Minor hysteresis loops for Bethe lattice. . . . . . . . . . . . . . . . . . . 39

4.2 Time order at which spins flip during a particular avalanche. . . . . . . . 41

4.3 Merging of different stable configurations . . . . . . . . . . . . . . . . . 47

5.1 Magnetization in the increasing field in 2 and 3 dimensions. . . . . . . . . 50

5.2 A three coordinated lattice in three dimensions . . . . . . . . . . . . . . 51

5.3 A snapshot of the up-spins just before the jump. . . . . . . . . . . . . . . 52

5.4 Distribution of the scaled coercive field on a square lattice for different

lattice size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.5 �� coer � vs. ��! #"%$ for square lattice. . . . . . . . . . . . . . . . . . . . 55

Synopsis

The study of nonequilibrium properties of systems driven by external forcing, has attracted

a considerable interest in the recent years. There is a wide variety of systems (e.g. sand

or rice piles, forest fires, earthquakes, vortices in dirty type II superconductors, solid on

solid friction, moving of interfaces in random media, disordered ferromagnets), which pass

from one metastable state to another metastable state, through avalanche-like dynamics,

in response to slowly varying external conditions (driving force). Due to the presence

of multiple metastable states, these systems exhibit history dependence and hysteresis.

The phenomenon of hysteresis was first studied in the context on magnets. An important

quantitative measure of this burst like response of the system to external perturbations is

the distribution of avalanche sizes.

If a ferromagnet is subjected to varying field, magnetization change occurs in small

jumps, give rise to a noise in the speaker attached to a pickup coil, called Barkhausen

noise. Experimentally it is observed (Spasojevic et al. 1996, and references therein) that

the distribution of magnetization jumps exhibits a power law behavior over a few decades

with a cutoff. Sethna et al. (1993) introduced the random field Ising model (RFIM) as

a simple theoretical model for hysteresis and Barkhausen noise in ferromagnets. In this

thesis, we consider the RFIM with ferromagnetic nearest neighbour interaction, evolving

under zero-temperature Glauber single-spin-flip dynamics, in the presence of an slowly

varying external magnetic field on a Bethe lattice an on regular lattices. We study the

distribution of avalanche sizes and minor hysteresis loops on a Bethe lattice and the slow

variation of coercive field with system size on regular lattices.

The advantage of working on the Bethe lattice is that the usual BBGKY hierarchy of

equations for correlation functions closes, and one can hope to set up exact self-consistent

equations for the correlation functions. The fact that Bethe’s self-consistent approximation

becomes exact on the Bethe lattice is useful as it ensures that the approximation will not vi-

v

Synopsis vi

olate any general theorems, e.g. the convexity of thermodynamic functions, sum rules etc.

In the presence of disorder, in spite of the closure of the BBGKY hierarchy, the Bethe ap-

proximation is still very difficult, as the self-consistent equations become functional equa-

tions for the probability distribution of the effective field. These are not easy to solve, and

available analytical results in this direction are mostly restricted to one dimension (Derrida

et al. 1978, Puma et al. 1978, Vilenkin 1974), or to models with infinite-ranged interac-

tions (Sherrington and Kirkpatrick 1975). The RFIM model on a Bethe lattice is special

in that the zero-temperature nonequilibrium response to a slowly varying magnetic field

can be determined exactly (Dhar et al. 1997). To be precise, the average non-equilibrium

magnetization in this model can be determined exactly if the magnetic field is increased

very slowly, from & ' to ()' , in the limit of zero temperature. It thus provides a good the-

oretical model to study the slow relaxation to equilibrium in glassy systems. The dynamics

is governed by the existence of many metastable states, with energy barriers separating

different metastable states.

In this thesis, we derive the exact self-consistent equations for the avalanche distribu-

tion function in the RFIM on a Bethe lattice (Sabhapandit et al. 2000). We solve these

equations explicitly for the rectangular distribution of the quenched field, for the linear

chain �*�,+ , and the 3-coordinated Bethe lattice and also get the exact hysteresis loop for

this distribution. For more general coordination numbers, and general continuous distribu-

tions of random fields, we argued that for very large disorder, the avalanche distribution

is exponentially damped, but for small disorder, for continuous unbounded distributions

of random fields, generically, one gets a jump in magnetization, and near the jump the

avalanche distribution function has a power-law tail.

A rather surprising feature of the hysteresis loops on Bethe lattice is that the behavior

when the coordination number � is�

is very different from that if �.- �. For �/� �

, for

continuous unbounded distributions of random fields, the hysteresis loops show no jump

discontinuity of magnetization even in the limit of small disorder, but for higher � they do.

In general, the Bethe approximation is expected to work well away from the critical region,

and hence in the hysteresis problem. Therefore, we ask the question: is this � -dependence

of hysteresis loops due to the peculiarities of the Bethe lattice, or even persists on regular

lattices? We study the hysteresis in the limit of low disorder in two and three dimensions,

and we relate the spin flip process to bootstrap percolation (Sabhapandit et al. 2002). We

find that the � -dependence of the hysteresis loop persists on regular lattices. However, the

value of coercive field is not correctly predicted by the Bethe approximation.

We also study the minor hysteresis loops in this model on Bethe lattice, for the case

Synopsis vii

where we start with external field �0�,& ' , where all the spins are down, then increase the

field to �� , subsequently reverse to �21 , again reverse to �43 and so on. We obtain the exact

expression for magnetization on the minor loops as a function of external field for arbitrary

distribution of quenched random fields.

There is a controversy in literature whether the Barkhausen noise is an example of

self-organized criticality (SOC). The power-law tail in the event-size distribution was in-

terpreted by Cote and Meisel (1991) as an example of SOC. But from simulation results

in RFIM, Perkovic, Dahmen and Sethna (1995) have argued that large bursts are expo-

nentially rare, and the approximate power-law tail of the observed distribution comes from

crossover effects due to nearness of a critical point. Our analytical results about the behav-

ior of the avalanche distribution function also answer this question.

We now define the model precisely and describe the results obtained in more details.

Definition of the model

In RFIM, the Ising spins 57698!�;:<�7= interacts with nearest neighbors through a ferromag-

netic interaction > . There are quenched random fields �28 at each site ? drawn independently

from a continuous distribution @A��48B� . The entire system is placed in an externally applied

uniform field � . The Hamiltonian of the system isC �D&E>GFH 8#I JLK 6M8B6NJO&PF 8 �Q8B6M82&R�SF 8 6M8UTThe system evolves under the zero-temperature Glauber single-spin-flip dynamics (see, for

example: Kawasaki 1972): a spin-flip is allowed only if the process lowers energy. We

assume that the rate of spin-flips is much larger than the rate at which � is changed, so that

all flippable spins may be said to relax instantly, and any spin 6�8 always remains parallel to

the net local field V�8 at the site:698�� sign �WV�8X�Y� sign �Z>\[FJL]�� 6NJA(^�Q8Q(^�2�_TDue to the nearest neighbour interaction, a flipping spin often causes one or more neigh-

bours to flip also, thereby originating a whole avalanche of spin flips. As the external field

is increased adiabatically (the field is constant during the individual avalanche) from & 'to ' , the total number of spins flip at a field � , determines the increase in magnetization

and avalanche statistics at that particular field.

Synopsis viii

Distribution of avalanche sizes on a Bethe lattice

The usual way to solve a problem on Bethe lattice is to consider the problem on a Cayley

tree and calculate all the thermodynamic quantities in the deep inside of the tree. We con-

sider a uniform Cayley tree of height � where each non-boundary site has a coordination

number � (see Fig. 0.1). The level � consists of a single vertex ` , the root of the tree. Foracb � , the level �d�e& a � has �4��Y&f�g�ihkj � vertices. At each vertex there is a Ising spin. Since

in the limit �ml ' , all sites deep inside the tree are equivalent, the magnetization at the

root ` determines the mean magnetization in the deep inside the tree and hence on Bethe

lattice. nnooppqq rrss

ttuu vvww xxyy zz{{ ||}}~~��

��

o

r = 1

r = 2

r = 3

r = 4

Figure 0.1: A Cayley tree of coordination number �� and height � �¢¡ .We start with �0�;&S' , when all spins are down and slowly increase � . As we increase� , some sites where the quenched random field is large positive will find the net local

field positive, and will flip up. Flipping a spin makes the local field at neighboring sites

increase, and in turn may cause them to flip. Thus, the spins flip in clusters of variable

sizes. If increasing � by a very small amount causes 6 spins to flip up together, we shall

call this event an avalanche of size 6 . As the applied field increases, more and more spins

flip up until eventually all spins are up, and further increase in � has no effect. Consider

the state of the system at external field � , and all the flippable sites have been flipped. We

increase the field by a small amount £¤� till one more site becomes unstable and calculate

the probability that this would cause an ‘avalanche’ of ¥ spin flips.

We define, ¦)§ be the probability that an avalanche propagating in a subtree flips exactly¥ more spins in the subtree before stopping. We derive the exact self-consistent equation

for ¦*�B¨�� , the generating function of ¦E§ . This is polynomial equation in ¦*�B¨�� of degree

Synopsis ix

��&©� , in which the coefficients depend on the external field � , and the distribution of the

quenched random fields. Finally, we express the probability distribution of avalanches of

various sizes ª¬«M�� , when the external field is increased from � to �(®£¤� in terms of ¦*�B¨�� .Therefore, in principle, we can find the avalanche distribution for any given distribution of

the quenched random fields for all coordination number.

¯©° ±¯° ± ²\³!´¯µ° ².± ¶·³!´

¸ ¹º»

¼ ½¿¾

ÀÁ Â

ÃÄÅÆÇÈ

ÃÄ Å Æ ÇÈÉ ÇÉ ÆÉ Å

Figure 0.2: Behavior of RFIM in the magnetic field - disorder ( ÊOË�Ì ) plane for a linear chain. The

regions A-D correspond to qualitatively different responses. In region A all spins are down and in

region D all are up. The avalanches of finite size occur in region B and C.

We consider the special case of a rectangular distribution of the random field. In this

case, we explicitly solve the self-consistent equations for Bethe lattices with coordination

numbers ��µ+ and�. Figure 0.2 shows the different regimes showing different qualitative

behavior of the hysteresis loops for a linear chain ( �Í�Î+ ). We find that in region B, the

magnetization is a linear function of � and the function ¦G�X¨�� is independent of the applied

field � . The distribution function ª�«��Ï�2� was shown to have a simple dependence on 6 of

Synopsis x

the form ªE«��Ï�2�Ð�ÑE�k6)Ò >ÓÕÔ «�Öwhere ÑE� is a constant, that depends only on >4� Ó , and does not depend on 6 or � . In region

C, the mean magnetization is a nonlinear function of � and we show that ª<«_�� is of the

form ª¬«_�Ï�2�Ð�Ø×ÙÑ�Ú � 6(RÑÛÚ1ÝÜ Ò >ÓÕÔ «�Ö for 6 b +ÞTHere Ñ Ú � and Ñ Ú 1 have no dependence on 6 but are explicit functions of � . The integrated

distribution ßà« also has the formß<«Y�á×ÙÑS1â6(äãS1 Ü Ò >Ó Ô « ÖkåBæ�ç 6 b +ÞTThe qualitative behavior of solution for �f� �

is very similar to the linear chain. We

again get regions A-D as before, but the boundaries are shifted a bit. As before, in region

B, the average magnetization is a linear function of � , and the avalanche distribution is

independent of � . We find that in regime B, the distribution of avalanche sizes is given byª¬«��Ï�2�Ð�©�éè �Ï+ê6��âë�Ï6Û&^��_ë#�Ï6(^+ê�âëXì �Z�Û&ä>2� Ó � « Ò >ÓÍÔ «where � is a normalization constant independent of 6 and � . In the region C, we find that

the avalanche distribution is of the formª¬«��Ï�2�Y�µ�íÚ è �Ï+ê6��âë�Ï6î&ï��_ë#�Ï6(^+ê�âëXìñð «where � Ú is a normalization constant independent of 6 , and ð is a cubic polynomial in the

external field � .

We also analyse the self-consistent equations to determine the form of the avalanche

distribution for some other unimodal continuous distributions of the random field. We find

that in each case for coordination number � b , the magnetization shows a first order

jump discontinuity as a function of the applied field at some field-strength ��òó8�«dô , for weak

disorder. For �¬�© , we find that, near the discontinuity the avalanche distribution varies asª¬«��Ï�2�YõD6 j÷öø¬ù �O(áú 1Nû�ü 1�ý j « Ö for large 6 .Where û �µ�þòó8�«�ô4&f� , and ú and ü are constants. The avalanche distribution integrated over

a cycle then varies as 6�j¤ÿ�� 1 for large 6 . However, this can not be called SOC as the field is

swept across its critical value, while in SOC, we should have criticality without fine-tuning

of any parameter.

Synopsis xi

Hysteresis and bootstrap percolation

We also study the hysteresis in the RFIM in the limit of low disorder, on regular lattices,

in two and three dimensions, with an asymmetric distribution of quenched random field,

given by @A�Ï�Q8X�Y� �Ó�� Z& �Q8d� Ó ��þ��þ8X�_TWe show that the hysteresis loops on lattices with coordination number three are qualita-

tively different from those with �*- � .We find that, for �� , for fixed

Ó� > , for system size $^- �� ó>2� Ó � , the behavior

of hysteresis loops becomes independent of $ . We argue that the magnetization does not

undergo a single macroscopic jump, but many small jumps and coercive field � coer, tends to> asÓ

tends to � . Also, in our numerical simulation, we see that there is no macroscopic

jump-discontinuity for any non-zeroÓ

, for �� on 2-d (hexagonal) and 3-d lattices.

On square lattice, we observe that, at low disorder, as the external field is increased, the

domains of up spins grow in rectangular clusters and at a critical value of external field one

of them suddenly fills the entire lattice. We also see that � coer decreases with the increasing

system size. We relate this instability to the bootstrap percolation process BP � (for a

review, see: Adler 1991) with � �m+ . In the process BP � , the initial configuration is

prepared by occupying lattice sites independently with a probability � and the resulting

configuration is evolved by the following rules:

(a) the occupied sites remain occupied forever,

(b) an unoccupied site having at least � occupied neighbours, becomes occupied.

For � �,+ , on a square lattice, in the final configuration, the sites which are occupied

form disjoint rectangles. It has been proved that in the thermodynamic limit of large $ , for

any initial concentration �ä-�� , with probability � , in the final configuration all sites are

occupied (Aizenman and Lebowitz 1988).

We show that the leading $ dependence of � coer, to the lowest order inÓ

is given by,

� coer � Q>í&R+ Ó #"%$ Ö for � #"%$ >2� Ó�� +¤>í& Ó #" � 3� ø �d "î$¢&^>2� Ó �� Ö for >2� Ó� "î$ �� Ï+¤>4� Ó �_TOn a cubic lattice, we relate the spin flip process to the three dimensional BP 3 (Cerf

and Cirillo 1999). We show that the leading $ dependence of � coer, to the lowest order in

Synopsis xiiÓis given by,� coer � � >í& � Ó #"%$ Ö for � #"î$ ��+¤>2� � Ó � �� Q>í&R+ Ó #" �d #"î$¿& 1�3�� Ö for �Ï+¤>4� � Ó � #"î$ �� Ï+¤>4� Ó � �� +¤>í& Ó #"% #" �d "î$ & 1�3�� Ö for �� Ï+¤>4� Ó � "î$ �� Ï+ >2� Ó �k�âTThe bootstrap instability does not occur on lattices with �P� �

. On such lattices, if

the density of unoccupied sites is large enough so that they percolate, there are infinitely

extended lines of unoccupied sites in the lattice. These cannot not become occupied by

bootstrapping under BP 1 . Thus the critical threshold for BP 1 on such lattices is not � .Minor hysteresis loops on a Bethe lattice

Suppose the system is on the lower hysteresis curve at some external field � � , i.e. start

with external field �P�\& ' , where all the spins are down, then increase the field to � � .Then decreasing the field from �� to some field �41 and then again increasing to � � we

obtain the first minor loop. Similarly starting from the first minor loop at some field ��3and decreasing the field to �� , and then increasing to �23 , we obtain the second minor loop

➤

➤

➤

➤

➤

➤

➤

➤

"! $# &% (')+*-,$.0/214365874.05:9 ;=<?>

@:AB CDEFHGAEF�IC

JLKNMMOKNMP JLKNM

QRKSM

MOKSM

P QRKSM

Figure 0.3: Minor hysteresis loops

Synopsis xiii

and so on. In general, the ¥ -th minor loop for ¥e- � is obtained from the lower half of�d¥ &,�g� -th minor loop by decreasing the field from ��1U§ j � to �þ1U§ and then increasing to�þ1U§UT��WV�þ1U§ j � . This involves 57�þ§¤=YX;�Q§ Ö �Q§ j � Ö T T T Ö �� , the history of all the turning points

from �� to �Q§ . In this thesis we determine the expression for all the minor hysteresis loops

for arbitrary distribution of random fields.

The change in magnetization is determine by the fraction of spins which flip back on

reversal of external field. We see that, the spin which was the initiator of the avalanche

(which flipped at time step 0) can flip back at � only at the end, after all the spins of that

avalanche flip back and in this flip-back avalanche the spins flip exactly in the reverse time

order to the previous avalanche. This property will be called the time ordering property of

spin-flip-back process. We use this property to find the fraction of spins which flip back,

which is the probability that a spin 6g8 flips up at � and flips back at � Ú , when the the field

is changed from � to � Ú . This is related to the probability that a neighbour spin flips at �before 698 and the probability that a neighbour flips at � after 6�8 and flips back at � Ú before6M8 . We find the exact recursions for these probabilities, which can be solved for a given

distribution of random fields and hence the magnetization can be determined on minor

loops.

We see that, the reverse magnetization curve starting from � , meets the upper major half

at �¬& +¤> and merge with it for � Ú V �¬& +¤> . Therefore we consider the first minor loop in

the range × �c&P+¤> Ö � Ü . We also generalize the property of merging of minor hysteresis loop

to the major loop, to general graphs.

We show that for RFIM on a connected general graph ª , all the stable configurations at� go to unique stable configuration Z*�Ï��: +7�7� > � when the field is monotonically changed

to ��: +�� > ; where � � is the minimum number required such that, any connected subgraph[ of ª has at least one vertex such that, the number of edges in [ connected to that vertex

is \ �g� .

Bibliography

Adler J. (1991), Physica A 171, 453.

Aizenman M., and Lebowitz J. L. (1988), J. Phys. A: Math. Gen. 21, 3801.

Cerf R., and Cirillo E. N. M. (1999), Ann. Probab. 27, 1833.

Cote P. J, and Meisel L. V. (1991), Phys. Rev. Lett.67, 1334.

Derrida B., Vannimenus J., and Pomeau Y. (1978), J. Phys. C 11, 4749.

Dhar D., Shukla P., and Sethna J. P. (1997), J. Phys. A: Math. Gen.30, 5259-5267.

Kawasaki K. (1972) in C. Domb and M. S. Green (eds.), Phase Transition and Critical

Phenomena vol 2, Academic Press, London.

Perkovic O., Dahmen K. A., and Sethna J. P. (1995), Phys. Rev. Lett.75, 4528.

Puma M., and Fermandez J. F. (1978), Phys. Rev. B 18, 1391.

Sabhapandit S., Shukla P., and Dhar D. (2000), J. Stat. Phys. 98, 103.

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(1993), Phys. Rev. Lett.70, 3347.

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xiv

List of Publications

1. “Distribution of Avalanche Sizes in the Hysteretic Response of Random Field Ising

Model on a Bethe Lattice at Zero Temperature”,

Sanjib Sabhapandit, Prabodh Shukla and Deepak Dhar,

Journal of Statistical Physics 98, 103 (2000).

2. “Hysteresis in the Random Field Ising Model and Bootstrap Percolation”,

Sanjib Sabhapandit, Deepak Dhar and Prabodh Shukla,

Physical Review Letters 88, 197202 (2002).

xv

Chapter 1

Introduction

In the recent years, there has been a growing interest in the study of nonequilibrium sys-

tems. The states of systems at thermal equilibrium, are a priori given by the Boltzmann-

Gibbs distribution i.e. there is an energy function ]0�^Z�� associated to every possible

configuration Z of the system and each configuration Z has a weight proportional to�� × &_]G�`Z¬�k�Lacb � Ü , where acb is Boltzmann’s constant and � is the temperature of the

system. Then all the thermodynamic quantities are obtain by averaging over all the config-

urations with respective weights. However, in nature there exist a wide variety of systems,

which are not in thermal equilibrium. The probabilities of different states of these systems

are not given by the Gibbs distribution, but are determined by the underlying microscopic

dynamical processes, and are often hard to determine due to lack of a general framework.

An important class consists of nonequilibrium systems, which when driven by slowly

varying external forcing, relax through avalanche-like dynamics in response to the exter-

nal perturbations. Examples include sand or rice piles, forest fires, earthquakes, vortices

in dirty type II superconductors, solid on solid friction, moving of interfaces in random

media, disordered ferromagnets and many others (see Jensen 1998, chap. 3). Depending

on the system, the avalanche is characterized by different physical quantities. For exam-

ple, in sand piles the system is driven by slowly adding sand grains to the system and the

avalanche is characterized by the number of sand grains displaced after adding a single

grain or the lifetime of the avalanche. In earthquakes, it is the energy release and in case of

ferromagnets it is the size of the domain that flips. The avalanches occurs in various sizes

in a random sequence, and one is generally interested in the distribution of the avalanche

sizes.

What is common in all the systems mentioned above is the existence of threshold and

multiple metastable states i.e. if the applied external force is less than a critical value the

system does not response and when the force exceeds the critical value the system passes

1

Chapter 1. Introduction 2

from one metastable state to another. Due to existence of multiple metastable states, the

state of a system at a given time depends on the history of evolution (path along which

the system is evolving in configuration space) and systems exhibit hysteresis in the zero

frequency limit of external forcing d .In this thesis, we study a spin model in the presence of disorder, called random field

Ising model, introduced by Sethna et al. (1993) in the context of Barkhausen noise and

hysteresis in disordered ferromagnets. In this model, as the external field increases, the

magnetization increases as groups of spins flip up together. The dynamics is governed

by the existence of many metastable states, with large energy barriers separating different

metastable states. We hope that this study of non-equilibrium response in this model would

help in the more general problem of understanding the statistical mechanics of metastable

states in glassy systems.

The remainder of this chapter is organized as follows. section 1.1 contains a brief re-

view of theoretical studies of hysteresis in ferromagnets. In section 1.2, we briefly discuss

Barkhausen effect. In section 1.3, we define the random field Ising model with zero tem-

perature dynamics, and discuss some earlier results. In section 1.4, we discuss some of

the equilibrium properties of random field Ising model. Section 1.5 gives an outline of the

remaining chapters.

1.1 Hysteresis in ferromagnets

The studies of hysteresis in magnetic materials has been there in various branches of sci-

ence, for a long time (see Bertotti 1998). Apart from the intellectual interest, it also has

wide range of technological applications, from designing transformer cores to memory

devices.

Physicists have been looking for a convincing general theory to interpret the phe-

nomenon of hysteresis in magnets since the time of Rayleigh (1887), who gave the first

phenomenological theory where the experimental magnetization curves at small field were

approximated by parabolas. Starting from the demagnetization state (zero magnetization

in the absence of external field), the magnetizations e+f at small fields :¬� , are expressedgSystems also show hysteresis under periodic forcing. For example, when a ferromagnet is placed in

oscillation field, the magnetization lags behind its instantaneous equilibrium value and gives rise to hysteresis

loop. But the area of the hysteresis loop tends to zero in the zero frequency limit of the driving field (Dhar

and Thomas 1992)


➤

➤

➤

➤

h

ijlk�mon:p k mon:p

j h mon:p

h mon:p

Figure 1.1: Rayleigh hysteresis loop

as eqff�srÞ��:ut_� 1 Ö (1.1a)

and when the field is cycled between small : C max, the lower and the upper curves of the

hysteresis loop are represented by

evf®�á�`rS(wt C max �k�¬: �+ t �Ï� 1 & C 1max �_T (1.1b)

In Fig. 1.1, we have shown the magnetization curves starting from the demagnetization

state given by Eq. (1.1a) and the Rayleigh hysteresis loop given by Eq. (1.1b).

In Weiss’s (1907) theory of ferromagnetism, he postulated the existence of a powerful

internal “molecular field” in ferromagnet materials, which would tend to tries to align the

magnetic moments along one direction. It agrees with some of the experimental cases

where it is possible to attain a large saturation magnetization by the application of a very

weak magnetic field [see Fig. 1.2]. However, it did not explain the fact that, it is also

possible for the magnetization to be zero (or nearly zero) in the absence of a magnetic field.


Figure 1.2: Magnetization curve of single crystal of silicon iron. (Williams and Shockley 1949).

Weiss further made an assumption that, a ferromagnet can be subdivided into regions called

magnetic domains. In each individual domain, the the magnetic moments are aligned along

the molecular field, but the orientation of the spontaneous magnetizations in each domain

distributed randomly inside the sample and hence the resultant magnetization could be zero

in the absence of a external field, even at very low temperature. If a external field is applied

opposite to the magnetization direction, a domain reverses the direction of magnetization

when the external field exceeds a critical valueC ô . Therefore, if a gradually increasing

external field is applied, domains whose magnetization vectors are at an angle �yx�&8�� with

the external field, will suddenly reverse direction when the external field exceedsC ôZ�(z æL{ � .

This results a finite bulk magnetization for external field �®- C ô . A comprehensive review


of the physical principles of domain theory and of the crucial experiments which bear

directly on the foundation of the subject, may be found in an article by Kittel (1949).

Preisach (1935) introduced a modified domain model, in which he assumed that mate-

rial is composed of many small domains and each of them possesses a rectangular hystere-

sis loop. The interaction between domains are represented by a local field acting on each

domain. Thus each domain has two different coercive fields | and ú for the increasing

and decreasing branches respectively. The ensemble of domains is then described by the

distribution function �*�`| Ö ú � of the values of | and ú and hysteresis loops are obtained by

taking the weighted sum of magnetization in all the domains.

Sethna et al. (1993) proposed the random field Ising model with the zero temperature

dynamics as a simple theoretical model for the Barkhausen noise and hysteresis in dis-

ordered ferromagnets. In this model, magnetic domains are represented by Ising spins

( 6M8í� :c� ) and the external field is coupled to these spin. In contrast to the Preisach

model of hysteresis, where interactions between the individual hysteresis units (domains)

are ignored, in the random field Ising model the spins interact ferromagnetically with their

neighbors. The homogeneities and disorder in materials are modeled by introducing a un-

correlated random field acting on each domain, chosen at random from some distribution.

Since the domains interact ferromagnetically, flipping of a domain at some external field

may force the neighboring domains to flip as well in the same direction, thus leading to an

avalanche of domain flips, which is analogue of a Barkhausen pulse in real magnets (for a

comparison, see Fig. 1.5 and Fig. 1.6).

In the models discussed above, the hysteresis does not depends on the rate at which the

external field is varied i.e. relaxation time from one metastable state to another is much

larger to the rate at which the system is driving. In contrast, there are also other models

studied in the context of rate-dependent hysteresis, where the system exhibits hysteresis

only when it is driven at a finite rate (see Chakrabarti and Acharyya 1999, for a recent

review). Hysteresis in the � -vector model was widely studied by many authors (Rao

et al. 1990, Dhar and Thomas 1992, 1993, Somoza and Desai 1993). It was shown that

in all dimensions £µ- + , for � b + at low frequency } and low amplitudeC�~

of the

driving field the area of the hysteresis loop scales as � C�~ }O� � � 1 with logarithmic corrections.

At high frequencies the area varies asC 1~ �U} . For any

C�~, there is a dynamical phase

transition separating these two frequency regimes. Above the critical frequency }S� C4~ � , the

hysteresis loop does not posses inversion symmetry. Using the nucleation theory Dhar and

Thomas (1993) showed that for �\�;� and £G- � , the area of the hysteresis loop scales as� �® #" � C�~ }O� � j � �H� ò j �`� for } C-~.


1.2 Barkhausen effect

The first evidence for the existence of ferromagnetic domains was from an experiment

by Barkhausen (1919). His experiment consists in amplifying the voltage induced in a

secondary pick-up coil wound around a ferromagnetic sample, while the sample is being

magnetized by a continuous variation of external magnetic field [Fig. 1.3]. He observed a

noise induced in the pick-up coil, corresponds to a sudden, discontinuous jumps in magne-

tization [Fig. 1.4]. These jumps are interpreted as discrete changes in the size or rotation

of ferromagnetic domains. An elementary introduction of the Barkhausen effect may be

found in the textbook by Feynman et al. (1977).

Figure 1.3: Barkhausen effect

In the recent years, there has been a great interest in the study of the statistical prop-

erties of the Barkhausen noise. A typical train of barkhausen noise signals observed in

experiments is shown in Fig. 1.5. Three basic physical quantities that describe a single

Barkhausen noise signal in an experiment, are signal duration, area of the signal and the

energy released during the signal occurrence. It is observed that distribution of these quan-

tities follow power law over a few decades with a cut-off as shown in Fig. 1.7 (see Spaso-

jevic et al. 1996, and references therein). This power law tail in the Barkhausen avalanche

distribution was interpreted by Cote and Meisel (1991) as an example of self-organized


�

�➤

➤

Figure 1.4: Barkhausen jumps

criticality d . But Perkovic et al. (1995) have argued that large bursts are exponentially rare,

and the approximate power-law tail of the observed distribution comes from crossover ef-

fects due to nearness of a critical point.

Barkhausen effect is also widely used as a noninvasive material characterization tech-

nique for ferromagnetic materials (see Sipahi 1994, for an overview).

1.3 Hysteresis in random field Ising model

The nonequilibrium random field Ising model was proposed by Sethna et al. (1993) as a

model for Barkhausen noise and hysteresis in ferromagnets. The model is defined on a

lattice. At each lattice site ? , there is a Ising spin 6g8î��:<� , which interacts with nearest

neighbors through a ferromagnetic exchange interaction ( >ï-�� ). Spins 576 8�= are coupled

to the on-site quenched random magnetic field �28 and the external field � . The Hamiltonian

of the system is given by C �D&E> FH 8#I JLK 6M8B6NJO& F 8 �Q8B6M82&R� F 8 6M8 Ö (1.2)

where �d? Ö��=� denotes that the sum runs over nearest neighbor pairs of spins on sites ? and�. We assume that 57�þ8Ï= are quenched independent identically distributed random variablesg

In self-organized critically, systems exhibit critical behavior (power law correlations), without fine tun-

ing any parameter (for an overview, see: Dhar 1999, Jensen 1998, Bak 1997).


Figure 1.5: An example of experimental Barkhausen signal (voltage pulse produce from a pickup

coil around a ferromagnet subjected to a slowly varying applied field) (Urbach et al. 1995).

��`��U�c��R�

�� ¡¢ £�¡

¤�¥§¦R¦¤©¨R¦§¦¤«ªR¦R¦¤�¬§¦R¦¤«®¦§¦¤�¯°¦R¦¤�±§¦R¦

ª®¦§¦¬R¦§¦®¦§¦¯§¦§¦±R¦§¦²®¦§¦¤�¦§¦¦

Figure 1.6: Time series of the avalanches (the number of spin flips at a given field) in the random

field Ising model on a square lattice of size ³«´U´WµY³«´U´ . From one avalanche to the next avalanche is

considered as one time step.


Figure 1.7: Experimental data for the distribution of Barkhausen signal durations, areas and ener-

gies (Spasojevic et al. 1996).


with the probability that the value of the random field at site ? lies between ��8 and �Q8ê( £Þ�Q8being @A��þ8X�þ£Þ�Q8 .

When the external field is changed, the system relaxes to a stable spin configuration

through zero-temperature Glauber single-spin-flip dynamics (Kawasaki 1972, see), which

is specified by the transition rates

Rate × 6M8�l &S6M8 Ü �·¶¹¸ ifÓ ]�\º�� otherwise

(1.3)

whereÓ ] is the change of energy in the system as a result of the spin flip. Therefore, a

spin-flip is allowed only if the process lowers energy. We assume that the external field

is increased adiabatically, i.e. ¸+» } , the rate at which the magnetic field � is increased.

Thus if the spin-flip is allowed, it relax instantly, so that the spin 6�8 in a stable configuration

is parallel to the net local field VM8 at the site:6M8 � sign �XV_8B�Ð� sign ¼½�>\[FJL]�� 6NJA(^�Q8Q(^��¾¿�T (1.4)

Note that the limit }O� ¸ l � is taken after the limit �Øl � . If the limits are taken in

the reverse order, the state of the the system at each � , is the equilibrium state for all finite� and the hysteresis loop area goes to zero.

We start with �0�;&S' , when all spins are down and slowly increase � . As we increase� , some sites where the quenched random field is large positive will find the net local

field positive, and the spin at that site will flip up. Flipping a spin makes the local field

at neighboring sites increase, and in turn may cause them to flip. Thus, the spins flip in

clusters of variable sizes. If increasing � by a very small amount causes 6 spins to flip

up together, we shall call this event an avalanche of size 6 . As the applied field increases,

more and more spins flip up until eventually all spins are up, and further increase in � has

no effect.

As an illustrative example, consider a four by four square lattice with periodic bound-

ary condition and a particular realization of quenched random fields, which is shown in

Fig. 1.8(a). We set >Õ�Ø� . Now start with � �Ø&S' , when all spins are down [Fig. 1.8(b)]

and slowly increase it. A spin with � up neighbors, flips up at � , if the quenched random

field at the particular site �þ8÷-ïî&.+§�,&.� . Therefore, when the external field just exceeds

the value �êT � , the local field at the site where �48O�Î+ÞTÁÀ , becomes positive and the spin at

that site flips up. This increases the local fields at its neighboring sites by +¤> and as a result

spins at some of these sites flip up [Fig. 1.8(c)]. These process continues till there is no

more sites where the local field is positive at that external field. In the figure, we denote


ÂÄÃRÅÇÆ�Â&ÈUÅ Ã�ÂÄÃ®Å É Ã®ÅÇÈÊ Å Ë ÃRÅÇÆ È®Å Ì Ê Å ÍÂÄÃRÅ Ë Ê ÅÇÆ È®Å É ÂÄÍRÅ ÎÂ&Î®Å È Ê Å Ã Â&ÎUÅ Í Ã®Å Ê

(a) (b) ÏWÐÒÑÔÓ (c) ÏÕÐ×Ö�ØÙÖÛÚ

(d) ÏÜÐÝÖ�Ø Þ�Ú (e) ÏÕÐÝÖ�Ø ß�Ú (f) ÏÜÐáà®Ø â�Ú

(g)

ã ä å æ(h)

ç

èêéë ìíîïñðéîïÙòì

ó`ô õö`ô ÷ö^ô õøùô ÷øyô õõ�ô ÷

øùô õõ`ô ÷õ`ô õú õ`ô ÷ú øùô õ

(i)

Figure 1.8: (a) Quenched random fields at various sites. (b) - (f) Stable spin configurations at

various external fields. The black spins are inactive spins at that particular field and the colored

spins are part of the avalanche. The colors specify the order in which spins flip during the avalanche.

(g) Clusters of spins, which flip during one avalanche. (h) Color map showing the order of events.

(i) Magnetization curve, corresponding to evolution.


active spins with different color according to order at which they flip during the avalanche.

As shown in Fig. 1.8(b) - (f), the system passes from one stable configuration to another,

as the external field is increased, till all the spins in the system flip up. In Fig. 1.8(g), we

show the different clusters of spins, which flip at different fields. Fig. 1.8(i) shows the

corresponding magnetization.

Sethna et al. (1993) studied the model with the infinite-range interaction (mean field

theory), where every spin is coupled to all � other spins with coupling >2�M� . They found

that there exists a critical value of disorderÓ ô (which in the case of a Gaussian distribution

of random fields is �üû �Ï+ê�®x÷�â> ), below which the hysteresis curve displays a jump due

to an infinite avalanche of spin flips, which spans the system. Above the critical disorder

systems show smooth magnetization curve without macroscopic jumps. However, this

mean field theory does not show any hysteresis for disorderÓ b Ó ô . Dahmen and Sethna

(1993, 1996) studied the hysteresis loop critical exponents expanding about mean field

theory in � & û dimensions. A power-low distribution with avalanche of all sizes is seen only

at the critical value of the disorder. However, the numerical simulations by Perkovic et al.

(1995) indicate that the critical region is remarkably large: almost three decades of power-

law scaling in the avalanche size distribution remain when measured 40% away from the

critical point. Therefore, they argued that several decades of scaling seen in experiments

need not be self-organized criticality, as many of the samples might have disorders within

40% of the critical value.

Interestingly the model can be solved exactly on a Bethe lattice for the magnetization

on the hysteresis curve for arbitrary distribution of random fields (Dhar et al. 1997). In

contrast to the infinite-ranged mean field theory, the calculation on Bethe lattice shows

hysteresis even for large disorder. Another interesting result of the Bethe lattice calculation

is that, the first order jump in the magnetization disappear for coordination number of the

lattice less than 4. Only for coordination number 4 and above, there exists a critical value

of disorder below which there is a jump discontinuity in the magnetization.

1.4 Equilibrium properties of random field Ising model

In this thesis we are interested in the nonequilibrium properties of the random field Ising

model. However, it is useful to recall the equilibrium properties of this model, which

has been an important problem in statistical physics for a long time. This model has a


number of interesting realization in nature. A recent review of earlier work on this model

may be found in an article by Nattermann (1998). This model was first studied by Imry

and Ma (1975), in the context of possible destruction of long-range order by arbitrarily

weak quenched disorder. The pure Ising model with nearest neighbor interactionC4~ �&E>þý H 8 I JkK 6M8X6âJ , is known to have a ferromagnetically ordered phase in all dimensions £G-©� .

When the random field term & ý �Q8X698 is introduced, it acts against the order. Imry and Ma

(1975) argued that arbitrarily weak disorder destroys long-ranged ferromagnetic order in

dimensions £�\©+ . The argument goes as follows:

L

Figure 1.9: Domain of reverse spins.

If we consider a domain of reverse spins [Fig. 1.9], of linear size õe$ , the domain wall

energy is õ·$ ò j � . However, according to the central limit theorem, if the random field

has short-range spatial correlations, the fluctuation in the magnetic field energy in such

domains is õ $ ò � 1 . Thus, by splitting into domains of size $ , the system will gain bulk

energy of ÿí�d$ ò � 1 � per domain, and loose a surface energy a surface energy of ÿ ��$ ò j � �per domain. Thus, whenever £?\ + , there will exists a large enough $ , for an arbitrarily

small random field, where it will become energetically favorable to the system to break

into domains of that size.

The argument by Imry and Ma (1975) suggests that the lower critical dimension d is£�� b + , rather than £��)� + , because other mechanisms could destroy long-range order

in higher dimensions. It is widely believed that the upper critical dimension�

is £��í� � ,instead of £�� © for the pure Ising system. However, whether the lower critical dimension£�� + or £��

, was a matter of a long controversy, but has now been established thatgThe dimension below which long-range ferromagnetic order cannot exist.�The dimension above which the critical exponents are those of the Gaussian fixed point.


£�� á+ . Imbrie (1984) showed that if the disorder is small, the model in dimension £ � �exhibits long-range order at zero temperature. Aizenmann and Wehr (1989) rigorously

proved uniqueness of the Gibbs state in £R� + , i.e. absence of any phase transition, in

agreement with the Imry-Ma prediction.

As far as an exact calculation of thermodynamic quantities is concerned, there are only

a few results. For example, Bruinsma (1984) studied the random field Ising model on a

Bethe lattice in the absence of an external field and for a bivariate random field distribu-

tion. There are no known exact results for the average free energy or magnetization, for a

continuous distribution of random field, even at zero temperature and in zero applied field.

1.5 Outline of this thesis

In this thesis, we study the nonequilibrium ferromagnetic random field Ising model with

zero temperature Glauber single flip dynamics. The remaining chapters of the thesis are

organized as follows:

In chapter 2, we discuss some special properties of the model that makes the analytical

treatments possible. We briefly recapitulate the derivation of self-consistent equations for

the magnetization in the model.

In chapter 3, we use a similar argument to construct the generating function for the

avalanche distribution for arbitrary distribution of the quenched random field. In sec-

tion 3.2, we consider the special case of a rectangular distribution of the random field.

In this case, we explicitly calculate the probability distribution of avalanches, for the for

Bethe lattices with coordination numbers �c�D+ and�. In section 3.3, we analyse the self-

consistent equations to determine the form of the avalanche distribution for some general

unimodal continuous distributions of the random field. In chapter 4, we derive the self-

consistent equations for the magnetization on minor hysteresis loops on a Bethe lattice,

when the external field is varying cyclically with decreasing magnitudes. We also discuss

some properties of stable configurations, when the external field is varying.

In chapter 5, we study the model with an asymmetric distribution of quenched fields,

in the limit of low disorder in two and three dimensions. We relate the spin flip process to

bootstrap percolation, and find nontrivial dependence of the coercive field on the coordina-

tion number of the lattice.

Chapter 6 contains a discussion of our results, and some concluding remarks.


Some algebraic details of the analytical solution for the distribution of avalanche sizes,

for the rectangular distribution of quenched fields on Bethe lattice are relegated to ap-

pendix A.

Part of this work has appeared in journals as refereed papers. Though it is mostly a

repetition of the material presented in chapter 3 and chapter 5, for the convenience of the

reader, we have reproduced these papers as an appendix (reprints) to this thesis.

Chapter 2

Earlier exact results on hysteresis

in random field Ising model

The difficulty of solving various mathematical equations describing actual physical situa-

tions leads to various approximation methods. These approximation method can be clas-

sified into two categories: One in which the approximation is made in the mathematical

equations itself and another in which the physical system is simplified. Into the second

category fall many lattice model systems. Again in higher dimensions the lattices contain

closed circuits which makes the model difficult to solve. Thus, one considers the problem

in the mean-field theory, where the underlying lattice structure becomes irrelevant or on a

different lattice where it can be solved exactly. Bethe lattice or Cayley tree is one in which

there is no circuits at all which makes the model easier to solve. The simplicity of the

lattice motivates one to study various systems on a Bethe lattice.

In this chapter we briefly discuss derivation of hysteresis curve in the random field

Ising model in the mean field theory (infinite-range interaction) and on the Bethe lattice.

In sections 2.1 and 2.2 we discuss two properties of zero temperature random field Ising

model, namely the return point memory effect and the abelian nature of spin-flips, which is

used to set up self-consistent equation to determine magnetization on the Bethe lattice and

later in other chapters.

2.1 Return point memory

Sethna et al. (1993) showed that the RFIM exhibits the following return point memory

effect: Suppose we start with � �é& ' , and all spins down at �à� � . Now we change

the field slowly with time, in such a way that � ��L�4\�� <� , for all times �4V� . Then

16

Chapter 2. Earlier exact results on hysteresis in random field Ising model 17

��

(a)

➤

➤

➤

➤

➤

➤

��

(b)

Figure 2.1: (a) change of external field with time. (b) magnetization curve vs. external field given

by (a). When the external field returns to the previous extremal value, the magnetization returns to

the value at that field i.e. ��XÊ�� !�"�XÊ#�� 1$� � .the configuration of spins at the final instant �S�% does not depend on the detailed time

dependence of � ��L� , and is the same for all histories, so long as the condition � ��L�o\� �<�for all earlier times is obeyed. In particular, if the maximum value �÷��<� of the field was

reached at an earlier time �â� , then the configuration (and hence the magnetization) at time is exactly the same as that at time �N� [Fig. 2.1].

Consider two spin configurations Z<576ê� Ö 6g1 Ö T9T9T Ö 6M§Þ= and Z Ú 576 Ú � Ö 6 Ú 1 Ö TMT9T Ö 6 Ú§ = . If 698 b 6 Ú8for each site ? , the configurations Z and Z Ú are called partially ordered, Z b Z Ú . Let two

configurations ZG�&�L� and Z Ú ��L� be evolve under the field �÷�&�L� and � Ú �&�L� respectively. Suppose

the initial configurations Z*�`� � and Z Ú �^� � are partially ordered such that ZG�^� � b Z Ú �`� �and the fields satisfy � �&�L� b � Ú ��L� . Then if a spin 6 Ú8 ��L� is up in configuration Z Ú ��L� , the

corresponding spin 698Z�&�L� in configuration ZG�&�L� must be up, since the local field V98ó��L� inZG�&�L� can not be less than V Ú8 ��L� in Z Ú �&�L� . Therefore the configurations Z*��L� and Z Ú ��L� will

always remain partially ordered, Z*��L� b Z Ú �&�L� . This is the no passing property of the

system. An earlier treatment of “no passing” rule was given by Middleton (1992) in the

context of charged-density waves.

Let us consider the Fig. 2.2. The configuration Ñ is reached by increasing the field from

a lower value to �� . On increasing the field monotonically from � � to �41 , configuration ãis reached. Naturally, the configurations Ñ and ã are partially ordered such thatã b Ñ�T (2.1a)


' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '

' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '

' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '' ' ' ' ' ' ' ' '' ' ' ' ' ' ' ' '' ' ' ' ' ' ' ' '' ' '

' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '

' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '

' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '(

(

()+* )-,

.

/

0➤

➤

➤

Figure 2.2: Partial ordering between configurations, when the field is (1) increased from Ê � to Ê 1 ,(2) then decreased from Ê 1 to Ê � , (3) again increased from Ê � to Ê 1 .Similarly when the field is decreased monotonically from ��1 to �� , partial ordering exists

between the configuration ã and the final configuration Z such that

Z�\ ã*T (2.1b)

Since during the evolution from Ñ to Z , the field � �&�L� satisfies � ��L� b � � ,Z b Ñ�T (2.1c)

Now suppose the configuration Z evolve to a configuration ß when the field is again

increased monotonically from � � to �þ1 . Since the partial ordering is preserved by dynamics,

from Eq. (2.1b), ß \ã Ö (2.1d)

and from Eq. (2.1c), ß b ã Ö (2.1e)

as Ñ evolves to ã , when the field is increased from � � to �þ1 . From Eq. (2.1d) and Eq. (2.1e),

we must have ß �µã Ö (2.1f)


i.e. the system returns exactly to the same earlier configuration, when the field is decreased

from a maximum value �21 and then increased to the same value. The same memory effect

extends to subcycles within the cycles and so on.

2.2 Abelian property

Because of the previous property, we may choose to increase the field suddenly from &S'to �÷�X�S� in a single step. Then, once the field becomes �0�e� �X� � , several spins would have

positive local fields. Suppose there are two or more such flippable sites. Then flipping any

one of them up can only increase the local field at other unstable sites, as all couplings are

ferromagnetic. Thus to reach a stable configuration, all such spins have to be flipped, and

the final stable configuration reached is the same, and independent of the order in which

various spins are relaxed. This is the abelian property of relaxation (Dhar et al. 1997).

Using the symmetry between up and down spins, it is easy to see that the abelian property

also holds whether the new value of field � Ú Ú is greater or less than its initial value � Ú so

long as one considers transition from a stable configuration at � Ú to a stable configuration

at � Ú Ú .2.3 Hysteresis in the infinite-range interaction model

In this section, we will briefly discuss the results obtained by Sethna et al. (1993), on the

hysteresis in the random field Ising model with infinite-range interaction. In this mean field

theory, every spin is coupled to all � other spins with coupling >4�M� . The Hamiltonian is

given by C �;& >� ù F 8 6M8 ý 1 &R�SF 8 6M82&RF 8 �þ8B698ÏT (2.2a)

Now the interaction of a spin with other spins is replaced by its interaction with the mag-

netization eá�Ï�2� of the system. The Hamiltonian then takes the formC �D& F 8 �Z>$e (ä��(^�Q8X�k6M8 Ö (2.2b)

i.e. the effective local field at site is >$e ( �í( �48 . The spin at this site will flip up if

this field is positive i.e. the quenched random field �28 at this site exceeds &E>$e &ä� . This


happens with probability 132j �54 � � � j � @ �Ï�Q8B�þ£¤�Q8UTTherefore, the average magnetization satisfies the self-consistent equationeá��Ð�e+ 162j �54 � � � j � @A��þ8X�þ£¤�þ82&^�êT (2.3)

Note that, for symmetric distributions of random fields, eá�^� �O� � is the trivial solution at�� . Now, if eá�^� �� is the only solution at �� , then there is no hysteresis. To

have other nontrivial solutions for eá�`� � , the slope of the expression on the right hand side

of Eq. (2.3), as a function of eá�^� � must be greater than unity at eá�`�ê�à� � . At �ä� �and near eá�^� � � � , the right hand side of Eq. (2.3) can be approximated as +°eá�`� �N> @ �`� � .Therefore, the condition that the Eq. (2.3) to has multi-valued solution is@ �`� � b �+¤> T (2.4)

This condition corresponds to a critical disorder strengthÓ ô (width of the random field

distribution), above which there is no hysteresis i.e. the magnetization follows the same

curve in the increasing and decreasing field. BelowÓ ô , the magnetization curves in the

increasing and decreasing field are different near �0� � , i.e. the system exhibits hysteresis.

Moreover, there is a critical field, where the magnetization jumps from one solution to

another one. In a specific case, where the random field distribution is Gaussian,@ �Ï�Q8B�Y� �7 +Rx Ó �� ù & � 18+ Ó 1 ý Ö (2.5)

using the condition given by Eq. (2.4), the critical value of disorder is obtained asÓ ô!� û �Ï+ê�®x÷�N>ÞT (2.6)

Figure 2.3 shows the magnetization curves for this mean-field at various values of disorderÓ V Ó ô , Ó � Ó ô andÓ - Ó ô for Gaussian distribution of random fields. Note that

hysteresis and jump in the magnetization exist only below a critical disorder [Fig. 2.5(c)].

Sethna et al. have studied in detail the case of critical disorder, and the power-law

divergence of various quantities at this critical point. The special value of disorder does

not seem to be particularly important and we shall not discuss it here.

2.4 Hysteresis on the Bethe lattice

The shortcoming of the treatment discussed in the previous section is that the pair couplings

are weak and no correlations and short-range order. One can keep mean field theory, but


8:9;8=<

>@?�A

BC DE

FHGJIIKGMLIKGJIN IKGJLN FHGJI

FHGJIIKGMLIKGJI

N IKGMLN FHGJI

(a)

OQPRO=S

T@U�V

WX YZ

[H\J]]K\M^]K\J]_ ]K\J^_ [H\J]

[H\J]]K\M^]K\J]

_ ]K\M^_ [H\J]

(b)

` ab`dc

egfih

jk lm

npoJqq�oMrq�oMqs q�oMrs ntoJq

ntoMqq�ourq�oMq

s q�ours ntoMq

(c)

Figure 2.3: Magnetization curve for the random field Ising model with infinite-range interaction

at various values of disorder: (a) Ì��wv , (b) Ì�� Ì ô ��x �ù³ty{z � v and (c) Ì\��´�|J}~v , for the

Gaussian random field distribution given by Eq. (2.5). The dashed line in (c) shows the third root

of the self-consistent equation for magnetization, given by Eq. (2.3).


add correlations by working on a Bethe Lattice.

The advantage of working on the Bethe lattice is that the usual BBGKY hierarchy of

equations for correlation functions closes, and one can hope to set up exact self-consistent

equations for the correlation functions. The fact that Bethe’s self-consistent approximation

becomes exact on the Bethe lattice is useful as it ensures that the approximation will not vi-

olate any general theorems, e.g. the convexity of thermodynamic functions, sum rules etc.

In the presence of disorder, in spite of the closure of the BBGKY hierarchy, the Bethe ap-

proximation is still very difficult, as the self-consistent equations become functional equa-

tions for the probability distribution of the effective field. These are not easy to solve, and

available analytical results in this direction are mostly restricted to one dimension, or to

models with infinite-ranged interactions. On the Bethe lattice, for short-ranged interac-

tions with quenched disorder, e.g. in the prototypical case of the :�> random-exchange

Ising model, the average free energy is trivially determined in the high temperature phase,

but not in the low-temperature phase. It has not been possible so far to determine even the

ground-state energy exactly despite several attempts.

��

��

�� ¡¡¢¢ ££¤¤ ¥¥¦¦§§¨¨

©©ªª

o

r = 4

r = 3

r = 2

r = 1

Figure 2.4: A Cayley tree of coordination number 3 and 4 generations.

The random field Ising model model on a Bethe lattice is special in that the zero-

temperature nonequilibrium response to a slowly varying magnetic field can be determined

exactly (Dhar et al. 1997). To be precise, the average non-equilibrium magnetization in

this model can be determined if the magnetic field is increased very slowly, from & ' to()' , in the limit of zero temperature. It thus provides a good pedagogical model to study

the slow relaxation to equilibrium in glassy systems.


The usual way to solve a problem on Bethe lattice is to consider the problem on a

Cayley tree and calculate all the thermodynamic quantities in the deep inside of the tree.

Consider a uniform Cayley tree of ¥ generations where each non-boundary site has a coor-

dination number � and the boundary sites have coordination number � [see Fig. 2.4]. The

first generation consists of a single vertex. The a -th generation has �4�d�Û& ��khkj 1 vertices foracb + . At each vertex there is a Ising spin.

Because of the return point memory, to find the magnetization at field � in the lower

half of the hysteresis loop, we start with �.�Î&S' , when all spins are down and increase

the field to � in a single step. Now at this field, since the spins can be relaxed in any order

(abelian property), we relax them in this: First all the spins at generation ¥ (the leaf nodes)

are relaxed. Then spins at generation ¥Í&� are examined, and if any has a positive local

field, it is flipped. Then we examine the spins at generation ¥f&ä+ , and so on. If any spin

is flipped, its descendent are reexamined for possible flips d . In this process, clearly the

flippings of different spins of the same generation a are independent events.

Let �� h � �Ï�2� be the probability that a spin on the a -th generation will be flipped when

its parent spin at generation a &;� is kept down, the external field is � , and each of its

descendent spins has been relaxed. As each of the �*& � direct descendents of a spin is

independently up with probability �-� h T��`� , the probability that exactly � of them are up is« [ j ��¬ × � � h T��`� Ü � ×��O& � � h T��`� Ü [ j � j � . Suppose we pick a site at random in the tree away from

the boundary, the probability that the local field at this site is positive, given that exactly �of its neighbors are up, is precisely the probability that the local field ��8 at this site exceeds× �d�E&R+R�/�â>í&R� Ü . We denote this probability by � � �Ï�2� . Clearly,� �S��Ð� 132

� [ j 1��(� � j � @ �Ï�Q8B�þ£¤�Q8�T (2.7)

Now it is straightforward to write down a recursion relation for �� h � in terms of �� h T��`� :� � h � �Ï�2�Ð� [ j �F�A] ~ ù � & �� ý � � � h T��`� �Ï�2�� î&P� � h T��`� �� [ j � j � � � �Ï�2�_T (2.8)

Given a value of � , we can determine � �S�Ï�2� using Eq. (2.7). Then using Eq. (2.8), and the

initial condition � � §U� �Ø� ~ �Ï�2� , � � h � can be determined for all a V ¥ . For a ¥ , thesegThis step is not really necessary if we are only interested in determining the magnetization at the site ® .

Skipping this step leads to considerable simplification of the relaxation process: First the spins of generation¯ are examined, then those of ° ¯ Ñ Ö²± etc. till we finally examine the spin at ® . No spin is checked more

than once. The resulting configuration is not fully relaxed, but it is easy to prove that further relaxation will

not change the state of the spin at ® . The argument can be extended to show that the probability that an

avalanches starting at ® is of size ³ also is the same in this partially relaxed state as in the fully relaxed state.


probabilities tend to limiting value, �´�µ�§H¶ 2 �� h � �� , which satisfies the equation� � ��2�Y� [ j �F�!] ~ ù �E&^�� ý × � � �Ï�2� Ü � × �î&�� Ï�2� Ü [ j � j � � �S�Ï�2�_T (2.9)

This is a polynomial equation in � ��2� , which can be solved in terms of 5N� �S�Ï�2�N= .Finally, for the spin at ` , there are � downward neighbors, and the probability that it is up

is given by

Prob ��6t·®�e(�� 2� � [F�!] ~ ù �� ý × � � �Ï�2� Ü � × �î&P� � ��2� Ü [ j � � � ��2�_T (2.10)

Because all spins deep inside the tree are equivalent, Prob ��6K·f�µ(�� 2� determines the

average magnetization for all sites deep inside the tree. This determines the lower half of

the hysteresis loop. The upper half is obtained similarly.

For the three coordinated ( �� ) Bethe lattice, the self-consistent equation satisfied by� � [Eq. (2.9)] is quadratic and from physical arguments at least one root must vary between

0 and 1 continuously with � for any value of disorder strengthÓ

. Hence the magnetization

given by Eq. (2.10) is also a continuous function of � . This is also the case with a linear

chain ( ��µ+ ), where the self-consistent equation [Eq. (2.9)] is linear.

On the other hand, the situation is quite different for � b . For example, for �/� ,Eq. (2.9) is cubic, which has either one or three real roots which will vary with � . Figure 2.5

shows this variation for two values of disorder of the random field distribution given by@A��Q8B�Ð� �+ Ó sech1 �Ï�Q8B� Ó �âT (2.11)

Note that for large disorder, there is only one real root which vary continuously from 0

to 1, giving rise to a continuous magnetization curve as shown in Fig. 2.6. But for small

disorder, � � ��2� as a function of � shows a “S” shaped curve, where at some value of � ,

two real root merge to becomes imaginary and disappear from the real plane. Therefore,

as we vary � , on the physical ground initially � � ��2� takes the lower value till the point

where it becomes complex and at that point it jumps to the upper value, giving rise to a

jump discontinuity in the corresponding magnetization curve [Fig. 2.6].

This can be generalized to higher coordination number, where the mechanism of two

real solutions of the polynomial equation Eq. (2.9) merging and both becoming complex is

still the same.


-0.5 0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

¸º¹¼»

½¾¿ÁÀÃÂÅÄ ÆÃÇ&È

¿ÉÀÃÂÊÄ ËÌÇÍÈ

Figure 2.5: Variation of Î � �XÊ � with Ê for the Bethe lattice with �� ¡ , and the random field

distribution given by Eq. (2.11).

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-0.5 0.0 0.5 1.0 1.5 2.0 2.5ÏÑÐ6Ò

Ó²ÔÕ Ö×ØÙÛÚÔØÙÝÜÖ Þàßâá!ãbä�å�æ Þ=ßçáèãêéëåìæ

Figure 2.6: Magnetization as a function of increasing field for the Bethe lattice with � �¢¡ and the

random field distribution given by Eq. (2.11).

Chapter 3

Distribution of avalanche sizes

on the Bethe lattice

In this chapter, we study the distribution of avalanche sizes in the random field Ising model

on a Bethe lattice (Sabhapandit et al. 2000). This chapter is organized as follows. In

section 3.1, we set up a self-consistent equation for the generating function ¦*�B¨�� of the

probability ¦)§ , that an avalanche propagating in subtree flips exactly ¥ more spins in the

subtree before stopping, for arbitrary distribution of quenched random fields. Then we

expressed the generating function ªG�X¨�� of distribution of avalanche sizes, in terms of ¦*�B¨�� .In section 3.2, we consider the special case of a rectangular distribution of the random

field. In this case, we explicitly solve the self-consistent equations for Bethe lattices with

coordination numbers ��µ+ and�. However, this case is non-generic. For small strength of

disorder, the magnetization jumps from &E� to (�� at some value of the field, but for larger

disorder, when the system shows finite avalanches, there is no jump in magnetization and

the distribution function decays exponentially for large 6 . In section 3.3, we analyse the

self-consistent equations to determine the form of the avalanche distribution for unimodal

continuous distributions of the random field. We find that for coordination number � b ,the magnetization shows a first order jump discontinuity as a function of the applied field at

some field-strength � disc, for weak disorder. Just below �G�µ� disc, the avalanche distribution

has a universal �i& � �ê+�� power-law tail.

3.1 Generating function for avalanche distribution

Consider a Cayley tree rooted at ` , of � generation [Fig. 2.4]. We will be interested in

the portion of the tree where generation a � , in the limit � l ' . Now consider the

26

Chapter 3. Distribution of avalanche sizes on the Bethe lattice 27

íîííîíïîïïîïðððððñññññ

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÷øúù

Figure 3.1: A sub-tree û � formed by ü and its descendents. The sub-tree is rooted at ü and ý is

the parent spin of ü .

state of the system at external field � , and all the flippable sites have been flipped. We

increase the field by a small amount £¤� till one more site becomes unstable. We would like

to calculate the probability that this would cause an ‘avalanche’ of ¥ spin flips. Since all

sites deep inside are equivalent, we may assume the new susceptible site is the site ` .

It is easy to see that this avalanche propagation is somewhat like propagation of infec-

tion in the contact process on the Bethe lattice. The ‘infection’ travels downwards from the

site ` which acts as the initiator of infection. If any site is infected, then it can cause infec-

tion of some of its descendents. If the descendent spin is already up, it cannot be flipped;

such sites act as immune sites for the infection process. If the descendent spin is down, it

can catch infection with a finite probability. Furthermore, this probability does not depend

on whether the other ‘sibling’ sites catch infection. Infection of two or more descendents

of an infected site are uncorrelated events. Thus, we can expect to find the distribution

of avalanches on the Bethe lattice, as for the size distribution of percolation clusters on a

Bethe lattice (Stauffer and Aharony 1992). However, a precise description in terms of the

contact process is complicated, as here the infection spreads in a correlated background of

‘immune’ (already up) spins, and the probability that a site catches infection does depend

on the number of its neighbors that are already up.

We start with the initial configuration of all spins down. Now increase the external field

to the value � . Consider a site � at some generation a - � of the Cayley tree [Fig. 3.1].

We call the subtree formed by � and its descendents � � , the subtree rooted at � . We keep

its parent spin þ at generation a &�� down, and relax all the sites in � � at the uniform field


� . If � is far away from the boundary, the probability that spin at � is up is �¬�7�Ï�2� , which

is obtained by solving the self-consistent equation given by Eq. (2.9). The conditional

probability that spin at a descendant of � is up, given that the spin at � is down is also� � �Ï�2� . We measure the response of �� to external perturbation by forcibly flipping the

spin at þ (whatever the local field there) and see how many spins in this subtree flip in

response to this perturbation. Let ¦E§ be the probability that the spin at � was down whenþ was down and ¥ spins on the subtree �� flip up if ÿ�� is flipped up. Here allowed values

of ¥ are � Ö � Ö + Ö T9T9T . Clearly, we have� � ( 2F§9] ~ ¦ §¬�,�êT (3.1)

We define ¦G�X¨�� be the generating function of ¦E§ as,¦*�B¨��Ð� 2F§g] ~ ¦ §�¨ § T (3.2)

Clearly, ¦*�B¨í�s� � � ¦ ~ Ö (3.3)¦*�B¨í�D�� î&P� � T (3.4)

It is straight forward to write the self-consistent equation for ¦*�B¨�� . Let us first relax all

spins on �� keeping � and þ down. The probability that exactly � the descendents of �are turned up in this process be denoted by Pr �^�/� . Clearly

Pr �y�/� � ù �E& �� ý � � � �i�î&�� [ j � j � T (3.5)

For a given � , the conditional probability that local field at � is such that spin remains

down, even if þ is turned up is ��&Õ��(T�� . Summing over � , and using the expression for

Pr �^�/� above, we get¦ ~ � [ j �F�A] ~ ù �)& �� ý � � � �Z�%&P� � � [ j � j � × �î&Í� ��T�� Ü T (3.6)

We can write down an expression for ¦<� similarly. In this case, if � of the direct

descendents of � are up when þ is down, the local field at all the remaining ��&<�Þ&Y� direct

descendents must be such that they remain down even if � is flipped up. This probability

is« [ j ��!¬ � � � ¦ [ j � j �~

. The local quenched field at � must satisfy ��)&R+§�/�â>/&R�®-�4�-��)&R+§� &R+ê�â>í&P� . The probability for this to occur is ��(T��A&f� � . Hence we get¦�� Î[ j �F�!] ~ ù �E& �� ý � � � ¦ [ j � j �~ � � �(T��A&f� �ñ�_T (3.7)


The equation determines ¦E§ for higher ¥ can be written down similarly. It only involves

the probabilities ¦Y� with �üVä¥ for the descendent spins. Formally we can write

¦ §¬� [ j �F�!] ~ ù �E&^�� ý � � � �� 2F� §��U] ~ [ j � j �8�]�� ¦)§�� Þ� ý ¥�8 Ö ¥/&^��<�� T��A&f� �ñ� Ö (3.8)

where �Q�B? Ö�� X ¶ � for ? � �� for ?�� is the Kroneker delta.

These recursion equations are expressed more simply in terms of the generating func-

tion ¦G�X¨�� . Multiplying both sides by ¨ § and then summing over ¥ , we see that the self-

consistent equation for ¦G�X¨�� is

¦G�X¨��Ð�µ¦G�X¨í� �ê� (P¨ [ j �F�!] ~ ù �E&ï�� ý � � � ¦*�B¨�� [ j � j � �� T��A&f� �ñ�_T (3.9)

This is a polynomial equation in ¦G�X¨�� of degree �&.� , whose coefficients are functions

of � through � ��2� and � � ��2� . Using Eq. (2.9) and Eq. (3.6), it is easily checked that for¨P� � , the ansatz ¦*�B¨P� �� &^� � satisfies the equation, as it should. To determine¦G�X¨�� for any given external field � , we have to first solve the self-consistent equation for� � [Eq. (2.9)]. This then determines ¦*�B¨^� � � using Eq. (3.6), and then, given �E� and¦G�^� � , we solve for ¦*�B¨�� by solving the �d�E&ï�g� -th degree polynomial equation Eq. (3.9).

Finally, we express the relative frequency of avalanches of various sizes when the ex-

ternal field is increased from � to �î(Õ£¤� in terms of ¦G�X¨�� . Let ª�«M��þ£¤� be the probability

that avalanche of size 6 is initiated at ` . We also define the generating function ª*�B¨ � �� asª*�B¨ � �2�Y� 2F«Ï]�� ªE«M��ó¨ « T (3.10)

Consider first the calculation of ª�«��Ï�2� for 6�� . Let the number of descendents of `that are up at field � be � . For the spin at site ` to be down at � , but flip up at �à( £Þ� ,

the local field �ç· must satisfy × ��&ï+§�/�â>Í& �Ï�à( £¤�� Ü VØ�ç· VÎ× ��c& +R�/�â>Í& � Ü . This

occurs with probability @ ��Þ>f&^+R�Í>f&ä�2�þ£¤� . Each of the ��E&w�/� down neighbors of `must not flip up, even when 6p· flips up. The conditional probability of this event is ¦ [ j �~ .

Multiplying by the probability that � neighbors are up, we finally getª��M�Ï�2�Y� [F�A] ~ ù �� ý � � � ¦ ~ [ j � @A��¤>®&�+§�Í>/&P�2�_T (3.11)


Arguing similarly, we can write the equation for ª�«M�� for 6 �©+ Ö � etc. These equations

simplify considerably when expressed in terms of the generating function ªG�X¨ � �2� , and we

get ª*�B¨ � ��Ð� ¨ [F�A] ~ ù �� ý � � � ¦*�B¨�� [ j � @ ��Þ>/&R+§�Í>í&R�2�âT (3.12)

In numerical simulations, and experiments, it is much easier to measure the avalanche

distribution integrated over the full hysteresis loop. To get the probability that an avalanche

of size 6 will be initiated at any given site ` in the interval when the external field is

increased from �� to �41 , we just have to integrate ª*�B¨ � �� in this range. For any � , the

value of £ ª¬��£ê¨ at ¨Í� � is proportional to the mean size of an avalanche, and thus to the

average slope of the hysteresis loop at that � .

3.2 Explicit calculation for the rectangular distribution

While the general formalism described in the previous section can be used for any distri-

bution, and any coordination number, to calculate the avalanche distributions explicitly, we

have to choose some specific form for the probability distribution function. In this section,

we shall consider the specific choice of a rectangular distribution: The quenched random

field is uniformly distributed between & Ó andÓ

, so that

@A�Ï�Q8X�Y� �+ Ó Ö for & Ó \�Q8Ô\ Ó T (3.13)

In this case, the cumulative probabilities � �S�Ï�2� become piece wise linear functions of� , and � -dependence of the distribution is easier to work out explicitly. We shall work out

the distributions for the linear chain ( ��µ+ ), and the 3-coordinated Bethe lattice.

3.2.1 The linear chain ( �¬�e+ )The simplest illustration is for a linear chain. In this case the self-consistent equation, for

the probability � � [Eq. (2.9)] becomes a linear equation. This is easily solved, and explicit

expressions for ¦ ~ , and ¦*�B¨�� are obtained (see Appendix A.1). The different regimes

showing different qualitative behavior of the hysteresis loops are shown in Fig. 3.2

For � VÎ+ >.& Ó (region A), all the spin remain down. For � - Ó, all spins are up

(region D). ForÓ Vµ> , we get a rectangular loop and the magnetization jumps discontinu-


��

��

!

" #%$

&' (

)*+,-.

)*+,-.

/ -/ ,/ +

Figure 3.2: Behavior of RFIM in the magnetic field - disorder ( ÊOË�Ì ) plane for a linear chain. The

regions A-D correspond to qualitatively different responses. In region A all spins are down and in

region D all are up. The avalanches of finite size occur in region B and C.

ously from &E� to (�� in a single infinite avalanche, and we directly go from region A to D

as the field is increased. ForÓ -©> , we get nontrivial hysteresis loops.

The hysteresis loops for different values ofÓ � �QT10 Ö ��T20 and +ÞT10 are shown in Fig. 3.3.

IfÓ

is sufficiently large (Ó - > ), we find that the mean magnetization is a precisely

linear function of the external field for a range of values of the external field � (region B in

Fig. 3.2). For larger � values, the magnetization shows saturation effects, and is no longer

linear (region C).

The explicit forms of the generating function ¦G�X¨�� are given in the Appendix A.1.

We find that in region B, the function ¦G�X¨�� is independent of the applied field � . The


-2 -1 1 2

-1

-0.5

0.5

1

354768

(a)

-2 -1 1 2

-1

-0.5

0.5

1

95:7;<

(b)

-3 -2 -1 1 2 3

-1

-0.5

0.5

1

=5>7?@

(c)

Figure 3.3: Hysteresis loops for the linear chain for the rectangular distribution of quenched fields

with different widths. (a) Ì ypvG��´�|J} , (b) Ì ypvG�%A5|J} and (c) Ì ypvG�+³K|J}distribution function ª�«_�� has a simple dependence on 6 of the formªE«��Ï�2�Ð�ÑE�k6)Ò >Ó Ô «�Ö (3.14)

where ÑE� is a constant, that depends only on >2� Ó , and does not depend on 6 or � ,ÑE�Y� �+ Ó �i�î&ï>4� Ó � 1�Z>4� Ó � T (3.15)

In region C, the mean magnetization is a nonlinear function of � . But ¦*�B¨�� is still a

rational function of ¨ . From the explicit functional form of ¦G�X¨�� and ª*�B¨ � �2� are given in

the appendix A.1, we find that ª�«M�� is of the formª¬«_�Ï�2�Ð�á×ÙÑ�Ú � 6(RÑÛÚ1ÝÜ Ò >ÓÕÔ « Ö for 6 b +ÞT (3.16)


Here Ñ Ú � and Ñ Ú 1 have no dependence on 6 but are explicit functions of � .

Integrating over � from & ' to ' we get the integrated avalanche distribution ß0« ,ß<«Y� 132j 2 ªE«M��2�þ£Þ� T (3.17)

It is easy to see from above that the integrated distribution ßG« also has the formß<«Ð�Ø×ÙÑS1_6(äãS1 Ü Ò >Ó Ô «�Ö for 6 b + Ö (3.18)

where the explicit forms of the coefficients Ñ 1 and ãS1 are given in the Appendix A.1.

3.2.2 The case �� The analysis for the case � � �

is very similar to the linear case. In this case, the self-

consistent equation for � ��Ï�2� [Eq. (2.9)] becomes a quadratic equation. The qualitative

behavior of solution is very similar to the earlier case. Some details are given in Ap-

pendix A.2. We again get regions A-D as before, but the boundaries are shifted a bit, and

are shown in Fig. 3.4. As before, in region B, the average magnetization is a linear function

of � , and the avalanche distribution is independent of � .

We find that in regime B, the distribution of avalanche sizes is given byªE«M��2� �� è �Ï+�6 �âë�Ï6î&ï��âëW��6O(^+��_ë ì �i�î&ï>2� Ó � « Ò >Ó Ô «�Ö (3.19)

where � is a normalization constant given by� � �+ Ó �Z�Û&ï>4� Ó � 1 ��ó>2� Ó � T (3.20)

It is easy to see that for large 6 , ª�«��Ï�2� varies asª¬«!õ 6 j ö ø ð « Ö (3.21)

where ð �4�i�î&ï>2� Ó ��Z>2� Ó �âT (3.22)

In region B, >2� Ó is always less than �� , and so this function always has an exponential

decay for large 6 .In the region C, we find that the avalanche distribution is of the formªE«��Ï�2�Ð�©�/Ú è �Ï+ê6��_ë�Ï6Û& �g�_ë#�Ï6O(ä+ê�_ë#ì%ð « Ö (3.23)


BDC E F GBHC E F IJGBHC FKE L IJG

MN

OP

Q R%S

TU V

WXYZ[\]

XYZ[\]

^ \^ [^ Z

Figure 3.4: Behavior of RFIM in the magnetic field - disorder ( ÊcËÕÌ ) plane for Bethe lattice of

coordination number 3. The qualitative behavior in different regions A-D is similar to that of a

linear chain (Fig. 3.2).

where � Ú is a normalization constant independent of 6 , and ð is a cubic polynomial in the

external field � :

ð � �_ �i�î&P+¤>4� Ó � 1 �a` À &b0 � �Z>2� Ó � ( �ê�UÀ4�ó>2� Ó � 1 & �U�dcþ�ó>2� Ó � 3�e( ` &50î(©�U�þ�Z>2� Ó ��(�ê� �ó>2� Ó � 1fe �Ï�2� Ó �( ` � & Àþ�Z>2� Ó � 1ge �Ï�2� Ó � 1 (µ��2� Ó � 3 �÷T (3.24)

For any fixed 6 , the integrated distribution ß*« can be evaluated explicitly, but become

lengthy even for small 6 .


3.3 General distributions

The analysis of the previous section can, in principle, be extended to higher coordination

numbers, and other distributions of random fields. However, the self-consistent equations

become cubic, or higher order polynomials. In principle, an explicit solution is possible

for �?\�0 , but it is not very instructive. However, the qualitative behavior of solutions is

easy to determine, and is the same for all � b . We shall take �*�, in the following for

simplicity. Since we only study the general features of the self-consistent equations, wehjilknm�o

p

k mFigure 3.5: A schematic plot of a unimodal random field distribution which asymptotically go to

zero at qJr .

need not pick a specific form for the continuous distributions of random field distribution@A��þ8X� . We shall only assume that it has a single maximum around zero and asymptotically

go to zero at :E' , as shown in Fig. 3.5.

For small width (Ó

) of the random field distribution i.e. for weak disorder the mag-

netization shows a jump discontinuity as a function of the external uniform field, which

disappears for a larger values ofÓ

(section 2.4). For fields � just lower than the value

where the jump discontinuity occurs, the slope of the hysteresis curves is large, and tends

to infinity as the field tends to the value at which the jump occurs. This indicates that large

avalanches are more likely just before the first order jump in magnetization.

For �� , the self-consistent equation for � ��Ï�2� [Eq. (2.9)] is cubic

r � � 3 ( t_� � 1 (�s_� � (ä£�� (3.25)


where r Ö t Ö s and £ are functions of the external field � , expressible in terms of the cumula-

tive probabilities �48 Ö ?÷� � to�, rc��43 & � �41 ( � ��A&Í� ~ Ö (3.26a)t%� � �41Ð& � �� ( � � ~ Ö (3.26b)sî� � ��A& � � ~ & � Ö (3.26c)£��R� ~ T (3.26d)

This equation will have � or�

real roots, which will vary with � . We have shown this

variation for the real roots which lie between 0 and 1 in Fig. 2.5 for the case where @A��8B�is a simple distribution given by Eq. (2.11).

We have also solved numerically the self-consistent equation for �E� for other choices

of �÷��þ8X� , like the gaussian distribution, and for higher �4��µ Ö 0 Ö � � . In each case we find that

the qualitative behavior of the solution is very similar. Note that the rectangular distribution

discussed in the previous section is very atypical in that both the coefficients r and t vanish

for an entire range of values of � .

In the generic case, we find two qualitatively different behaviors: For larger values ofÓ, there is only one real root for any � . For

Ósufficiently small, we find a range of �

where there are�

real solutions. There is a critical valueÓ ô of the width which separates

these two behaviors. For the particular distribution chosen [Eq. (2.11)],Ó ôutD+ÞT �U� � _ + .

In the first case, the real root is a continuous function of � , and correspondingly, the

magnetization is a continuous function of � . This is the case corresponding toÓ �D+ÞT10 in

Fig. 2.5.

For smallerÓ V Ó ô , for large :¬� there is only one root , but in the intermediate

region there are three roots. The typical variation is shown forÓ � �êT10 in Fig. 2.5. In the

increasing field the probability � � ��2� initially takes the smallest root. As � increases, at a

value �0�e� disc, the middle and the lower roots become equal and after that both disappear

from the real plane. At �� disc the probability � ��2� jumps to the upper root. Thus

forÓ V Ó ô there is a discontinuity in � � �Ï�2� which gives rise to a first order jump in the

magnetization curve.

The field � disc where the discontinuity of magnetization occurs, is determined by the

condition that for this value of � , the cubic equation [Eq. (3.25)] has two equal roots. The

value of � � at this point, denoted by � �òó8�«�ô , satisfies the equation� r ~ � � 1òó8�«dô (ä+°t ~ � �òó8�«�ô (Ks ~ � � Ö (3.27)

where r ~ Ö t ~ and s ~ are the values of r Ö t and s at �G�µ� disc.


We now determine the behavior of the avalanche generating function ª<«��Ï�2� for large 6and � near � disc. The behavior for large 6 corresponds to ¨ near � . So we write ¨/�,��&b� ,with � small, and �G�e� disc & û . Near � disc, r Ö t Ö T9T9T vary linearly with û and� � � � �òó8�«�ô & | 7 û (ä`0� û � Ö (3.28)

where | is a numerical constant.

Since ¦G�X¨¿� ��Û� �S&^� � �Ï�2� , if ¨ differs slightly from unity ¦*�B¨�� also differs from� & � �7�Ï�2� by a small amount. Substituting ¨í�,�÷&v� and ¦*�B¨/�D�÷&v��Y�;� & � ��&xw*� û Ö ��in the self-consistent equation for ¦*�B¨�� [Eq. (3.9)], where both � and w are small, using

Eq. (3.27), we get to lowest order in � , û and ww 1 ( ú 7 û wµ& ü 1 �)� � Ö (3.29)

where ú and ü are some constants. Thus, to lowest orders in û and � , w is given bywD�;�i��+ê��y û ú 1 û (P�ü 1 �S& ú 7 û{z T (3.30)

Thus ¦*�B¨�� has leading square root singularity at ¨¢� �î(D| ø�}��~ ø . Consequently, ª*�B¨ � ��will also show a square root singularity ¨/�,�4(�| ø�}��~ ø . This implies that the Taylor expansion

coefficients ª¬«��Ï�2� vary asª¬«��Ï�2�YõD6 j öø ù �O( ú 1Nû�ü 1 ý j « Ö for large 6 . (3.31)

At û �s� , we get ªE«M�� disc �Ðõ 6 j÷öø T (3.32)

Thus at �0�e� disc the avalanche distribution has a power law tail.

To calculate the integrated distribution ßG« , we have to integrate Eq. (3.31) over a range

of û values. For large 6 , only û V ~ ø| ø « contributes significantly to the integral, and thus we

get ß<«Ðõ 6 j��ø , for large 6êT (3.33)

Thus the integrated distribution shows a robust �i&�0ê�ê+�� power law for a range of disorder

strengthÓ

.

Chapter 4

Minor hysteresis loops

on the Bethe lattice

In this chapter, we derive exact self-consistent equations to obtain magnetization on the

minor loops as a function of external field for arbitrary distribution of quenched random

fields on a Bethe lattice. The return hysteresis loops for the linear chain was obtained by

Shukla (2000). In sec. 2.4, we have discussed how to obtain the magnetization on the

lower hysteresis curve, i.e. if we start with �^� & ' , when all the spins are down, and

then slowly increase the external field. Now suppose the system is on the lower hysteresis

curve at some external field � � . Decreasing the field from � � to some field �41 and then

again increasing to � � , we obtain the first minor loop. Similarly starting from the first

minor loop at some field �43 and decreasing the field to �� , and then increasing to �23 , we

obtain the second minor loop and so on. Figure 4.1 shows two minor loops. In general,

the ¥ -th minor loop for ¥¿- � is obtained from the lower half of �B¥/& �g� -th minor loop by

decreasing the field from �41U§ j � to �41U§ and then increasing to �21U§«T��oV �þ1U§ j � . This involves57�Q§Þ=ÝX·�Q§ Ö �Q§ j � Ö T T T Ö �� , the history of all the turning points from � � to �Q§ . In the next

section we will obtain the exact expressions for the magnetizations on the minor loops for

arbitrary distributions of random fields. Similar results were later obtained independently

by Shukla (2001).

4.1 Magnetization on minor loops

In sec. 2.4, we have determined the average magnetization in the deep inside the Cayley

tree, on the lower hysteresis loop for arbitrary distributions of random field distributions.

The average magnetization is equivalent to the magnetization at the root ` of the Cayley

38

Chapter 4. Minor hysteresis loops on the Bethe lattice 39

➤

➤

➤

➤

➤

➤

➤

➤

�� 7�� v�%��f� ��

�f�� ¡¢�£�¡¢¥¤�

¦¨§ª©©«§ª©¬ ¦§®©

¯°§ª©

©«§ª©

¬ ¯°§ª©

Figure 4.1: Minor hysteresis loops for Bethe lattice.

tree [Fig. 2.4], in the limit the number of generations, ±�² ³ . We obtained the magneti-

zation at ´ as,

Prob µ·¶t·¹¸�º¼» ��½¾ ¸ ¿ÀÁÃÂ ~ÅÄ7ÆÇxÈÊÉÌË5Í µ ½¾{Î Á É »JÏ Ë5Í µ ½¾{Î ¿ÑÐ ÁÓÒ Á µ ½¾¥Ô (2.10)

where Ë Í µ ½¨¾ is the limiting value ( Õ×Ö ± , and the limit ±¹² ³ ) of conditional probabilityË �ÙØ � , that a spin on the Õ -th generation will be flipped when its parent spin at generationÕÚÏÛ» is kept down, the external field is½

, and each of its descendent spins has been relaxed,

and is obtained by solving the polynomial equationË5Í µ ½¾ ¸Ü¿ÝÐßÞÀÁÃÂ ~ Ä Æ ÏK»Ç ÈÊÉÌË�Í µ ½¾{Î Á É »JÏ Ë5Í µ ½¾{Î ¿ÝÐßÞàÐ ÁáÒ Á µ ½¾¥Ô (2.9)

andÒ Á µ ½¾ is the probability that a spin flips up, given that exactly Ç neighbors are up,

which is obtained by integrating the random field distribution âuµ ½ãa¾ as,Ò Á µ ½¨¾ ¸ 132� ¿ÝÐåä Á �çæ Ðåè â�µ ½°ãa¾êéå½°ã·ë (2.7)

Similarly for the upper half of the hysteresis loop, when the external field is decreased

from ³ , we can define ì-�ÙØ � µ ½¾ to be the conditional probability that a spin on the Õ -th


generation will be flipped down when its parent spin at generation ÕíÏî» is kept up, the

external field is decreased from ³ to½

, and each of its descendent spins has been relaxed.

The limiting value ì Í µ ½¾ also satisfies self-consistent equation

ì Í µ ½¾ ¸Ü¿ÝÐßÞÀÁuÂ ~ Ä Æ Ïï»Ç È É »JÏbì Í µ ½¾{Î Á É ì Í µ ½¾{Î ¿ÑÐßÞàÐ Á É »JÏ Ò Á T Þ µ ½¾{ÎðÔ (4.1)

and in terms of ì Í µ ½¨¾ the upper half of the major loop can be obtained. SinceÒ Á T Þ µ ½¾ ¸Ò Á µ ½ ºòñåó ¾ , the recursion relation satisfies by »ôÏïì �1Ø � µ ½ Ïõñöó ¾ , is same as the relation

satisfies by Ë �1Ø � µ ½¨¾ which is given by Eq. (2.8). Therefore, we conclude that ì��ÙØ � µ ½ Ï5ñöó ¾ ¸»JÏ Ë �ÙØ � µ ½¾ .4.1.1 First minor hysteresis loop

Suppose the system is on the lower hysteresis curve at some external field½ Þ . Now if

the field is decreased from½ Þ to some field

½ ä and then again increased to½ Þ , return point

memory [section 2.1] ensures that the loop closes. This is the first minor loop [see Fig. 4.1].

Now when the applied field is increased from Ï5³ to½ Þ and then decreased to a field

½ ä , to

find out the spins which can flip down we need to consider only about the subset of spins

which flipped up at field½ Þ . Suppose a spin at a randomly chosen site flips up at field

½ Þ .As a result, the net local field at each of its nearest neighbors increases by an amount ñåóand some of down neighbors might become unstable. We flip up those spins at time step

1. After flipping them more of their neighbor might become unstable. We flip them up in

time step 2 and so on. This process will be continued till the avalanche stop. Figure 4.2

shows the order at which spins flip during a particular avalanche. Now in this avalanche

if a spin ¶ ã flips up at time step � and as a result, if Ç of its neighbors flip at time step�Úºò» , then the local field at ÷ will increase by ñ Ç ó . Therefore when the field is decreased

to½ äùø ½ Þ Ïõñåó , ¶ ã can not flip back at

½ ä unless all the neighbors which had flipped at

time step �Úºú» after ¶ ã flipped up, again flip back at½ ä . Therefore, the spin which was the

initiator of the avalanche (which flipped at time step 0) can flip down at½ ä only at the end,

after all the spins of that avalanche flip back and in this flip-back avalanche the spins flip

exactly in the reverse time order to the previous avalanche. This property will be called the

time ordering property of spin-flip-back process.

Consider the case, when the system is on the lower half of the major loop at field½ Þand then the field is decreased to

½ Þ Ï�ñåó . Then all the neighbors of a vertex ÷ which had

flipped up at½ Þ after ¶ ã flipped up will flip back at

½ Þ Ïõñåó , since they flipped up when


ûûüüýýþþ

ÿÿ��

��

2

1

2 2

3 3

1

0

Figure 4.2: Time order at which spins flip during a particular avalanche.

their local fields had been increased by ñöó . Therefore, the conditional probability that a

spin is down at½ Þ ÏKñåó , given its parent spin is up is same as the conditional probability

that a spin is down at½ Þ , given its parent spin is down, which is »ôÏ Ë �ÙØ � µ ½ Þ ¾ . The later

is again equal to ì��1Ø � µ ½ Þ Ïïñöó ¾ , the conditional probability that a spin is down when the

field is decreased from ³ to½ Þ Ï ñöó , given its parent spin is kept up. Therefore the reverse

magnetization curve starting from½ Þ , meets the upper major half at

½ Þ Ïíñåó and merge with

it for½ ä � ½ Þ Ï ñåó . This result can be generalized for arbitrary graphs, which is discussed

in section 4.2. Thus, we can consider the first minor loop in the range É ½ Þ Ï�ñåó Ô ½ Þ Î . Since

in this range of external field the spin-flip-back process obeys time ordering, if a spin ¶ ãflips up at

½ Þ and flips down at½ ä , then the probability that a neighbor of it at generationÕ is up before ¶ ã flips back at

½ ä is same as the neighbor was up before ¶ ã flipped up at½ Þ ,given by Ë �ÙØ � µ ½ Þ ¾ . The probability that a neighbor is down before ¶ ã flips down at ¶ ã can

be splitted into two parts:

(1) it didn’t flip up after ¶ ã flipped up at½ Þ and

(2) it flipped up after ¶ ã flipped up at½ Þ and flips back at

½ ä before ¶ ã flips back.

Consider a site � at some level Õ of the Cayley tree [Fig. 3.1]. We call the subtree

formed by � and its descendents �� , the subtree rooted at � . We keep its parent spin þat generation ÕnÏ�» down, and relax all the sites in �� at the uniform field

½ Þ . Let � �1Ø �Ð µ ½ Þ ¾be the probability that ¶�� remains down after ¶g� turned up at½ Þ . For ÕùÖ ± , in the limit


±¹² ³ , these probabilities tends to limiting value � Í Ð µ ½ Þ ¾ , given by

� Í Ð µ ½ Þ ¾ ¸ »nÏ ¿ÝÐßÞÀÁÃÂ ~ ÄdÆ Ïï»Ç È ÉÌË5Í µ ½ Þ ¾{Î Á É »JÏ Ë5Í µ ½ Þ ¾{Î ¿ÝÐßÞàÐ Á Ò Á�� Þ µ ½ Þ ¾ ë (4.2)

Let � �1Ø �Ð µ ½ ä Ô ½ Þ ¾ be the conditional probability that:

(a) ¶�� was down at½ Þ , given ¶g� was down,

(b) ¶�� flipped up at½ Þ after ¶�� flipped up and

(c) ¶�� flips back at½ ä , given ¶g� is still up.

Then a recursion relation for � �ÙØ �Ð µ ½ ä Ô ½ Þ ¾ in terms of � �ÙØ � Þ �Ð µ ½ ä Ô ½ Þ ¾ can be written as

� �1Ø �Ð µ ½ ä ÔÑ½ Þ ¾ ¸ ¿ÝÐßÞÀÁÃÂ ~ ÄdÆ Ï%»Ç È É Ë Í µ ½ Þ ¾{Î Á�� ÙØ � Þ �Ð µ ½ Þ ¾ º�� ÙØ � Þ �Ð µ ½ ä Ô ½ Þ ¾�� ¿ÝÐßÞàÐ Á� É Ò Á�� Þ µ ½ Þ ¾ Ï Ò Á�� Þ µ ½ ä ¾àÎdÔ (4.3)

and its limiting value � Í Ð µ ½ ä Ô ½ Þ ¾ , satisfies the self-consistent equation

� Í Ð µ ½ ä Ô ½ Þ ¾ ¸ ¿ÝÐßÞÀÁuÂ ~ Ä Æ Ïï»Ç È É Ë�Í µ ½ Þ ¾àÎ Á � � Í Ð µ ½ Þ ¾ º�� Í Ð µ ½ ä Ô ½ Þ ¾ � ¿ÑÐßÞàÐ Á� É Ò Á�� Þ µ ½ Þ ¾ Ï Ò Á�� Þ µ ½ ä ¾{Î ë (4.4)

This is a polynomial equation in � Í Ð µ ½ ä Ô ½ Þ ¾ of degree Æ Ïõ» , whose coefficients are func-

tions of½ Þ and

½ ä through Ë Í µ ½ Þ ¾ , � Í Ð µ ½ Þ ¾ , Ò Á µ ½ Þ ¾ andÒ Á µ ½ ä ¾ . To determine � Í Ð µ ½ ä Ô ½ Þ ¾for any given pair of external fields

½ Þ and½ ä , we have to first solve the self-consistent

equation for Ë Í µ ½ Þ ¾ [Eq. (2.9)]. This then determines � Í Ð µ ½ Þ ¾ using Eq. (4.2), and then,

given Ë Í µ ½ Þ ¾ and � Í Ð µ ½ Þ ¾ , we solve for � Í Ð µ ½ ä Ô ½ Þ ¾ by solving the µ Æ Ïï» ¾ -th degree poly-

nomial equation Eq. (4.4). Now the decrease in magnetization, when the field is decreased

from½ Þ to

½ ä , is determined by the probability that a spin at ´ was up at½ Þ and turns down

at½ ä , given by,

Prob µ ¶H· ¸ Ï�» � ½ ä � ¶H· ¸�º¼» � ½ Þ ¾ ¸¿ÀÁÃÂ ~ Ä ÆÇ ÈÊÉÌË5Í µ ½ Þ ¾{Î Á�� Í Ð µ ½ Þ ¾ º�� Í Ð µ ½ ä Ô ½ Þ ¾ � ¿ÝÐ Á É Ò Á µ ½ Þ ¾ Ï Ò Á µ ½ ä ¾àÎdë (4.5)

This determines the upper half of first minor loop.

Similarly when the field is again reversed from½ ä to

½"! � ½ Þ , using the symmetry

between up an down spins it is easy to see that again the time ordering property holds.

Therefore the probability that the neighbor of a spin ¶ ã is down before ¶ ã flips up at½"!

isÉ � �ÙØ �Ð µ ½ Þ ¾ º#� �1Ø �Ð µ ½ ä ÔÑ½ Þ ¾{Î . The probability that a neighbor is up at½$!

before ¶ ã flips up is

given by sum of two probabilities � �1Ø �� µ ½ ä ÔÑ½ Þ ¾ and � �ÙØ �� µ ½"!gÔ ½ ä Ô ½ Þ ¾ ; where � �ÙØ �� µ ½ ä Ô ½ Þ ¾ is


the probability that: (a) ¶%� is up at½ Þ , given that ¶g� is kept down and &� is relaxed, (b)¶�� flips up at

½ Þ and (c) ¶�� remains up after ¶g� flips down at½ ä and $� is relaxed and� �ÙØ �� µ ½"!gÔ ½ ä Ô ½ Þ ¾ is the probability that: (a) ¶%� is up at

½ Þ , given that ¶g� is kept down and &� is relaxed, (b) ¶g� flipped up at½ Þ , (c) ¶�� flipped down after ¶g� flipped down at

½ ä and

(d) ¶�� flips back at½'!

, given ¶g� is still down.� �ÙØ �� µ ½ ä Ô ½ Þ ¾ is the equal to the probability that the spin is up at½ Þ minus the probability

that it becomes down at½ ä . Its limiting value is given by

� Í � µ ½ ä Ô ½ Þ ¾ ¸ Ë�Í µ ½¨¾ Ï ¿ÝÐßÞÀÁuÂ ~ Ä Æ Ï%»Ç È ÉÌË5Í µ ½ Þ ¾{Î Á � � Í Ð µ ½ Þ ¾ º�� Í Ð µ ½ ä Ô ½ Þ ¾ � ¿ÝÐßÞàÐ Á� É Ò Á µ ½ Þ ¾ Ï Ò Á µ ½ ä ¾{Î ë (4.6)

The limiting value � Í � µ ½"!�Ô ½ ä Ô ½ Þ ¾ satisfies the self-consistent equation

� Í � µ ½"!gÔ ½ ä Ô ½ Þ ¾ ¸ ¿ÝÐßÞÀÁÃÂ ~ Ä Æ Ïï»Ç È � � Í � µ ½ ä Ô ½ Þ ¾ º�� Í � µ ½'!gÔÑ½ ä Ô ½ Þ ¾�� Á� � � Í Ð µ ½ Þ ¾ º�� Í Ð µ ½ ä Ô ½ Þ ¾ � ¿ÝÐßÞàÐ Á É Ò Á µ ½"! ¾ Ï Ò Á µ ½ ä ¾àÎdë (4.7)

Solving the above self-consistent equation [Eq. (4.7)] we determine � Í � µ ½"!gÔ ½ ä Ô ½ Þ ¾ and

then the increase in magnetization, when the field is increased from½ ä to

½'!is determined

in terms of the following probability:

Prob µ ¶H· ¸òºí»(� ½"!_� ¶H·¹¸ Ï�» � ½ ä ¾ ¸¿ÀÁuÂ ~nÄ7ÆÇ È � � Í � µ ½ ä Ô ½ Þ ¾ º�� Í � µ ½"!gÔ ½ ä Ô ½ Þ ¾ � Á � � Í Ð µ ½ Þ ¾ º�� Í Ð µ ½ ä ÔÑ½ Þ ¾ � ¿ÝÐ Á� É Ò Á µ ½"! ¾ Ï Ò Á µ ½ ä ¾àÎdÔ (4.8)

which determines the lower half of first minor loop.

4.1.2 General minor hysteresis loops

In the previous sub section, we obtained the first minor loop. The other minor loops can be

obtained similarly. In all the minor loops the spin-flip-back process obeys time ordering.

In general, the ± -th minor loop for ±*) » is obtained from the lower half of µ ± Ïî» ¾ -thminor loop by decreasing the field from

½ ä,+ ÐßÞ to½ ä,+ and then increasing to

½ ä,+ � Þ � ½ ä,+ ÐßÞ .For convenience, we will use the notation - ½ + .0/î½+Ô ½+ ÐßÞ Ô�ë ë ëçÔ ½ Þ for the history of all the

turning points from½ Þ to

½+ .

On the upper half of the ± -th minor loop ( ±1) » ), when the field is decreased from½ ä,+ ÐßÞ to½ ä,+ , the probability that a neighbor of a spin ¶ ã is up before ¶ ã (which is deep


inside the tree) flips down at½ ä,+ is É � Í � µ2- ½ ä,+ Ðåä . ¾ º3� Í � µ�- ½ ä,+ ÐßÞ . ¾{Î . The probability that a

neighbor of ¶ ã is down before it flips down is given by É � Í Ð µ2- ½ ä,+ ÐßÞ . ¾ º4� Í Ð µ�- ½ ä,+ . ¾{Î , where� Í Ð µ2- ½ ä,+ ÐßÞ . ¾ is given by,

� Í Ð µ2- ½ ä,+ ÐßÞ . ¾ ¸ É � Í Ð µ�- ½ ä,+ Ð !�. ¾ º�� Í Ð µ�- ½ ä,+ Ðåä . ¾àÎÏ ¿ÝÐßÞÀÁÃÂ ~ÅÄdÆ Ïï»Ç È�É � Í � µ�- ½ ä,+ Ðåä . ¾ º�� Í � µ�- ½ ä,+ ÐßÞ . ¾{Î Á� É � Í Ð µ2- ½ ä,+ Ð !�. ¾ º�� Í Ð µ2- ½ ä,+ Ðåä . ¾àÎ ¿ÝÐßÞàÐ Á� É Ò Á�� Þ µ ½ ä,+ ÐßÞ ¾ Ï Ò Á�� Þ µ ½ ä,+ Ðåä ¾{Î·Ô (4.9)

and � Í Ð µ2- ½ ä,+ . ¾ satisfies the self consistent equation

� Í Ð µ2- ½ ä,+ . ¾ ¸ ¿ÝÐßÞÀÁuÂ ~ Ä Æ Ï%»Ç È É � Í � µ�- ½ ä,+ Ðåä . ¾ º�� Í � µ�- ½ ä,+ ÐßÞ . ¾{Î Á� É � Í Ð µ2- ½ ä,+ ÐßÞ . ¾ º�� Í Ð µ2- ½ ä,+ . ¾àÎ ¿ÝÐßÞàÐ Á� É Ò Á�� Þ µ ½ ä,+ ÐßÞ ¾ Ï Ò Á�� Þ µ ½ ä,+ ¾àÎ ë (4.10)

Therefore the decrease in magnetization, when the field is decreased from½ ä,+ ÐßÞ to

½ ä,+ , is

obtained from

Prob µ ¶H· ¸ Ï » � ½ ä,+ � ¶H·¹¸�º¼» � ½ ä,+ ÐßÞ ¾¸ ¿ÀÁuÂ ~�Ä7ÆÇvÈ�É � Í � µ2- ½ ä,+ Ðåä . ¾ º�� Í � µ2- ½ ä,+ ÐßÞ . ¾{Î Á� É � Í Ð µ�- ½ ä,+ ÐßÞ . ¾ º�� Í Ð µ�- ½ ä,+ . ¾àÎ ¿ÝÐ Á É Ò Á µ ½ ä,+ ÐßÞ ¾ Ï Ò Á µ ½ ä,+ ¾{Î·ë (4.11)

Similarly on the lower half, the increase in magnetization, when the field is increased

from½ ä,+ to

½ ä,+ � Þ , is obtained from

Prob µ ¶H· ¸òºí»(� ½ ä,+ � Þ � ¶t·¹¸ Ï�» � ½ ä,+ ¾¸ ¿ÀÁuÂ ~ Ä7ÆÇ È�É � Í � µ2- ½ ä,+ . ¾ º�� Í � µ2- ½ ä,+ � Þ . ¾{Î Á� É � Í Ð µ�- ½ ä,+ ÐßÞ . ¾ º�� Í Ð µ�- ½ ä,+ . ¾àÎ ¿ÝÐ Á É Ò Á µ ½ ä,+ � Þ ¾ Ï Ò Á µ ½ ä,+ ¾{Î·Ô (4.12)

where � Í � µ�- ½ ä,+ . ¾ is given by

� Í � µ�- ½ ä,+ . ¾ ¸ É � Í � µ2- ½ ä,+ Ðåä . ¾ º�� Í � µ2- ½ ä,+ ÐßÞ . ¾{ÎÏ ¿ÑÐßÞÀÁuÂ ~ Ä Æ Ïï»Ç È�É � Í � µ2- ½ ä,+ Ðåä . ¾ º�� Í � µ2- ½ ä,+ ÐßÞ . ¾{Î Á� É � Í Ð µ�- ½ ä,+ ÐßÞ . ¾ º�� Í Ð µ�- ½ ä,+ . ¾{Î ¿ÝÐßÞàÐ Á� É Ò Á µ ½ ä,+ ÐßÞ ¾ Ï Ò Á µ ½ ä,+ ¾àÎ Ô (4.13)


and � Í � µ2- ½ ä,+ � Þ . ¾ is obtained by solving the self consistent equation

� Í � µ2- ½ ä,+ � Þ . ¾ ¸ ¿ÝÐßÞÀÁuÂ ~ Ä Æ Ï%»Ç È�É � Í � µ�- ½ ä,+ . ¾ º�� Í � µ�- ½ ä,+ � Þ . ¾{Î Á� É � Í Ð µ2- ½ ä,+ ÐßÞ . ¾ º�� Í Ð µ2- ½ ä,+ . ¾àÎ ¿ÝÐßÞàÐ Á� É Ò Á µ ½ ä,+ � Þ ¾ Ï Ò Á µ ½ ä,+ ¾{Î·ë (4.14)

In Fig. 4.1, we have plotted first two minor hysteresis loops, generated by solving

above equations, for three coordinated Bethe lattice. The random field distribution is given

by Eq. (2.11) and we choose 5�¸ » ë 0 .4.2 Merging of different stable configurations

In the previous section, we see that the two ends of minor loop are at major loop, with the

external field is differed by ñåó . In this section we generalize this to any stable configuration

on any graphs. We prove that, for RFIM on a connected general graph � , for a given

realization of random fields, all the stable configurations at external field½

go to unique

stable configuration 6 µ ½87 ñ Æ Í ó ¾ when the field is monotonically changed to½87 ñ Æ Í ó ;

where Æ Í is the minimum number required such that, any connected subgraph 9 of � has

at least one vertex such that, the number of edges in 9 connected to that vertex is : Æ Í . For

example, Æ Í ¸òñ for square lattice, and for Bethe lattice Æ Í ¸ » .Proof. — Consider two stable configurations 6 Þ µ ½¾ and 6 ä µ ½¾ at field

½. We can de-

compose vertices of the graph � , into sets: (1) ; �$� , up-spins in both configurations, (2); �=< , up-spins in 6 Þ and down-spins in 6 ä , (3) ;>< � , down-spins in 6 Þ and up-spins in 6 äand (4) ;><2< , down-spins in both configurations. Consider when the external field is in-

creased monotonically by ñ Æ Í ó . Since in the zero temperature dynamics, in the increasing

field field the spins flip only once, and the order in which various spins are relaxed does

not matter, we can increase the field in one step to½ ºáñ Æ Í ó and then first relax spins from

sets ; �=< and ;?<Û� . Now the set ; �=< can be written as union of disjoint subsets ; � Þ ��=< Ô ; � ä ��@< ë�ë�ë .Consider one such subset, which is on a subgraph 9 . On this subgraph, the local field at a

vertex ÷ in configurations 6 Þ and 6 ä at field½

are,

A �CBED�� è � �ã ¸ ½°ã º ½ º ÆGFã ó Ï ÆGHã ó º3I ã )#J Ô (4.15)ALK BNM K è=OPOã ¸ ½°ã º ½ Ï ÆGFã ó º ÆGHã ó º�I ã � J Ô (4.16)


where Æ Fã are the number of vertices in 9 connected to vertex ÷ , Æ Hã are the number of vertices

in the set ;><Û� connected to ÷ and I ã is the contribution to the local field from the sets ; � �and ;><2< . Since

ALK BED K è=OQOã Ï ALK BNM K è@OPOã )*J , from Eq. (4.15) and Eq. (4.16) we get Æ Fã Ï Æ Hã )RJor Æ Fã ø » as Æ Hã ø J . This means that, there can not be a subset of ; �=< which has only one

element.

Now when the external field is increased to½ º ñ Æ Í ó , the local field at vertex ÷ in

configuration 6 ä becomes,A Hã ¸1- ½°ã º ½ º Æ Íã óÛÏ ÆSHã ó º3I ãT. º#- µ Æ Íã Ï Æ Fã ¾ ó º�ñ ÆGHã ó .Åë (4.17)

Therefore, at all those vertices in 9 where Æ Fã : Æ Í (there is at least one such vertex in 9 ,

by the definition of Æ Í ), the local field will become positive. So, the spins in 6 ä , at those

vertices will flip up and the original subset will will shrink to a new one on a different

reduced subgraph 9 H and the same argument holds for it also. Therefore, after iterative use

of this relaxation procedure, all the subsets of ; �=< will become null sets and by the same

argument, it is also true for ;U< � . Therefore, after relaxing all the spins from the sets ; �=<and ;>< � , the resultant unstable configurations V6 Þ µ ½ ºjñ Æ Í ó ¾ and V6 ä µ ½ ºjñ Æ Í ó ¾ are identical,

and hence relaxation of the remaining unstable spins from the set ;W<X< , will lead to the same

final stable configuration. From the symmetry between up and down spins, it is obvious

that the configuration 6 Þ µ ½¾ and 6 ä µ ½¨¾ go to the same final configuration, when the field

is decreased by ñ Æ Í ó .

For a square lattice Æ Í ¸ ñ , as in any connected subgraph of it, there exists at least

one vertex form which the number of edges connected to the subgraph is :�ñ . Therefore

any two different stable configurations should merge to one configuration, when the field

is increased or decreased by Y ó . In Fig. 4.3, we consider two stable spin configurations

(a) and (b) at external field½ ¸ZJ , on a square lattice, for a given realization of random

fields. The lattice size is 0GJ � 0(J . The spins which are down in (a) and up in (b) are shown

in (c) in black color. Now when we increase the external field to Y°ó , configurations (a)

and (b) evolve to new stable configurations (d) and (e) respectively. We see that these two

configurations (d) and (e) are identical as seen from their difference configuration (c).


(a) (b) (c)

(d) (e) (f)

Figure 4.3: (a) and (b)are two different stable spin configurations at []\ ´ , with the same realiza-

tion of random fields, on a square lattice of size }«´Üµ }«´ . The up spins are represented by black and

down spins by white color. (c) shows the difference between (a) and (b). The spins, which are down

in (a) and up in (b) are represented by black color in (c). (d) and (e) are new stable configurations

obtained from (a) and (b) respectively, at [^\`_�v . (f) shows the difference between (d) and (e).

Chapter 5

Hysteresis on regular lattices

in the low disorder limit

In general, Bethe approximation is expected to work well for noncritical properties. Is the

Bethe approximation is a good approximation for regular lattices? This is the question we

address in this chapter. Surprisingly, for asymmetrical distribution, the answer can be no.

In this chapter, we discuss the low disorder limit of the hysteresis loop in the random

field Ising model (RFIM) on periodic lattices in two and three dimensions. We find that

the behavior of hysteresis loops depends nontrivially on the coordination number Æ (Sab-

hapandit et al. 2002). For Æ ¸ba , for continuous unbounded distributions of random fields,

the hysteresis loops show no jump discontinuity of magnetization even in the limit of small

disorder, but for higher Æ they do. This is exactly as found in the exact solution on the

Bethe lattice (Dhar et al. 1997).

As discussed in the introduction, random field Ising model was first studied in the

context of possible destruction of long range order by arbitrarily weak quenched disorder

in equilibrium systems. Accordingly the distribution of random field was assumed to be

symmetrical. However, in hysteresis problem, the symmetry between up and down spins

state is already broken by the specially prepared initial state (all down in our case), and the

symmetry of the distribution plays no special role.

The analytical treatment of self-consistent equations on the Bethe lattice is immediately

generalized to asymmetrical case. However, we find that for asymmetrical distributions

the behavior of hysteresis loops in euclidean lattices can be quite different from that on

the Bethe lattice. On hypercubical lattices iné

dimensions, there is an instability related

to bootstrap percolation, that is absent on the Bethe lattice. This reduces the value of

the coercive field½

coer away from the Bethe lattice value ÿxµ{ó ¾ to zero, where ó is the

exchange coupling. We note that the limit 5Ü² J is somewhat subtle, as the system size

48

Chapter 5. Hysteresis on regular lattices in the low disorder limit 49

c Í required for self-averaging diverges very fast for small 5 , and the finite-size corrections

to the thermodynamic limit tend to zero very slowly.

In the following, we shall assume that the distribution has a asymmetrical shape, given

by âuµ ½°ã ¾ ¸ »5edgfih µ{Ï ½°ãkj 5 ¾Xl µ ½êã ¾ � (5.1)

wherel

is the step function. The mean value of½ã

can be made zero by a shift in the

value of the external uniform field. Our treatment is easily extended to other continuous

unimodal distributions. The exact form of âuµnm ¾ is not important, and other forms like

d=fEh µ ÏomvÏqp Ðsr ¾ which fall sharply for negative m have the same behavior.

For a given distribution âuµ ½ßã ¾ , we defineÒ Á µ ½¾ with Jt: Ç : Æ as the conditional

probability that the local field at any site ÷ will be large enough so that it will flip up, if Çof its neighbors are up, when the uniform external field is

½. ClearlyÒ Á µ ½¾ ¸ 1 2

K ¿ÑÐåä Á O æ Ðåè â�µ ½°ãa¾Jéå½êã ë (2.7)

Clearly, for any given value of½

, the magnetization depends on the distribution âuµ ½«ãa¾only through

Ò Á µ ½¨¾ .5.1 Hysteresis on three coordinated lattices

Consider first the case of the two-dimensional hexagonal lattice with Æ ¸ua . For periodic

boundary conditions, if 5 ¸vJ , starting with a configuration with all spins down, clearly

one has½

coer ¸wa ó . For 5 �¸xJ , the site with the largest local field flips first, and then if½ )Üó ,Ò Þ µ ½¾ ¸ » , this causes neighbors of the flipped spin to flip, and their neighbors,

and so on. Thus, so long as there is at least one flipped spin, all other spins also flip, and

the magnetization is » . The largest local field in a system ofc ä spins is of order ñ(5�yQz c .

Once this spin turns up, other spin will flip also up, causing a jump in magnetization from

a value {�Ï » to a value º¼» in each sample. Hence the coercive field, (the value of½

where

magnetization changes sign) to lowest order in 5 , is given by½coer ¸ba ó Ïáñ(5�yQz c Ô for »5Ö|y}z c Ö ó j 5 ë (5.2)

Sample to sample fluctuations in the position of the jump are of order 5 . On averaging

over disorder, the magnetization will become a smooth function of½

, with the width of the

transition region being of order 5 .


~ � ��2��~ � ��^�

� �

�� 4�

�L�� =��

�]�&�C��]�&�P ��¢¡��P£�]��¥¤

¦ �¢��P ��¢�

��P � ¦ �¢�

Figure 5.1: Magnetization in the increasing field. The curves for the two values of § coincide.

Curves A is for hexagonal lattice of size _L¨L©%ª ä and B is for a three coordinated lattice in three

dimensions [see Fig. 5.2] of size «%¬%ª ! .

For a fixed 5HÖ ó , ifc

is increased to a value near d=fEh µ{ó j 5 ¾�/ c Íhex,½

coer decreases

to a value near ó . For½ { ó ,

Ò Þ µ ½¾ is no longer nearly » , butÒ® µ ½¾ t¯J , and

Ò ä µ ½¾ tÒ ! µ ½¾ t » . The value of magnetization depends only onÒ Þ µ ½¾ , which is a function of° ½ ¸�µ ½ ÏKó ¾±j 5 . As

° ½increased from Ï�³ ,

Ò Þ µ ½¾ increases continuously from J to » .Note that for 5 ¸²J ë J°»ðó ,

c Íhex ³ »%JG´ ! . Therefore it is impossible to study the largec

limit with the available computers. To avoid the problem of probability of nucleation being

very small for½

near ó , we made the local field at a small fraction of randomly chosen sites

very large, so that these spins are up at any½

. The number of such spins we choose to be

of orderc

, so that their effect on the average magnetization is negligible. Introduction of

these “nucleation centers” makesc Í {²µxµ�¶ c ¾

( the average separation between centers),

and½

coer drops to a value near ó , so that, we can study the largec

limit with available

computers. Forc ) c Íhex, the behavior of hysteresis loops becomes independent of

c.

In Fig. 5.1, curve A shows the result of a simulation on the hexagonal lattice withc ¸²YNJ · ¸ , and periodic boundary condition. We see that magnetization no longer under-


goes a single large jump, but many small jumps. In the figure, we also show the plot of

magnetization when the random field at each site is decreased by a factor 10. This changes

the value 5 from J ë »ðó to J ë J°» ó . However, plotted as a function of° ½

, the magnetization for

these two different values (for small 5 ) fall on top of each other for the same realization

of disorder (except for the overall scale 5 ). Thus we can decrease 5 further to arbitrarily

small values, and the limit of 5 ² J is straightforward for each realization of disorder.

Then, averaging over disorder, for a fixed 5 , we see that½

coer tends to the value ó as 5tends to J . Also, we see that there is no macroscopic jump-discontinuity for any non-zero5 .

Figure 5.2: A three coordinated lattice ( ¹U\tº ) in three dimensions.

We also show in Fig. 5.1 [curve B], the results of simulation of a a -dimensional lattice

with Æ ¸ba [shown in Fig. 5.2] of size ñð0(¸ ! with periodic boundary condition. The behavior

is qualitatively same as that in two dimensions. The value of½

coer ¸ ó in the limit 5 ²J is same for symmetrical distribution, and also is the same as predicted by the Bethe

approximation.


5.2 Bootstrap instability in RFIM on square lattice

On the square lattice also, the value of½

coer is determined by the need to create a nucleation

event. Arguing as before, we see that½

coer to lowest order in 5 is given by½coer {RY°ó ÏbñG5�y}z c Ô for »�Ö y}z c Ö ó j 5 ë (5.3)

Adding a small number of nucleation sites suppresses this slow transient, and lowers½

coer

from Y°ó to a value near ñåó . However, in this case, even after adding the nucleation centers,

the system shows a large single jump in magnetization, indicating the existence of another

instability. We observed in the simulation that at low 5 , as½

is increased, the domains of

up spins grow in rectangular clusters [see Fig. 5.3] and at a critical value of½

coer, one of

them suddenly fills the entire lattice. This value½

coer fluctuates a bit from sample to sample.

Figure 5.3: A snapshot of the up-spins just before the jump ( [»\îA�¼½©%©%¾%«�_SºG¿ ). The lattice size is«�¨%¨ÁÀ8«�¨%¨ and §Â\Ã¨ ¼Ä¨%¨dA�¿ . Initial configuration is prepared with ¨ ¼Ä¨L¬LÅ up-spins.


Æ Ç È�É�È?È�Ê�Ë

ÌÎÍ ÏÑÐÌÒÍ Ó�Ô�ÕÌÎÍ Ô�Ö�ÏÌÎÍ ÖoÓ�ÔÌÎÍ Ó�×�ÔØÐ

ÙÛÚÂÜ'ÝÑÞ$ßáà âÃã�äæåèçéGêëìíî ïbðñòóôõö÷ø

ùú

û�ü�ý¥þû ÿiý �û ÿ"ý��û � ý¢üû � ý��û � ý¥þ

ü&ý � �ü&ý ÿ þü&ý ÿ �ü&ý¢ü��ü&ý¢üiü

Figure 5.4: Distribution of the scaled coercive field on a square lattice for different lattice size � ä .In Fig. 5.4 we have plotted the distribution of the scaled variable V ½� ¸ µ ½ coer Ï�ñåó ¾±j 5 for

different system sizesc

, for 5�¸ J ë J J »ðó . The number of different realizations varies from»%JG´ (for the largestc

) to »%J� (for the smallestc

). Note that the distribution shifts to the left

with the increasing system size, and becomes narrower.

This instability can be understood in terms of bootstrap percolation process BP Á (see

Adler 1991, for a review). Bootstrap percolation was first considered by Chalupa et al.

(1981) (also Kogut and Leath 1981) and was subsequently studied by many others in a

variety of contexts. The process BP Á is define as follows: On aé Ï dimensional lattice,

sites are independently occupied with a probabilityÒ

and the resulting configuration is

taken as the initial configuration, which is evolved by the following rules:

(a) the occupied sites remain occupied forever,

(b) an unoccupied site having at least Ç occupied neighbors, becomes occupied.

For Ç ¸ ñ , on a square lattice, in the final configuration, the sites which are occupied

form disjoint rectangles, like the cluster of up-spins in Fig. 5.3. It has been proved that

in the thermodynamic limit of largec

, for any initial concentrationÒ ) J , in the final

configuration all sites are occupied with probability » (Aizenman and Lebowitz 1988).

In the random field Ising model on a square lattice, for the asymmetric distribution


[Eq. 5.1] for½ )uJ ,

Ò Á ¸ » for Ç ø ñ , and any spins with more than one up-neighbors

flips up. Therefore, stable clusters of up spins are rectangular in shape. The growth of

domains of up spins is same as in the bootstrap percolation process BP ä .Consider a rectangular cluster of up spins, of length � and width Ç . Let Ë µ � Ô Ç ¾ be

the probability that, if this rectangle is put in a randomly prepared background of densityÒ Þ µ ½¾ , this rectangle will grow by the BP ä process to fill the entire space. The probability

that the random fields at any sites neighboring this rectangle will be large enough to cause

it to flip up isÒ Þ µ ½¾ . The probability that there is at least one such site along each of two

adjacent sides of length � and Ç of the rectangle is µ »7Ï�� ¾ µ »7Ï�� Á ¾ , where ��¸ »7Ï Ò Þ µ ½¾ .Once these spins flip up, this induces all the other spins along the boundary side to flip up

and the size of the rectangle grows to µ �ßºú» ¾ � µ Ç ºõ» ¾ . ThereforeË µ � Ô Ç ¾ ø µ »JÏ�� ¾ µ »JÏ�� Á ¾ Ë µ��êºõ» Ô Ç ºú» ¾ ë (5.4)

Thus the probability of occurrence of a nucleation which finally grows to fill the entire

lattice is Ë nuc ø Ò' µ ½¾�� Â Þ µ »JÏ�� ¾ ä{ Ò' µ ½¾ �� Â Þ É »nÏ d=fEh -öÏ Ò Þ µ ½¾��".gÎ ä{ Ò' µ ½¾ d=fEh Ä Ï � äa Ò Þ µ ½¾ È Ô for small

Ò Þ µ ½¾¥ë (5.5)

The condition to determine½

coer is that for this value of½

, Ë nuc becomes of order » j c ä ,so that we get Ò' µ ½ coer

¾ d=fEh Ä Ï � äa Ò Þ µ ½ coer¾ È { »c ä ë (5.6)

This equation can be solved for½

coer for any givenc

. For the distribution given by Eq. (5.1),

this becomes

d=fEh Ä ½ coer ÏÃY°ó5 È dgfih� Ï ñ � äa dgfih Ä Ï ½ coer º�ñåó5 È�� { »c ä Ô for

½coer

� ñåó ë (5.7)

Therefore, the leadingc

-dependence of½

coer, to lowest order in 5 is given by½coer {îñåóÛÏt5�y}z y a� ä µkyQz c Ï%ó j 5 ¾ z Ô for ó j 5�Ö|y}z c Ö d=fEh µ ñöó j 5 ¾¥ë (5.8)

This agrees with our observation that the scaled critical field V½�� shifts to the left with

increasing system size. The width of the distribution of over which the coercive field


� !#"$!%!#&('

) * +-,) * .0/21

) * /432+) * 35.0/) * .768/9,

:<;>=@? A

B�CD EGFHIJK

LNMPOQLNMPOSRLNMTRVULNMWRYX

L7MWRVUL7MWRYXL7MWRYOL7MZL\[

Figure 5.5: ] Þ_^ [ coer ` vs. A_aSbdc-� for square lattice.

varies, can be calculated from the width over which the probability of having at least one

nucleation in the entire lattice, i.e. »5Ï µ »JÏ Ë nuc¾fe M { »�Ï d=fEh µ{Ï Ë nuc

c ä ¾ , changes from

almost zero to almost unity: g ½coer ³ 5y}z c (5.9)

Therefore, for any fixed 5�)#J , the jump will smeared out on averaging over disorder. Only

in the limit 5�² J andc ² ³ , the average magnetization will show a jump discontinuity.

To test the validity of Eq. (5.6) in simulations, we putÒ& µ ½¾ ¸bJ ë J Jih independent of

½.

Eq. (5.6) then simplifies to Ò Þ µ ½ coer¾ { � ä¸�y}z c ë (5.10)

In Fig. 5.5, we have plottedÒ Þ for the mean

½coer from Fig. 5.4 versus » j y}z c . The graph

is approximately a straight line, which agrees with Eq. (5.10). The slope of the line isJ ë c(¸ih 7 J ë J J · , less than in Eq. (5.10), which only gives an upper bound to½

coer.

If½ )²J , we will have

Ò ä ¸Ü» , and bootstrapping ensures that so long asÒ& )eJ , we

will have all spins up in the limit of largec

. This implies that½

coer ¸bJ in this limit.

If there are sites with large negative quenched fields, the bootstrap growth stops at


such sites. Hence the bootstrap instability cannot be seen for symmetric distributions.

For a symmetrical distribution of random fields, the average distance from a nucleus to the

nearest spin, which does not flips up even if it has two up neighbors, isc Í ¸ » jYj »JÏ Ò ä µ ½¾ .Therefore the average area covered by a nucleus is

c ä e�� Â Þ µ »JÏ�� ¾ ä forc � c Í (5.11a)

and c Í ä elk�� Â Þ µ »JÏ�� ¾ ä forc ) c Í ë (5.11b)

The condition to determine½

coer is that for this value of½

, the average area due to the

growth becomes µxµ c ä ¾ , i.e.Ò" µ ½ coer¾ c ä � average area covered by a nucleus { µxµ c ä ¾¥ë (5.12)

From Eq. (5.11b) and Eq. (5.12), in the limitc ² ³ ,Ò' µ ½ coer

¾»JÏ Ò ä µ ½ coer¾ e k�� Â Þ µ{»JÏ�� ¾ ä { µxµ » ¾ ë (5.13)

NowÒ' µ ñåó ¾ ¸ »nÏ Ò ä µ ñåó ¾ , and the product m e@k� Â Þ µ{»nÏ�� ¾ ä is µ µ{» ¾ at

½ ¸ ñåó asÒ Þ µ ñöó ¾ ¸» j ñ . Therefore,

½coer { µxµ ñöó ¾ .

Even if the quenched fields are only positive, the instability does not occur on lattices

with Æ ¸ba . On such lattices, if the unoccupied sites percolate, there are infinitely extended

lines of unoccupied sites in the lattice. These cannot not become occupied by bootstrapping

under BP ä . Thus the critical threshold for BP ä on such lattices is not J .

5.3 Bootstrap instability in RFIM on cubic lattice

The arguments for large void instability can be easily extended to higher dimensions. Iné ¸ba , if½ )�J , then

Ò Á µ ½¾ ¸ » for Ç ø a , therefore the spin flip process is similar to the

spanning process of three dimensional BP!

(Cerf and Cirillo 1999). In this case, it is known

that for any initial non-zero density, in the thermodynamical limit, the final configuration

has all sites occupied with probability » . The clusters of up-spins grow as cuboids, and at

each surface of the cluster, the nucleation process is similar to that in two dimension. Let


n be the probability that, a nucleation occurs at a given point of a surface of the clusters of

up spins which sweeps the entire two dimensional plane at½

.n { Ò Þ µ ½¨¾ dgfEh Ä Ï � äa Ò ä µ ½¾ È ë (5.14)

The probability that, there exist at least one nucleation which sweeps the entire plane of

size � � � , is »�Ïîµ »×Ï n ¾ � M . Therefore, the probability Ë nuc, that a nucleation sweeps the

entire three dimensional lattice at½

satisfiesË nuc ø Ò" µ ½¾ �� Â Þ� »JÏõµ{»JÏ n ¾ � M � !

{ Ò" µ ½¾%�� Â Þ� »JÏ dgfEh µ{Ï n � ä ¾ � !{ dgfEh µ Ï�; j ¶ n ¾¥Ô for small n � (5.15)

where ; ¸ ! ä ¶ �po µTa j ñ ¾ . ½ coer is determined by the condition that Ë nuc must be of the order» j c ! : Ò' µ ½ coer¾ dgfEhrqs Ï ;j Ò Þ µ ½ coer

¾ d=fEh Ä � ä¸ Ò ä µ ½ coer¾ È�tu { »c ! ë (5.16)

The leadingc

-dependence of½

coer is different in different ranges of½

coer, depend-

ing on whether the strongest dependence of the left-hand side comes from variation ofÒ' µ ½¾ Ô Ò Þ µ ½¨¾ orÒ ä µ ½¾ .In the range Y°ó � ½

coer� ¸ ó :

Ò Á ¸ » , for Ç ø » . Then we must haveÒ® µ ½ coer

¾ ³» j c ! , which for the distribution given by Eq. (5.1) results½coer {R¸ ó Ïqa(5�y}z c ë (5.17a)

The corresponding range ofc

, for the validity of of above equation is »òÖ y}z c Öµ ñåó j a 5 ¾ .In the range ñöó � ½

coer� Y°ó :

Ò Á ¸ » , for Ç ø ñ . Then in Eq. (5.16) the left hand side

varies as d=fEh � Ï�; H jVj Ò Þ µ ½ coer¾ �

, which gives½coer {bY°ó Ï�ñ(5�y}z<vLyQz c Ï ñåóa 5�w Ô (5.17b)

which is valid in the range µ ñåó j a 5 ¾ Ö|y}z c Ö d=fEh µ ñöó j 5 ¾ .In the range J � ½

coer� ñåó :

Ò Á ¸ » for Ç ø a . Then from Eq. (5.16), to the lowest

order in 5 , we get ½coer { ñåó Ït5�y}z y}zxvLyQz c Ï ñåóa 5yw Ô (5.17c)


for d=fEh µ ñöó j 5 ¾ Ö yQz c Ö d=fEh µ dgfih µ·ñåó j 5 ¾ ¾ .In the limit

c{z c Ícub ¸ dgfEh µ d=fEh µ d=fEh µ ñåó j 5 ¾ ¾ ¾ , the loop becomes independent ofc

,

with½

coer ² J . We have also verified the existence of jump in numerical simulation forÆ ¸ Y (diamond lattice) in three dimensions.

Chapter 6

Discussion

Analytical treatment of problems having quenched disorder is usually difficult. There are

few models having nontrivial quenched disorder that can be solved exactly. In this thesis,

we set up exact self-consistent equations for the avalanche distribution function for the

RFIM on a Bethe lattice. We were able to solve these equations explicitly for the rectan-

gular distribution of the quenched field, for the linear chain Æ ¸�ñ , and the 3-coordinated

Bethe lattice. For more general coordination numbers, and general continuous distributions

of random fields, we argued that for very large disorder, the avalanche distribution is ex-

ponentially damped, but for small disorder, generically, one gets a jump in magnetization,

accompanied by a square-root singularity. For field-strengths just below corresponding to

the jump discontinuity, we showed that the avalanche distribution function has a power-law

tail of the form ¶ Ð !$| ä . The integrated avalanche distribution then varies as ¶ Ð | ä for large¶ .We have also studied the behavior the return loop, when the external field is increased

from Ï�³ to some value½ Þ , and then decreased to a lower value

½ ä and again increased

to the previous extremum value½ Þ . We set up exact self-consistent equations to determine

the magnetizations on all minor loops for arbitrary distributions of random fields.

Some unexpected features of the solution deserve mention. Firstly, we find that the

behavior of the self-consistent equations for Æ ¸va is qualitatively different from that forÆ )xa . The behavior for the linear chain ( Æ ¸ ñ ) is, of course, expected to be different

from higher Æ . One usually finds same behavior for all Æ )�ñ . Mathematically, the reason

for this unusual dependence is that the mechanism of two real solutions of the polynomial

equation merging, and both becoming unphysical (complex) is not available for Æ ¸ a .

Here the self-consistency equation is a quadratic, and from physical arguments, at least

one of the roots must be real. That a Bethe lattice may show non-generic behavior for

low coordination numbers has been noted earlier by Ananikyan et al. (1994) in their study

59

Chapter 6. Discussion 60

of the Blume-Emery-Griffiths model on a Bethe lattice. These authors observed that the

qualitative behavior for Æ � ¸ is different from that for Æ ø ¸ . To find out whether this

unusual Æ dependence of the hysteresis loop persists on regular lattices, we study it on the

regular lattices in two and three dimension, in the limit of low disorder. We find that for

asymmetrical distributions of random fields, there is a instability which is not present in

three coordinated lattices, and hence the hysteresis curve is continuous for such lattices.

The second point we want to emphasize is that here we find that the power-law tail in

the distribution function is accompanied by the first-order jump in magnetization. Usu-

ally, one thinks of critical behavior and first-order transitions as mutually exclusive, as

first-order jump pre-empts a build-up of long-ranged correlations, and all correlations re-

main finite-ranged across a first-order transition. This is clearly not the case here. In fact,

the power-law tail in the avalanche distribution disappears, when the jump disappears. A

similar situation occurs in equilibrium statistical mechanics in the case of a Heisenberg

ferromagnet below the critical temperature. As the external field½

is varied across zero,

the magnetization shows a jump discontinuity, but in addition has a cusp singularity for

small fields } . But in this case the power-law tail is seen on both sides of the transition.

Note that for most values of disorder, and the external field, the avalanche distribution

is exponentially damped. We get robust power law tails in the distribution, only if we

integrate the distribution over the hysteresis cycle across the magnetization jump. But, in

this case, the control parameter½

is swept across a range of values, in particular across

a (non-equilibrium) phase transition point! In this sense, while no explicit fine-tuning

is involved in an experimental setup, this is not a self-organized critical system in the

usual sense of the word. Recently Pazmandi et al. (1999) have argued that the hysteretic

response of the Sherrington-Kirkpatrick model to external fields at zero temperature shows

self-organized criticality for all values of the field. However, this seems to be because of

the presence of infinite-ranged interactions in that model.

In chapter 3, we discussed the behavior of avalanche distribution for various distribu-

tions of random fields. A general question concerns the behavior of the avalanches for

more general probability distributions. Clearly, ifÒ µ ½ã ¾ has a discrete part, it would give

rise to jumps inÒ ã

as a function of½

, and hence give rise to several jumps in the hysteresis

loop. These could preempt the cusp singularity mechanism which is responsible for the~Below �� , the magnetization goes as,��f��\�P�@� � �5��4� ��P�� S� as � �¢¡ �¤£4¥<¦§¥#¨

(see Parisi 1988) ©

Chapter 6. Discussion 61

power-law tails. If the distributionÒ µ ½ßã ¾ is continuous, but multimodal, then it is possible

to have more than one first order jump in the magnetization } . This is confirmed by ex-

plicit calculation in some simple cases. IfÒ µ ½ßãa¾ has power-law singularities, these would

also lead to power-law singularities inÒ ã

, and hence in Ë Í µ ½¾ . Even for purely continuous

distributions, the merging of two roots as the magnetic field varies need not always occur.

For example, it is easy to check that for the rectangular distribution, even for Æ ø Y , we

do not get a power law tail for any value of 5 . The precise conditions necessary for the

occurrence of the power-law tail needs to be investigated further.

Finally, we would like to mention some other open questions. Our analysis relied heav-

ily on the fact that initial state was all spins down. Of course, we can start with other initial

conditions. For example, start with the equilibrium state at temperature and field½

, and

then quench to zero temperature. Our present treatment cannot be applied to these cases

as finite temperature brings about very nontrivial coorrelations between spins. It would be

interesting to set up self-consistent field equations for them. In case of minor loops also,

we have always started with½ ¸ Ï�³ and then vary the field cyclically. Moreover, to

find the magnetization at some particular point of the hysteresis curve, we start with the

previous extremal field and change the field to the new value in one jump and then relax

the system. It would be useful to find out some dynamical relations by which system can

be evolved from any state by changing the field in infinitesimal steps.

Another extension would be to make the rate of field-sweep comparable to the single-

spin flip rate (still assuming zero temperature dynamics). This would mean some large

avalanches in different parts of the sample could be evolving simultaneously. Then one

could study the sweep-rate dependence of the hysteresis loops, and the frequency depen-

dence of the Barkhausen noise spectra. This is perhaps of some relevance in real experi-

mental data, and would also make contact with other treatments of Barkhausen noise that

focus on the domain wall motion.

Another case of some interest is other type of disorder e.g. the site-dilution case dis-

cussed by Tadic (1996). It seems plausible from the structural stability of the mechanism

which leads to the cusp singularity just before the jump-discontinuity in magnetization,

in our model, introduction of site dilution would not change the qualitative behavior of

solutions.

We hope that many of these issues will be resolved in the next few years.~This would happen if ª2« �P�@� as a function of � shows a ‘double S’ curve. Then there must be at least

¨values of � for which the slope of the curve is infinite. This is possible only if the equation determining ª¬«�®�¯ �[variant of Eq. (3.27)] is at least a quartic, hence only if °§±#² .

Appendix A

A.1 Avalanche distribution on a linear chain

For the case of a linear chain, the self-consistent equation, for the probability Ë Í [Eq. (2.9)]

is a linear equation, whose solution is,Ë Í µ ½¾ ¸ Ò"É »JÏúµ Ò Þ Ï Ò' ¾{Î ë (A.1)

For½ � ñöó Ï#5 ,

Ò'is zero, and hence Ë Í µ ½¾ is zero, and all the spin remain down

(region A in Fig. 3.2).

For½ ) ñåó Ï 5 , and 5 � ó ,

Ò Þ is » wheneverÒ®

is nonzero. Then from Eq. A.1,Ë Í µ ½¾ becomes » . Thus, for 5 � ó , we get a rectangular loop and the system changes

from all spins down to all spins up state in a single big avalanche.

For 5�) ó ,Ò Þ Ï Ò' equals ó j 5 and is independent of

½, in the range ñåó×Ï 5 � ½ � 5 .

Thus Ë Í µ ½¾ is a linear function of½

in this range, increasing from J to » .Defining n ¸ »ñ Ä »7º ½

5 Ï ñöó5 È Ô (A.2)

we obtain the expression for Ë Í as

Ë�Í µ ½¾ ¸ ³´´µ ´´¶J for n � J ,}ÞàÐ æ |®· for J�: n : »nÏ%ó j 5 ,» for n ) »JÏ%ó j 5 .

(A.3)

Using Eq. (3.6), the expression for ì is,ì ¸ µ »JÏ Ò Þ ¾ Ïúµ Ò ä Ï Ò Þ ¾ Ë�Í µ ½¨¾ ë (A.4)

The generating function ì µ m ¾ obtained from the self-consistent equation [Eq. (3.9)] is,ì µ m ¾ ¸ ì ºqm Ë Í µ Ò ä Ï Ò Þ ¾»JÏ`m�µ Ò Þ Ï Ò' ¾ Ô(A.5)

62

Appendix A. 63

and the generating function � µnm2¸ ½¾ given by Eq.( 3.12) becomes ,

� µnm2¸ ½¨¾ ¸#m ` É ì µ m ¾{Î ä âuµ ñöóxÏ ½¾ ºKñ Ë�ÍðÉ ì µ m ¾{Î âuµ Ï ½¾ º Ë�Í ä âuµ{Ï5ñåó Ï ½¾ e ë (A.6)

Now if 5�)õñåó , and Ï>5îº�ñåó � ½ � 5 Ï�ñåó (region B in Fig. 3.2),µ Ò ä Ï Ò Þ ¾ ¸�µ Ò Þ Ï Ò' ¾ ¸ ó j 5 Ôâuµ ñöó Ï Ç ó Ï ½¨¾ ¸ »ñ(5 for Ç ¸ J Ô » Ô ñE�and Ë Í º�ìº¹7¸ »JÏKó j 5 ë

Thus ì µnm ¾ ¸ »�Ïïµ{ó j 5 ¾»JÏúµ{ó j 5 ¾ m Ï Ë5Í Ô (A.7)

and � µ m2¸ ½¾ ¸ mñ(5 É Ë�Í º�ì µnm ¾àÎ ä ¸ mñ(5 µ{»JÏ%ó j 5 ¾ äµ »nÏÃm ó j 5 ¾ ä ë (A.8)

Expanding � µ m2¸ ½¾ in powers of m , we get the probability distribution of avalanches in

region B given by Eq. (3.14) of sec. 3.2.1.

In the region C,Ò ä saturates to value » , â�µ Ï�ñåó7Ï ½¾ becomes zero and µ Ò ä Ï Ò Þ ¾ becomesµ »JÏ%ó j 5 Ï n ¾ . Thus we get, ì ¸ µ »nÏ%ó j 5 Ï n ¾ äµ »JÏKó j 5 ¾ ë

(A.9)

In terms of Ë Í and ì we getì µnm ¾ ¸ ì ºtm Ë Í É »JÏbñêµàó j 5 ¾ Ï n Î»JÏúµàó j 5 ¾ m Ô(A.10)

and � µnm2¸ ½¨¾ ¸ mñ(5 ` ÉÌË5Í º�ì µ m ¾{Î ä Ï Ë�Í ä e ë (A.11)

Expanding � µnm2¸ ½¨¾ in powers of m we get , in region C

� Þ µ ½¾ ¸ »ñ(5 � µ Ë�Í º�ì ¾ ä Ï Ë5Í ä � Ô (A.12)

and �¼»¥µ ½¾ ¸ É ; H Þ ¶7º�; H ä Î v ó5 w » Ô for ¶ ø ñ ë (A.13)

Here ; ä and ½ ä have no dependence on ¶ but are explicit functions of½

; H Þ ¸ »ñ(5 � »µ{ó j 5 ¾ µ Ë Í º�ì ¾ ä º »µàó j 5 ¾ ä Ä »JÏ ñåó5 Ï ½5 È µ Ë Í º�ì ¾ Ë Íº »Yµ{ó j 5 ¾ ! Ä »JÏ ñåó5 Ï ½

5 È ä Ë5Í ä tu Ô; H ä ¸ Ï »5 qs »ñêµàó j 5 ¾ ä Ä »JÏ ñåó5 Ï ½

5 È µ Ë�Í º�ì ¾ Ë�Í º »Yµ{ó j 5 ¾ ! Ä »JÏ ñåó5 Ï ½5 È ä Ë5Í ä tu ë

Appendix A. 64

Integrating over½

from Ï�³ to ³ we get the integrated avalanche distribution ¾<» ,¾¿» ¸¢À �Ð � �¼»lµ ½¾ éå½�Ô (A.14)

where¾ Þ ¸ »µ »JÏKó j 5 ¾ ä� »JÏq¸Áv ó5 w ºú»�Y¿v ó5 w ä Ï Y ¸a v ó5 w ! º Yöc¸ v ó5 w ´ Ï · h v ó5 w � Ô

(A.15)

and, for ¶ ø ñ , ¾¿» ¸�µk; ä ¶ÅºÂ½ ä ¾ v ó5 w » Ô (A.16)

with

; ä ¸ »a Jêµ{ó j 5 ¾ � a JôÏ%» »�J¿v ó5 w ºõ»%aih�v ó5 w ä ÏrhSY�v ó5 w ! � Ô½ ä ¸ »»Ãhêµ{»nÏ%ó j 5 ¾ � h5Ïï»%JÁv ó5 w º�Y¿v ó5 w ä � ëA.2 Avalanche distribution on a three coordinated Bethe

lattice

For Æ ¸ba , the self-consistent equation. for Ë Í µ ½¾ [Eq. (2.9)] is a quadratic equation,É µ Ò ä Ï Ò Þ ¾ Ïúµ Ò Þ Ï Ò' ¾àÎ Ë�Í µ ½¨¾ ä º É ñ°µ Ò Þ Ï Ò' ¾ Ïï» Î Ë�Í µ ½¨¾ º Ò' ¸ÂJ ë (A.17)

For the rectangular distribution, the coefficient of Ë Í ä is zero for a range of½

-values, andË Í µ ½¾ is still a piece wise linear function of½

Ë�Í µ ½¨¾ ¸ ³´´µ ´´¶J for n � J ,}ÞàÐåä K æ |®· O for J8: n :î»JÏ�ñêµ{ó j 5 ¾ ,» for n ) »JÏ�ñêµàó j 5 ¾ , (A.18)

where n is defined as, n ¸ »ñ Ä »7º ½5 Ï aåó5 È ë (A.19)

The self-consistent equation for ì µ m ¾ [Eq. (3.9)] becomes,

m�µ Ò Þ Ï Ò' ¾ É ì µnm ¾àÎ ä º É ñSm Ë5Í µ Ò ä Ï Ò Þ ¾ Ïï» Î ì µ m ¾ ºqm Ë�Í ä µ Ò ! Ï Ò ä ¾ º�ì ¸ J Ô (A.20)

where ì is obtained [Eq. (3.6)] asì ¸�µ{»JÏ Ò Þ ¾ Ïbñêµ Ò ä Ï Ò Þ ¾ Ë�Í º É µ Ò ä Ï Ò Þ ¾ Ïúµ Ò ! Ï Ò ä ¾àÎ Ë�Í ä Ô (A.21)

Appendix A. 65

and the expression for � µnm2¸ ½¾ [Eq. (3.12)] becomes,

� µnm2¸ ½¾ ¸Âm ` É ì µ m ¾{Î ! â�µTa óvÏ ½¨¾ º�a É ì µ m ¾{Î ä Ë5Í â�µ{ó Ï ½¾ºWa É ì µnm ¾àÎ Ë�Í ä âuµ{Ï�ó Ï ½¾ º Ë�Í ! â�µ Ï?a óÛÏ ½¾ e ë (A.22)

Now in the region B, µ Ò ! Ï Ò ä ¾ ¸�µ Ò ä Ï Ò Þ ¾ ¸�µ Ò Þ Ï Ò' ¾ ¸ ó j 5 ÔâuµTaåó Ï�ñ Ç ó Ï ½¾ ¸ »ñ(5 for Ç ¸ J to a Ôand Ë5Í º�ìº¹7¸ »JÏKó j 5 ë

Solving Eq. (A.20) and choosing the root which is well behaved for m near J , we get

ì µnm ¾ ¸ »JÏ j »�ÏtYßµ{ó j 5 ¾ m�µ Ë Í º�ì ¾ñêµàó j 5 ¾ m Ï Ë Í Ô (A.23)

and the expression for � µnm2¸ ½¾ [Eq. (A.22)] becomes

� µ m2¸ ½¾ ¸ mñ(5 ÉÙË Í º�ì µnm ¾àÎ ! ë (A.24)

Expanding � µnm ¾ in power series of m , we obtain the Eq. (3.19) of sec. 3.2.2.

In the region C,Ò !

saturates to the value » , âuµ{Ï>a óvÏ ½¾ becomes zero and µ Ò ! Ï Ò ä ¾ is

no longer independent of½

. Substituting the appropriate expressions, we find that

ì µ m ¾ ¸ »�Ï j »JÏtYµ{ó j 5 ¾ m É µ »JÏtaßµàó j 5 ¾ Ï n ¾ ºòµ Ë Í º�ì ¾{Îñêµàó j 5 ¾ m Ï Ë�Í Ô (A.25)

and � µnm2¸ ½¨¾ ¸ mñ(5 ` ÉÌË5Í º�ì µ m ¾{Î ! Ï Ë�Í ! e ë (A.26)

We note that the term inside the radical sign in ì µ m ¾ , and also in � µnm2¸ ½¾ , is a simple

linear function of m . It is thus straightforward to expand it in powers of m using binomial

expansion. This gives us the Eq. (3.23) of sec. 3.2.2.

Reprints

In this appendix, we reproduce reprints of the papers that have already appeared in refereed

journals in the following order.

1. Distribution of Avalanche Sizes in the Hysteretic Response of Random Field Ising

Model on a Bethe Lattice at Zero Temperature,

Sanjib Sabhapandit, Prabodh Shukla and Deepak Dhar, J. Stat. Phys. 98, 103

(2000).

2. Hysteresis in the Random Field Ising Model and Bootstrap Percolation,

Sanjib Sabhapandit, Deepak Dhar and Prabodh Shukla, Phys. Rev. Lett. 88,

197202 (2002).

66

Journal of Statistical Physics 98, 103 (2000) 1

Distribution of Avalanche Sizes in the Hysteretic Response of Random Field Ising Modelon a Bethe Lattice at Zero Temperature

Sanjib Sabhapandit Ä Å , Prabodh Shukla Æ�Ç , Deepak Dhar ÄÉÈÊTheoretical Physics Group, Tata Institute of Fundamental Research,

Homi Bhabha Road, Mumbai-400005, India.ËPhysics Department, North Eastern Hill University,

Shillong-793 022, India.

We consider the zero-temperature single-spin-flip dynamics of the random-field Ising model on a Bethe latticein the presence of an external field Ì . We derive the exact self-consistent equations to determine the distributionProb ÍZÎ$Ï of avalanche sizes Î , as the external field increases from Ð�Ñ to Ñ . We solve these equations explicitlyfor a rectangular distribution of the random fields for a linear chain and the Bethe lattice of coordination numberÒ0ÓÁÔ , and show that in these cases, Prob ÍZÎ$Ï decreases exponentially with Î for large Î for all Ì on the hysteresisloop. We find that for Ò9ÕºÖ and for small disorder, the magnetization shows a first order discontinuity for severalcontinuous and unimodal distributions of the random fields. The avalanche distribution Prob ÍZÎ$Ï varies as ÎØ×ÚÙWÛ�Üfor large Î near the discontinuity.

Key Words: Random Field Ising Model, Hysteresis, Barkhausen noise, avalanches.

I. INTRODUCTION

Analytical treatment of problems having quenched disorderis usually difficult. There are few models having nontrivialquenched disorder that can be solved exactly. In this paper,we obtain exact results for the non-equilibrium properties ofthe random-field Ising model (RFIM) on the Bethe lattice. Weconsider the single-spin-flip Glauber dynamics of the systemat zero temperature, as the external magnetic field is slowlyvaried from ÝßÞ to à2Þ . As the field increases, the mag-netization increases as groups of spins flip up together. Thismodel has been proposed as a model of the Barkhausen noiseby Sethna et al [1] (see also [2]). In this paper, we set upthe exact self-consistent equations satisfied by the generatingfunction of the distribution of avalanche sizes, and analyzethese to determine the behavior of the avalanche distributionfunction on the Bethe lattice.

The study of the equilibrium properties of the RFIM hasbeen an important problem in statistical physics for a longtime. In 1975, Imry and Ma [3], showed that arbitrarily weakdisorder destroys long-ranged ferromagnetic order in dimen-sions áãâåä . The persistence of ferromagnetism in á<æ¢äwas a matter of a long controversy, but has now been estab-lished [4]. A recent review of earlier work on this model maybe found in [5]. As far as an exact calculation of thermody-namic quantities is concerned, there are only a few results.For example, Bruinsma studied the RFIM on a Bethe latticein the absence of an external field and for a bivariate randomfield distribution [6]. There are no known exact results for theaverage free energy or magnetization, for a continuous distri-bution of random field, even at zero temperature and in zero

applied field.

-3 -2 -1 1 2 3h

-1

-0.5

0.5

1

M

FIG. 1. The Hysteresis loop of magnetization ç versus the exter-nal field è for RFIM. The zoomed figure shows the small jumps inmagnetization that give rise to the Barkhausen noise

The non-equilibrium properties of the RFIM has attracteda lot of interest lately, arising from the observation by Sethnaet al [1] that its zero-temperature dynamics provides a sim-ple model for the Barkhausen noise and return point mem-ory. Barkhausen noise is the high frequency noise generateddue to the small jumps in magnetization observed when fer-romagnets are placed in oscillating magnetic fields [Fig. 1].Understanding and reduction of this noise is important forthe design of many electronic devices [7]. Experimentallyit is observed [8–11] that the the increase of the magnetiza-tion occurs in bursts that span over two decades of size andthe distribution of burst (avalanche) sizes seems to follow apower law in this range. Similar avalanche-like relaxationalevents are also observed in other systems, for example, thestress-induced martensitic growth in some alloys [12]. This

éE-mail address: [email protected]êE-mail address: [email protected]ëE-mail address: [email protected]


power-law tail in the event-size distribution was interpretedby Cote and Meisel [11] as an example of self-organized crit-icality. But Percovic et al [2] have argued that large burstsare exponentially rare, and the approximate power-law tail ofthe observed distribution comes from crossover effects due tonearness of a critical point. Recently Tadic [13] has presentedsome evidence from numerical simulations that the exponentsfor avalanche distribution can vary continuously with disor-der. Our results about the behavior of the avalanche distri-bution function also relate to this question whether any fine-tuning of parameters is required to see power-law tails in theavalanche distribution in the RFIM, and if the exponents canbe varied continuously with disorder.

The advantage of working on the Bethe lattice is that theusual BBGKY hierarchy of equations for correlation functionscloses, and one can hope to set up exact self-consistent equa-tions for the correlation functions. The fact that Bethe’s self-consistent approximation becomes exact on the Bethe latticeis useful as it ensures that the approximation will not violateany general theorems, e.g. the convexity of thermodynamicfunctions, sum rules. In the presence of disorder, in spite ofthe closure of the BBGKY hierarchy, the Bethe approxima-tion is still very difficult, as the self-consistent equations be-come functional equations for the probability distribution ofthe effective field. These are not easy to solve, and availableanalytical results in this direction are mostly restricted to onedimension [14], or to models with infinite-ranged interactions[15]. On the Bethe lattice, for short-ranged interactions withquenched disorder, e.g. in the prototypical case of the ìîírandom-exchange Ising model, the average free energy is triv-ially determined in the high temperature phase, but not in thelow-temperature phase. It has not been possible so far to de-termine even the ground-state energy exactly despite severalattempts [16].

Calculation of time-dependent or non-equilibrium proper-ties presents its own difficulties, even in the absence of disor-der. Usually, for ïÁð�ñ , one has to resort to the limit of co-ordination number becoming large, with interaction strengthscaled suitably with coordination number to give a nontriv-ial thermodynamic limit [17]. The large-d limit in the self-consistent field approximation for quantum-mechanical prob-lems is similar in spirit [18].

The RFIM model on a Bethe lattice is special in that thezero-temperature nonequilibrium response to a slowly varyingmagnetic field can be determined exactly [19]. To be precise,the average non-equilibrium magnetization in this model canbe determined exactly if the magnetic field is increased veryslowly, from òßó to ô2ó , in the limit of zero temperature.It thus provides a good theoretical model to study the slowrelaxation to equilibrium in glassy systems. The dynamicsis governed by the existence of many metastable states, withlarge energy barriers separating different metastable states.We hope that this study of non-equilibrium response in thismodel would help in the more general problem of understand-ing the statistical mechanics of metastable states in glassy sys-tems.

A brief summary of our results is as follows. We derivethe exact self consistent equations for the generating func-

tion of the avalanche size distribution function õ÷ödø@ù on theBethe lattice. This is a polynomial equation in õ§öZø@ù andø , in which the coefficients depend on the external field ú ,and the distribution of the quenched random fields. We cansolve these equations explicitly numerically and thus deter-mine the qualitative behavior of the distribution of avalanchesfor any distribution of the quenched random fields. The be-havior depends on the coordination number û , and on the de-tails of the distribution function. We work out the distribu-tion of avalanches explicitly for a rectangular distribution ofthe quenched fields, for the linear chain ( ûºü�ý ), and the 3-coordinated Bethe lattice. In both cases, one finds only expo-nential decay. We also studied other unimodal continuous dis-tributions, e.g. when the random field distribution is gaussian,or of the form Prob öPúlþÿù ü�� , also for large � . Wefind that, for �� , there is a regime of disorder strengths forwhich the magnetization shows a jump-discontinuity ( “firstorder transition”), but the avalanche distribution, averagedover the hysteresis loop also shows a power law tail of theform �� (“critical fluctuations”).

The paper is organized as follows. In section II, we definethe model precisely. In section III, we briefly recapitulate thederivation of self-consistent equations for the magnetizationin our model, and then use a similar argument to construct thegenerating function for the avalanche distribution for arbitrarydistribution of the quenched random field. We set up a self-consistent equation for the probability !#" , that an avalanchepropagating in subtree flips exactly $ more spins in the subtreebefore stopping. The probability distribution of avalanches isexpressed in terms of this generating function. In section IV,we consider the special case of a rectangular distribution ofthe random field. In this case, we explicitly solve the self-consistent equations for Bethe lattices with coordination num-bers �&%(' and ) . However, this case is non-generic. Forsmall strength of disorder * , the magnetization jumps from+�, to - , at some value of the field, but for larger disorder,when the system shows finite avalanches, there is no jump inmagnetization and the distribution function decays exponen-tially for large � . In section V, we analyse the self-consistentequations to determine the form of the avalanche distributionfor some other unimodal continuous distributions of the ran-dom field. We find that in each case for coordination number�.�/� , the magnetization shows a first order jump disconti-nuity as a function of the applied field at some field-strength0�1�24365

, for weak disorder. Just below0 % 071�28365 , the avalanche

distribution has a universal 9 + );: ' � power-law tail. Section VIcontains a discussion of our results, and some concluding re-marks. Some algebraic details of the analytical solution forthe rectangular distribution of quenched fields are relegated totwo appendices.

II. DEFINITION OF THE MODEL

We consider a uniform Cayley tree of $ generations whereeach non-boundary site has a coordination number � (seeFig. 2). The first generation consists of a single vertex. The

Journal of Statistical Physics 98, 103 (2000) 3< -th generation has =�>?=�@�A�B�CEDGF vertices for <IH&J .KKLLMMNN OOPPQQRR SSTT UUVV WWXX YYZZ[[\\ ]]^^ __`` aabb ccdd eeff gghh iijj kkll mmnn oopp qqrrssttuuvv

o

r = 4

r = 3

r = 2

r = 1

FIG. 2. A Cayley tree of coordination number 3 and 4 generations.

The RFIM on this graph is defined as follows : At each ver-tex there is a Ising spin wxzy|{~} which interacts with nearestneighbors through a ferromagnetic interaction � . There arequenched random fields �6x at each site � drawn independentlyfrom a continuous distribution �� x�� . The entire system isplaced in an externally applied uniform field � . The Hamilto-nian of the system is� y��/�� x�� E� w�x�wE�� x ��x�w�x7�� x w�x (1)

We consider the response of this system when the externalfield � is slowly increased from �� to � � . We assume thedynamics to be zero-temperature single-spin-flip Glauber dy-namics, i.e. a spin flip is allowed only if the process lowersenergy. We assume that if the spin-flip is allowed, it occurswith a rate � , which is much larger than the rate at which themagnetic field � is increased. Thus we assume that all flip-pable spins relax instantly, so that the spin w�x is always parallelto the net local field � x at the site:w x y�w��¡ ��x �¢y£w��; � �¥¤��¦�§ w � � � x � �¨� (2)

We start with �©y�� , when all spins are down and slowlyincrease � . As we increase � , some sites where the quenchedrandom field is large positive will find the net local field pos-itive, and will flip up. Flipping a spin makes the local field atneighboring sites increase, and in turn may cause them to flip.Thus, the spins flip in clusters of variable sizes. If increasing� by a very small amount causes w spins to flip up together,we shall call this event an avalanche of size w . As the appliedfield increases, more and more spins flip up until eventuallyall spins are up, and further increase in � has no effect.

III. THE SELF-CONSISTENT EQUATIONS

The special property of the ferromagnetic RFIM that makesthe analytical treatment possible is this: Suppose we start with��yª�� , and all spins down at « y¬ . Now we change thefield slowly with time, in such a way that � ��« �©®/� ��¯ � , for

all times «~°±¯ . Then the configuration of spins at the finalinstant « y ¯ does not depend on the detailed time depen-dence of � �?« � , and is the same for all histories, so long as thecondition � ��« �²®�� ?¯ � for all earlier times is obeyed. In par-ticular, if the maximum value � ��¯ � of the field was reachedat an earlier time «³§ , then the configuration at time ¯ is ex-actly the same as that at time «�§ . This property is called thereturn point memory [1]. We may choose to increase the fieldsuddenly from �� to � ��¯ � in a single step. Then, once thefield becomes �©y£� ��¯ � , several spins would have positive lo-cal fields. Suppose there are two or more such flippable sites.Then flipping any one of them up can only increase the localfield at other unstable sites, as all couplings are ferromagnetic.Thus to reach a stable configuration, all such spins have to beflipped, and the final stable configuration reached is the same,and independent of the order in which various spins are re-laxed. This property will be called the abelian property of re-laxation. Using the symmetry between up and down spins, itis easy to see that the abelian property also holds whether thenew value of field �¨´ is greater or less than its initial value �¶µso long as one considers transition from a stable configurationat �¨µ to a stable configuration at ��´ .

We first briefly recapitulate the argument of our earlier pa-per [19] which uses the abelian nature of spin-flips to deter-mine the mean magnetization for any field � in the lower halfof the hysteresis loop by setting up a self-consistent equation.

Since the spins can be relaxed in any order, we relax themin this: first all the spins at generation (the leaf nodes) arerelaxed. Then spins at generation ·�¸} are examined, andif any has a positive local field, it is flipped. Then we exam-ine the spins at generation ·�&¹ , and so on. If any spin isflipped, its descendant are reexamined for possible flips [20].In this process, clearly the flippings of different spins of thesame generation º are independent events.

Suppose we pick a site at random in the tree away fromthe boundary, the probability that the local field at this siteis positive, given that exactly » of its neighbors are up, isprecisely the probability that the local field � x at this site ex-ceeds ¼ �¾½ �¿¹ » �À�I�¿�ÂÁ . We denote this probability by �7Ã~� �¨� .Clearly, � Ã � �¨�¢yÅÄ.ÆÇ ¤�ÈGÉ ÃËÊ?Ì ÈGÍ �Î� �¨x¾�¶Ï¡�¨x (3)

Let Ð Ç8Ñ Ê � �� be the probability that a spin on the ¿� º -thgeneration will be flipped when its parent spin at generation Ò� º �Å} is kept down, the external field is � , and each ofits descendent spins has been relaxed. As each of the ½ �Å}direct descendents of a spin is independently up with proba-bility Ð ÇÓÑ È §6Ê , it is straightforward to write down a recursionrelation for Ð Ç8Ñ Ê in terms of Ð ÇÓÑ È §¶Ê . For º~Ô�Ô } , these proba-bilities tend to limiting value Ð�Õ , which satisfies the equation[19]

Ð Õ � ��Öy×¤�È §�ÃË¦ÙØ�Ú ½ ��}» Û ¼ Ð Õ � ��Á Ã ¼ }Ü� Ð Õ � �¨�ÝÁ ¤�È § È Ã ��Ã~� �¨�(4)


For the spin at Þ , there are ß downward neighbors, and theprobability that it is up is given by

Prob à�á�â�ã�ä~å;æ�ç�è¢ãéêë�ì7í�î ßïñð#ò ó�ô à¾ç�è�õ ë ò åÜö ó�ô à�ç¨èÝõ é�÷Gë�ø ë à¾ç�è (5)

Because all spins deep inside the tree are equivalent,Prob à�á â ãùä~å¡æ�ç�è determines the average magnetization forall sites deep inside the tree. Using Eqs. (4-5), we can de-termine the magnetization for any value of the external fieldç . This determines the lower half of the hysteresis loop. Theupper half is obtained similarly.

Now consider the state of the system at external field ç , andall the flippable sites have been flipped. We increase the fieldby a small amount ú¡ç till one more site becomes unstable. Wewould like to calculate the probability that this would causean ‘avalanche’ of û spin flips. Since all sites deep inside areequivalent, we may assume the new susceptible site is the siteÞ .

It is easy to see that this avalanche propagation is some-what like propagation of infection in the contact process onthe Bethe lattice. The ‘infection’ travels ü�ý�þzÿ¨þ��ü�� from thesite Þ which acts as the initiator of infection. If any site is in-fected, then it can cause infection of some of its descendents.If the descendent spin is already up, it cannot be flipped; suchsites act as immune sites for the infection process. If thedescendent spin is down, it can catch infection with a finiteprobability. Furthermore, this probability does not dependon whether the other ‘sibling’ sites catch infection. Infectionof two or more descendents of an infected site are uncorre-lated events. Thus, we can expect to find the distribution ofavalanches on the Bethe lattice, as for the size distributionof percolation clusters on a Bethe lattice [21]. However, aprecise description in terms of the contact process is compli-cated, as here the infection spreads in a correlated backgroundof ‘immune’ (already up ) spins, and the probability that a sitecatches infection does depend on the number of its neighborsthat are already up.

We start with the initial configuration of all spins down.Now increase the external field to the value ç . Consider a site�

at some generation ��/å of the Cayley tree [Fig. 3]. Wecall the subtree formed by

�and its descendents � , the sub-

tree rooted at�

. We keep its parent spin � at generation � öñådown, and relax all the sites in at the uniform field ç . If�

is far away from the boundary, the probability that spin at�is up is ó ô à¾ç�è . The conditional probability that spin at a

descendant of�

is up, given that the spin at�

is down is alsoó ô à¾ç�è . We measure the response of to external perturba-tion by forcibly flipping the spin at � (whatever the local fieldthere) and see how many spins in this subtree flip in responseto this perturbation. Let �� be the probability that the spin at�

was down when � was down and û spins on the subtree� flip up if � � is flipped up. Here allowed values of û are�� å ��. Clearly, we haveó ô ä �ê� ì7í ��ã�å (6)

We define

� à �GèÖã �ê� ì7í � � � �(7)

!"!!"!#"##"#$$$$$%%%%%

&"&"&"&"&"&'&"&"&"&"&&"&"&"&"&"&'&"&"&"&"&&"&"&"&"&"&'&"&"&"&"&&"&"&"&"&"&'&"&"&"&"&&"&"&"&"&"&'&"&"&"&"&&"&"&"&"&"&'&"&"&"&"&&"&"&"&"&"&'&"&"&"&"&

("("("("("('("("("("(("("("("("('("("("("(("("("("("('("("("("(("("("("("('("("("("(("("("("("('("("("("(("("("("("('("("("("(("("("("("('("("("("()"))")*"**"*

+,

-/.FIG. 3. A sub-tree 0�1 formed by 2 and its descendents. The

sub-tree is rooted at 2 and 3 is the parent spin of 2 .

Clearly, 465 798;:=<�8>4�? and 465@7A8CBD<�8CBFEAG�H . It isstraight forward to write the self-consistent equation for 465 7I< .Let us first relax all spins on J�K keeping L and M down. Theprobability that exactly N the descendents of L are turned upin this process be denoted by GFO�5@NP< . Clearly

GFO�5 N�<�8RQTS E9BN U G HDV 5WBXEYG H <WZ�[�\][ V (8)

For a given N , the conditional probability that local fieldat L is such that spin remains down, even if M is turned upis BXE�^ V`_ \ . Summing over N , and using the expression forGFO�5@NP< above, we get

4F?a8 Z�[ \bV`c ? Q S E9BN U G H V 5WBdEeG H <WZ�[�\][ Vgf BdEh^ Vi_ \kj (9)

We can write down an expression for 4 \ similarly. In thiscase, if N of the direct descendents of L are up when M isdown, the local field at all the remaining S ElB�EmN directdescendents must be such that they remain down even if Lis flipped up. This probability is n Z�[ \Vpo Grq V 4 Z�[ \�[ V? . Thelocal quenched field at L must satisfy 5 S EtsuNP<]vAEtw/xw K xt5 S EysuNzEys{<�vgE�w . The probability for this to occur is^ V`_ \ Eh^ V . Hence we get

4 \ 8 Z�[�\bV`c ? 5|^ V`_ \ E}^ V < Q S E~BN U G H�V 4 Z�[�\][ V? (10)


The equation determines �� for higher � can be writtendown similarly. It only involves the probabilities �r� with�� for the descendent spins. These recursion equationsare expressed more simply in terms of the generating function�6�@�� . It is easily checked that the self-consistent equation for�6�@�� is�6�@��6� �}��=�� P�� `��a��F�9�� |� �i� � � � � ��F � �6�@�� (11)

This is a polynomial equation in �6� �I� of degree �P�¡� ,whose coefficients are functions of ¢ through � �£¢�� and��¤�£¢�� . It is easily checked that for �¥� � , the ansatz�6�@�9� � �� ¦� � satisfies the equation, as it should. Todetermine �6� �I� for any given external field ¢ , we have to firstsolve the self-consistent equation for � [Eq. 4]. This then de-termines �6� �h��=� using Eq. 9, and then, given � and �6�@�§� ,we solve for �6�@�� by solving the � ��6� � -th degree polynomialequation Eq. 11.

Finally, we express the relative frequency of avalanches ofvarious sizes when the external field is increased from ¢ to¢ �p¨ ¢ in terms of �6� �I� . Let ©rª��£¢�� ¨ ¢ be the probabilitythat avalanche of size « is initiated at ¬ . We also define thegenerating function ©6� �® ¢�� as©6� �¯ ¢��±°�ª²� � ©�ª��³¢��W� ª

(12)

Consider first the calculation of ©�ª��£¢�� for «´� � . Let thenumber of descendents of ¬ that are up at field ¢ be � . Forthe spin at site ¬ to be down at ¢ , but flip up at ¢ �h¨ ¢ , the lo-cal field ¢²µ must satisfy ¶ � ��· � ��¸ � �³¢ �¹¨ ¢��²º � ¢�µ � ¶ � � �· � ��¸ � ¢�º . This occurs with probability �¯� � ¸ �»· � ¸ � ¢�� ¨ ¢ .Each of the � �g� � � down neighbors of ¬ must not flip up,even when « µ flips up. The conditional probability of thisevent is � �� . Multiplying by the probability that � neigh-bors are up, we finally get© � �£¢�� i� �a�¼��}� � � �� ½� � ¸ �e· � ¸ � ¢�� (13)

Arguing similarly, we can write the equation for ©¹ª��³¢�� for«a� ·�¾k¿ etc. These equations simplify considerably when ex-pressed in terms of the generating function ©6�@�¯ ¢�� , and weget©6�@�¯ ¢��À�m� ��`��¦�¼��}� � � �6�@�� ½� � ¸ �e· � ¸ � ¢��

(14)

In numerical simulations, and experiments, it is much easierto measure the avalanche distribution integrated over the fullhysteresis loop. To get the probability that an avalanche ofsize « will be initiated at any given site ¬ in the interval whenthe external field is increased from ¢ � to ¢TÁ , we just have tointegrate ©6�@�¯ ¢�� in this range. For any ¢ , the value of

¨ ©rÂ ¨ �at �}� � is proportional to the mean size of an avalanche, andthus to the average slope of the hysteresis loop at that ¢ .

IV. EXPLICIT CALCULATION FOR THE RECTANGULARDISTRIBUTION

While the general formalism described in the previous sec-tion can be used for any distribution, and any coordinationnumber, to calculate the avalanche distributions explicitly, wehave to choose some specific form for the probability distri-bution function. In this section, we shall consider the specificchoice of a rectangular distribution : The quenched randomfield is uniformly distributed between �XÃ and Ã , so that

�½�³¢TÄ³�� Å=ÆÈÇ for É Æ>Ê9Ë�ÌÍÊmÆ(15)

In this case, the cumulative probabilities Î�Ï¹Ð Ë�Ñbecome

piecewise linear functions ofË

, andË

-dependence of the dis-tribution is easier to work out explicitly. We shall work outthe distributions for the linear chain ( Ò¡Ó Å

), and the 3-coordinated Bethe lattice.

A. The Linear Chain

The simplest illustration is for a linear chain. In this casethe self-consistent equation, for the probability ÔrÕ [Eq. 4] be-comes a linear equation. This is easily solved, and explicitexpressions for Ö�× , and Ö»Ð Ø Ñ

are obtained (see Appendix A).The different regimes showing different qualitative behaviorof the hysteresis loops are shown in Fig. 4

-8

-4

0

4

8

0 2 4 6 8 10

h/J

∆

A

B

CD

h=

h=

h=

− ∆ + 2

∆ − 2

∆

J

J

/J

FIG. 4. Behavior of RFIM in the magnetic field - disorder ( Ù�ÚaÛ )plane for a linear chain. The regions A-D correspond to qualitativelydifferent responses. In region A all spins are down and in region Dall are up. The avalanches of finite size occur in region B and C.


For Ü;ÝßÞáàAâtã (region A), all the spin remain down.For ÜYäåã , all spins are up (region D). For ãæÝlà , we get arectangular loop and the magnetization jumps discontinuouslyfrom â�ç to è¹ç in a single infinite avalanche, and we directlygo from region A to D as the field is increased. For ãéä;à ,we get nontrivial hysteresis loops.

The hysteresis loops for different values of ãëêíì�îðï�ñ�ç=î ïand Þ�î ï are shown in Fig. 5. If ã is sufficiently large ( ã>ä�à ),we find that the mean magnetization is a precisely linear func-tion of the external field for a range of values of the externalfield Ü (region B in Fig. 2). For larger Ü values, the mag-netization shows saturation effects, and is no longer linear (region C).

-2 -1 1 2h/J

-1

-0.5

0.5

1

(a)

M

-2 -1 1 2h/J

-1

-0.5

0.5

1

(b)

M

-3 -2 -1 1 2 3h/J

-1

-0.5

0.5

1

(c)

M

FIG. 5. Hysteresis loops for the linear chain for the rectangulardistribution of quenched fields with different widths (a) òFóDô»õ÷ö{ø ù ,(b) òFóDô»õ~ú�ø ù and (c) ò¦ó�ô6õ÷ûuø ù

The explicit forms of the generating function ü6ý@þ�ÿ aregiven in the Appendix A. We find that in region B, the func-tion ü6ý þIÿ is independent of the applied field � . The distri-bution function ��uý��ÿ has a simple dependence on � of theform � � ý��ÿ�� (16)

where �� is a constant, that depends only on �� , and doesnot depend on � or �� !"$#&%(' )+*�, #.-(/021436587 (17)

In region C, the mean magnetization is a nonlinear func-tion of 9 . But : 0<;=7 is still a rational function of

;. From the

explicit functional form of : 0>;?7 and @ 0<;BA 9 7 are given in theappendix A, we find that @DC 0 9 7 is of the form@ C 0 9 7�EGF HJIK�LNM+HOIPRQTS 1U�V�WBX f Y[Z+\�]_^a` (18)

Here b�cd and bJce have no dependence on \ but are explicitfunctions of f .

Integrating over f from g�h to h we get the integratedavalanche distribution i W ,i WNjlk+mn m

o Wqp f4r(stf (19)

It is easy to see from above that the integrated distribution i Walso has the formi WNjvu b e \xw+y e{zB|~}��(�>�6�� (20)

where the explicit forms of the coefficients �� and �� aregiven in the Appendix A.

B. The Case �J��The analysis for the case �� is very similar to the lin-

ear case. In this case, the self-consistent equation for �~�t��4�[Eq. 4] becomes a quadratic equation. The qualitative behav-ior of solution is very similar to the earlier case. Some detailsare given in Appendix B. We again get regions A-D as before,but the boundaries are shifted a bit, and are shown in Fig. 6.As before, in region B, the average magnetization is a linearfunction of � , and the avalanche distribution is independent of� .


-8

-4

0

4

8

0 2 4 6 8 10

h/J

∆/J

CD

B

A

∆ −

∆ − 3

J

J

J

− ∆ + 3

h=

h=

h=

FIG. 6. Behavior of RFIM in the magnetic field - disorder ( �B �¡ )plane for Bethe lattice of coordination number 3. The qualitative be-havior in different regions A-D is similar to that of a linear chain(Fig. 4).

We find that in regime B, the distribution of avalanche sizesis given by

¢�£q¤�¥4¦�§¨ª© ¤�«6¬[¦�®�¯O°²±´³�µ¶®�¯x·¹¸º³Rµ<» ®2± °¹¼4½6¾8³(¿�À ¼Á�Â�Ã (21)

where Ä is a normalization constant given byÄÆÅ ÇÈ$É²Ê2Ë Ì¹Í�Î É.Ï2Ð ËÑ2Ò4Ó6Ô.Õ (22)

It is easy to see that for large Ö , ×�Ø Ñ�Ù=Õ varies as×�Ø ÚÛÖºÜÞÝßtà4á (23)

where àãâ�ä=å2æ ç¹è�éºê.ë�åìè4éºê.ë (24)

In region B, è4é6ê is always less than æqéqí , and so this functionalways has an exponential decay for large î .

In the region C, we find that the avalanche distribution is ofthe form ï á å�ð=ëñâ�ò�óõô å�ö î ëR÷ø�ùJú¹û[üRýþø�ùxÿ��$üRý �� (25)

where �� is a normalization constant independent ofù, and

�is a cubic polynomial in the external field :

� � �� "!$#&% ��')(��*��)�+�-,.�/� % �0��)�+� �1�32�45�*��+�*6�7, # �8'",.�92��0��-,:��&�0��)�+� 7 �<;��+�=, # (>� % �0�� 7 �?;@�)�+� ,:�<;�� 6BA (26)

As C is not a very simple function of D , explicit expressionsfor the integrated distribution EGF are hard to write down.

V. GENERAL DISTRIBUTIONS

The analysis of the previous section can, in principle, beextended to higher coordination numbers, and other distribu-tions of random fields. However, the self-consistent equationsbecome cubic, or higher order polynomials. In principle, anexplicit solution is possible for H1IKJ , but it is not very in-structive. However, the qualitative behavior of solutions iseasy to determine, and is the same for all H+L1M . We shall takeHON:M in the following for simplicity. Since we only study thegeneral features of the self-consistent equations, we need notpick a specific form for the continuous distributions of ran-dom field distribution PRQ D*S?T . We shall only assume that it hasa single maximum around zero and asymptotically go to zeroat UWV .

For small width ( X ) of the random field distribution i.e. forweak disorder the magnetization shows a jump discontinuityas a function of the external uniform field , which disappearsfor a larger values of X [19]. For fields D just lower thanthe value where the jump discontinuity occurs, the slope of

the hysteresis curves is large, and tends to infinity as the fieldtends to the value at which the jump occurs. This indicatesthat large avalanches are more likely just before the first orderjump in magnetization.

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.5 0 0.5 1 1.5 2 2.5

M

h/J

∆ /∆ / J = 1.5J = 2.5

FIG. 7. Magnetization as a function of increasing field for theBethe lattice with Y[Z:\ and the random field distribution given byEq. 28.


For ]O^:_ , the self-consistent equation for `ba�c?d�e [Eq. 4] iscubic f

` aBgih1j ` aBkihml ` a�hmn ^:o (27)

where

fp j p l and n are functions of the external field d , ex-

pressible in terms of the cumulative probabilities qsr p�t ^uo tov, f

^�q gxw v q k h v qsy w q�zj ^ v q kxw|{ qsy h v q�zl ^ v qsy w v q�z w~}n ^�q�zThis equation will have } or

vreal roots, which will vary

with d . We have shown this variation for the real roots whichlie between 0 and 1 in Fig. 8 for the case where q�c?d@r?e is asimple distribution

q�c?d r ei^ }��|�9�9�3��)�<��0� �+� (28)

We have also solved numerically the self-consistent equationfor �W� for other choices of � �?�� , like the gaussian distribu-tion, and for higher � �<�.��5�� . In each case we find that thequalitative behavior of the solution is very similar. Note thatthe rectangular distribution discussed in the previous sectionis very atypical in that both the coefficients � and � vanish foran entire range of values of � .

In the generic case, we find two qualitatively different be-haviors: For larger values of

�, there is only one real root for

any � . For�

sufficiently small, we find a range of � wherethere are � real solutions. There is a critical value

��of the

width which separates these two behaviors. For the particulardistribution chosen

� �� 9 �/¡ � .In the first case, the real root is a continuous function of � ,

and correspondingly, the magnetization is a continuous func-tion of � . This is the case corresponding to

� � �5� � in Fig. 8.

For smaller�£¢u� �

, for large ¤ � there is only one root ,but in the intermediate region there are three roots. The typi-cal variation is shown for

� � �� in Fig. 8. In the increasingfield the probability �� ?� � initially takes the smallest root.As � increases , at a value �¥�¦�@§��©¨ � , the middle and thelower roots become equal and after that both disappear fromthe real plane . At �ª�u�@§��©¨ � the probability �W� �<� � jumps tothe upper root . Thus for

�«¢¬��there is a discontinuity in�>� �?� � which gives rise to a first order jump in the magnetiza-

tion curve .

-0.5 0 0.5 1 1.5 2

h/J

∆

∆

/J = 1.5

/J = 2.5

*

0.2

0.4

0.6

0.8

1

P

FIG. 8. Variation of "®°¯²±5³ with ± for the Bethe lattice with´"µ�¶ , and the random field distribution given by Eq. 28.

The field ·*¸�¹»º*¼ where the discontinuity of magnetization oc-curs, is determined by the condition that for this value of · , thecubic equation [Eq. 27] has two equal roots. The value of ½O¾at this point, denoted by ½b¾¸�¹»º*¼ , satisfies the equation¿�À&Á ½ ¾BÂ¸�¹»º*¼�Ã1Ä�Å Á ½ ¾¸�¹©º*¼ÆÃ�Ç ÁÉÈ.Ê (29)

whereÀ Á�Ë Å Á and Ç Á are the values of

À Ë Å and Ç at · È ·@¸�¹©º*¼ .We now determine the behavior of the avalanche generating

function ÌOº°Í?·�Î for large Ï and · near ·@¸�¹©º0¼ . The behavior forlarge Ï corresponds to Ð near Ñ . So we write Ð È Ñ�Ò�Ó , withÓ small, and · È ·�¸�¹»º*¼ÔÒ�Õ . Near ·�¸�¹©º�¼ , À Ë Å Ë3Ö×Ö3Ö vary linearlywith Õ and ½ ¾>Ø ½ ¾¸�¹©º�¼ Ò|Ù�Ú ÛÝÜ�Þ+ß$ÛBà (30)

where á is a numerical constant.Since â ß²ãåäçæ9àbäèæ>é�ê>ë�ß?ì�à , if ã differs slightly from

unity â ß²ã@à also differs from æWéåê ë ß$ì�à by a small amount.Substituting ãíä¥æÝéïî and â ß$ãðäñæÆéïî)àÝä¥æÝéïêWë�éïò�ß$Û9óôî)àin the self-consistent equation for â ß$ã�à [Eq. 11], where bothî and ò are small, using Eq. 29, we get to lowest order in î , Ûand ò òWõÔÜ�öÆ÷ ø�ùûúªüsý×þ>ÿ�� (31)

where � and ü are some constants. Thus, to lowest orders in øand þ , ù is given byù¥ÿ�� !#" (32)

Thus $&%('*) has leading square root singularity at ',+-/.10�24354687 . Consequently, 9&:(;�< =?> will also show a square root

singularity ;�@BADC1E 74FGIH�J . This implies that the Taylor expan-sion coefficients KMLNPO*Q vary asKRL�NSO?QUTWVYX[Z\^]`_[acb�dfegYhMikjRlMmon for large p . (33)

At q[r�s , we get tm8uPv*wyx�m#z|{U} p l�~� (34)


Thus at ��*�y�� the avalanche distribution has a power lawtail.

To calculate the integrated distribution � � , we have to inte-grate Eq. 33 over a range of � values. For large � , only �/��I�contributes significantly to the integral, and thus we get� �D�W��D�� , for large �� (35)

Thus the integrated distribution shows a robust �� Y¡Y¢ powerlaw for a range of disorder strength £ .

VI. DISCUSSION

In this paper, we set up exact self-consistent equations forthe avalanche distribution function for the RFIM on a Bethelattice. We were able to solve these equations explicitly forthe rectangular distribution of the quenched field, for the lin-ear chain ¤W¥ ¡ , and the 3-coordinated Bethe lattice. Formore general coordination numbers, and general continuousdistributions of random fields, we argued that for very largedisorder, the avalanche distribution is exponentially damped,but for small disorder, generically, one gets a jump in magne-tization, accompanied by a square-root singularity. For field-strengths just below corresponding to the jump discontinuity,the avalanche distribution function has a power-law tail of theform �Y¦M§4¨4© . The integrated avalanche distribution then variesas �Y¦*ªy¨4© for large � .

Some unexpected features of the solution deserve mention.Firstly, we find that the behavior of the self-consistent equa-tions for ¤«¥�¬ is qualitatively different from that for ¤®¯¬ .The behavior for the linear chain ( ¤°¥ ¡ ) is, of course, ex-pected to be different from higher ¤ . One usually finds samebehavior for all ¤± ¡ . Mathematically, the reason for thisunusual dependence is that the mechanism of two real solu-tions of the polynomial equation merging, and both becomingunphysical (complex) is not available for ¤±¥²¬ . Here theself-consistency equation is a quadratic, and from physical ar-guments, at least one of the roots must be real. That a Bethelattice may show non-generic behavior for low coordinationnumbers has been noted earlier by Ananikyan et al in theirstudy of the Blume-Emery-Griffiths model on a Bethe lattice.These authors observed that the qualitative behavior for ¤´³¶µis different from that for ¤´·¸µ [22].

The second point we want to emphasize is that here we findthat the power-law tail in the distribution function is accom-panied by the first-order jump in magnetization. Usually, onethinks of critical behavior and first-order transitions as mu-tually exclusive, as first-order jump pre-empts a build-up oflong-ranged correlations, and all correlations remain finite-ranged across a first -order transition. This is clearly not thecase here. In fact, the power-law tail in the avalanche distri-bution disappears, when the jump disappears. A similar sit-uation occurs in equilibrium statistical mechanics in the caseof a Heisenberg ferromagnet below the critical temperature.As the external field ¹ is varied across zero, the magnetization

shows a jump discontinuity, but in addition has a cusp singu-larity for small fields [23]. But in this case the power-law tailis seen on both sides of the transition.

Note that for most values of disorder, and the external field,the avalanche distribution is exponentially damped. We getrobust power law tails in the distribution, only if we integratethe distribution over the hysteresis cycle across the magne-tization jump. But, in this case, the control parameter ¹ isswept across a range of values, in particular across a (non-equilibrium) phase transition point! In this sense, while noexplicit fine-tuning is involved in an experimental setup, thisis not a self-organized critical system in the usual sense ofthe word. Recently Pazmandi et al have argued that the hys-teretic response of the Sherrington-Kirkpatrick model to ex-ternal fields at zero temperature shows self-organized critical-ity for all values of the field [24]. However, this seems to bebecause of the presence of infinite-ranged interactions in thatmodel.

The treatment of this paper may be extended to the site-dilution case discussed by Tadic [13]. From the structuralstability of the mechanism which leads to the cusp singular-ity just before the jump-discontinuity in magnetization, it isclear that in our model, introduction of site dilution would notchange the qualitative behavior of solutions.

A general question concerns the behavior of the avalanchesfor more general probability distributions. Clearly, if º �S¹M»S¢has a discrete part, it would give rise to jumps in º » as a func-tion of ¹ , and hence give rise to several jumps in the hysteresisloop. These could preempt the cusp singularity mechanismwhich is responsible for the power-law tails. If the distribu-tion º �S¹ » ¢ is continuous, but multimodal, then it is possible tohave more than one first order jump in the magnetization [25].This is confirmed by explicit calculation in some simple cases.If º �S¹ » ¢ has power-law singularities, these would also lead topower-law singularities in º » , and hence in ¼¾½ �S¹?¢ . Even forpurely continuous distributions, the merging of two roots asthe magnetic field varies need not always occur. For example,it is easy to check that for the rectangular distribution, evenfor ¤±·À¿ , we do not get a power law tail for any value of£ . The precise conditions necessary for the occurrence of thepower-law tail are not yet clear to us.

Finally, we would like to mention some open questions.Our analysis relied heavily on the fact that initial state wasall spins down. Of course, we can start with other initialconditions. It would be interesting to set up self-consistentfield equations for them. In particular, the behavior the returnloop, when the external field is increased from ��Á to somevalue ¹MÂ , and then decreased to a lower value ¹ © seems aninteresting quantity to determine. Another extension wouldbe to make the rate of field-sweep comparable to the single-spin flip rate (still assuming T=0 dynamics). This would meansome large avalanches in different parts of the sample couldbe evolving simultaneously. Then one could study the sweep-rate dependence of the hysteresis loops, and the frequency de-pendence of the Barkhausen noise spectra. This is perhapsof some relevance in real experimental data, and would alsomake contact with other treatments of Barkhausen noise thatfocus on the domain wall motion.


We thank M. Barma and N. Trivedi for their useful com-ments on the manuscript. DD would like to thank the PhysicsDepartment of North Eastern Hill University, for hospitality

during a visit there.

APPENDIX A

For the case of a linear chain, the self-consistent equation, for the probability ÃÅÄ [Eq. 4] is a linear equation, whose solution is ,Ã ÄÇÆSÈ?É[Ê Ë?ÌÍÏÎ/Ð¶ÑÓÒMÔ�Ð«ÒÖÕ8×ÙØ (A1)

For Ú�Û¶ÜkÝ ÐßÞ ,Ò?Õ

is zero, and hence àâá Ñ Ú × is zero, and all the spin remain down (region A in Fig. 4).

For Ú�ã¸ÜkÝ Ð�Þ , andÞ ÛWÝ ,

Ò ÔisÎ

wheneverÒ Õ

is nonzero. Then from Eq. A1, àRá Ñ Ú × becomesÎ. Thus, for

Þ Û�Ý , we get a

rectangular loop and the system changes from all spins down to all spins up state in a single big avalanche.

ForÞ ã�Ý ,

Ò�Ô�Ð�ÒÖÕequals Ý?ä Þ and is independent of Ú , in the range ÜåÝ Ð�Þ Û�ÚæÛ Þ . Thus àÅá Ñ Ú × is a linear function of Ú

in this range, increasing from ç toÎ.

Defining è�é ÎêUë�ì[íïîðòñ�ókôõ�ö (A2)

we obtain the expression for ÷âø as

÷ øåùPú öUûýüþ ÿ�� for �� for �� for �� !� (A3)

Using Eq. (9), the expression for "$# is,

" #&%(')��+* �-, ��'.*0/��1* �-,)243 '65 , (A4)

The generating function " '87 , obtained from the self-consistent equation [Eq.11] is,

" '97 , % " #;:<7 2 3 '.*�/��+* �-,=�>@?BADCFE;>1C G�H (A5)

and the generating function I A8?KJ L0H given by Eq. 14 becomes,

I A8?MJ L0HONP?RQ�S TUA9?0HWVYX�CBA[Z]\+>^L�H`_�Z�a4bcS TUA8?0HWV CBAd>eL0Hf_ga4bhXdCBA)>eZc\1>iL0Hkj(A6)

Now if lnm Zc\, and

> l _gZ]\poPL1o l >iZ]\( region B in Fig. 4 ),

ADC X >1CFEkHqN(ADCFEr>1C G�HON�\�s l+tCMA[Z]\+>@uv\1>iL0HqN =wyx for z|{~} �k�y� wand �4��g�$��{(�� x

Thus

�U�9�0�q{ �� x ��d� �!�� i�4�(A7)

and

� �9�M� ��r� ��y�n� �4 ;¡�¢U£8¤F¥W¦8§e¨ ¤©yª «)¬�g®�¯ ªR°)±²d³�´@µ`¶�·�¸�¹dº (A8)


Expanding »U¼9½M¾ ¿�À in powers of ½ , we get the probability distribution of avalanches in region B given by Eq. 16 of sec. IV A.

In the region C, Á0Â saturates to value Ã , Á ¼dÄeÅ]ÆÇÄi¿0À becomes zero and ¼ ÁÂ Ä ÁÈ À becomes ¼ Ã ÄgÆ�É!ÊËÄiÌ�À . Thus we get ,

ÍÏÎ&Ð ¼ Ã ÄgÆ�É�Ê(Ä@ÌhÀ ÂÑ)Ò�ÓgÔ�Õ!ÖR× (A9)

In terms of Ø4Ù and Ú4Û we get

Ú Ñ9Ü0×OÝ Ú Û;Þ Ü Ø Ùcß Ò�Ó^àáÑdÔ�Õ�Ö�×KÓ@â)ãä�å�æ�ç�è�é�ê�ë (A10)

and

ì æ9ëMí î�ê;ï ëð!ñ|ò�ó ô4õ�ög÷Uø9ù0ú�ûýü�þiô4õhüyÿ (A11)

Expanding � ø9ù�� ú in powers of ù we get , in region C

�� ø��ú�� ! (A12)

and "$# ��%&�('*) +-,.0/1�2+-,�43 5$6798-:<; for =?>A@CB (A13)

Here D-E and FGE have no dependence on = but are explicit functions of HDJIK�L MNPORQ ST�U&VXWZY T\[�]�^`_�abYdc�^ efhg&iXjZk�l f�m�n�o�p�q�k�m�nrf�sutwv gxzy|{~} xZ�� &��P�Z�� d�u�w� ��z�|�~� �Z��4��J��J� �?�� ¡ h¢&£P¤¦¥d§ \¨�©�ª`«�¬b¥d¨�© d®�¯ �C¢°²±|³µ´ °¦¶�· ¸¹&º�»&¼X½Z¾�¿µÀ�ÁbÂ º�Ã�ÄwÅ »ÆÈÇ�É~Ê ÆZË�Ì4Í

Integrating over É from Ç-Î to Î we get the integrated avalanche distribution ÏÑÐ ,Ï Ð�ÒÔÓ`ÕÖ ÕØ× ÐÚÙ É Ë�Û É (A14)

where

ÏÝÜ Ò Þßdà�á�â&ãXäZådæ|ç à�áéèÝê âë9ì`íïî0ðòñ$óô9õJöu÷ùøúûýü$þÿ�� (A15)

and, for "!$# , %'&)(+*-,�. �/10 .3254�67�8:9 (A16)

with ;=<=> ?@BADCFEHGJILKNM @BA:OQPBP3ASR ET�UQV$WYXBZ\[^]_�`=a)bdcfehg^ijlknmpoqnr�s tuwvyxzu|{Q}H~��'�� vn{�u3�S� }��Q��^��p�


APPENDIX B

For �� , the self-consistent equation. for �^��H� [Eq. 4] is a quadratic equation,� � �H¡5¢£�¥¤¦�§¢�� ¥¤�¢£�D¨f�ª©«� � ��H� ¡�¬ � �®�¥¤�¢h�D¨w�§¢�¯°©«� � ��H� ¬ �D¨=��±y² (B1)

For the rectangular distribution, the coefficient of �¥� ¡ is zero for a range of � -values, and �^��³�´� is still a piecewise linear

function of � � � �³�´�µ�·¶¸ ¹ ± for º�»Q±¼½�¾´¿¦ÀÂÁwÃÅÄÇÆ for È'É�Ê�ÉÌË5Í�ÎDÏFÐHÑ�Ò'ÓË for Ê�ÔÕË5ÍdÎyÏÖÐDÑJÒ'Ó (B2)

where Ê is defined as , Ê)× ËØÇÙzÚ�ÛÝÜÞàß�áãâäæå (B3)

The self-consistent equation for çSèêé å [Eq. 11] becomes,éëè®ì¥í)îhìHï åñð ç\èêé åªòêó�ôÕð õ éHö:÷è®ì ó îhìñí å î�ø ò ç'èêé åùô é´ö:÷ ó è®ì´ú5î£ì ó åûô ç:ï=üÕý (B4)

where ç"ï is obtained [Eq. 9] asç ï ü�èzø5îhì í å î õ è ì ó î£ì í å ö ÷ ôÕð è ì ó îhì í å îþè®ìHú)î£ì ó åªò ö ÷ ó (B5)

and the expression (14 ) for ÿ\èêé�� å becomes ,ÿ\è-é�� å ü$é�� ð ç\è-é åªò ú ìûè��hî� å ô � ð ç\è-é åªò ó ö:÷Åìëè�hî� åô � ð ç\èêé åªò ö:÷ ó ìûèzî��hî� åùô ö:÷ ú ìëèÖî�� £î�� å�� (B6)

Now in the region B , è®ì´ú)îhì ó å ü�è®ì ó îhì í å ü�è®ì í îhì ï å ü�� ä��ìûè�� hî õ�� hî� å ü ø�� for �� to !and "�#%$'&)( �+*-,'.0/ �

Solving Eq. B4 and choosing the root which is well behaved for 1 near � , we get

&32 1546� *-,87 9-:<;0=?>0@BADCE�=�F�GIH'J�KLCM�NO0PBQDRS T<U�V (B7)

and the expression for the integrated distribution ( B6 ) becomesW N�S�X Y0R6Z S[�\^] _�`%acb3dfe5ghfi (B8)

Expanding j d�e5g in power series of e , we obtain the Eq. 21 of sec. IV B.

In the region C, k i saturates to the value l , k d?m�n�o�mqp5g becomes zero and d k i m k0r g is no longer independent of p . Substituting

the appropriate expressions, we find that

bsdfetgvu l m8w x-y{z5|}0~B�D��|�x�y��|}0~��q��y<�� |��I�'��L��0�B�D�� (B9)

and � �f�� 5�v� ��¡ ^¢0£ ¤�¥I¦'§s¨f©tª�«¬-®<¤�¥�¬°¯ (B10)

We note that the term inside the radical sign in §s¨�©5ª , and also in ± ¨f©�² ³5ª , is a simple linear function of © . It is thus straight-

forward to expand it in powers of © using binomial expansion. This gives us the Eq. 25 of sec. IV B.


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[18] A. Georges, G. Kotliar, W. Krauth and M. Rozenberg, Rev.Mod. Phys. 68, 13 (1996).

[19] D. Dhar, P. Shukla, and J. P. Sethna, J Phys A: Math. Gen.30,5259-5267 (1997).

[20] This step is not really necessary if we are only interested in de-termining the magnetization at the site ´ . Skipping this stepleads to considerable simplification of the relaxation process:first the spins of generation µ are examined, then those of¶ µ¸·º¹�» etc. till we finally examine the spin at ´ . No spinis checked more than once. The resulting configuration is notfully relaxed, but it is easy to prove that further relaxation willnot change the state of the spin at ´ . The argument can be ex-tended to show that the probability that an avalanches startingat ´ is of size ¼ also is the same in this partially relaxed state asin the fully relaxed state.

[21] D. Stauffer and A. Aharony, Introduction to Percolation The-ory, Taylor and Francis, London, pp 26-34 (1992).

[22] N. S. Ananikyan, N. Sh. Izmailyan and R. R. Shcherbakov,JETP Lett. 59, 71 (1994); Pis’ma Zh. Eksp. Fiz. 59, 71 (1994).

[23] Below ½�¾ , the magnetization goes as

¿ÁÀ ¼ÃÂÄ µ ¶fÅ »�Æ ¿qÇÉÈËÊÍÌ Å Ì ÎÐÏÒÑ�ÓÔÖÕ×Ó×ØÚÙ asÅÜÛ�Ý Ù

See G. Parisi, Statistical Field Theory, Addison-Wesley, p 195(1988)

[24] F. Pázmándi, G. Zaránd, and G. T. Zemányi, preprint cond-mat/9902156 (1999).

[25] This would happen if Þ�ß ¶fÅ » as a function ofÅ

shows a ‘doubles’ curve. Then there must be at least à values of

Åfor which the

slope of the curve is infinite. This is possible only if the equa-tion determining Þ-ßÏâáäã ¾ (variant of Eq. 29) is at least a quartic,hence only if å�æ<ç .

VOLUME 88, NUMBER 19 P H Y S I C A L R E V I E W L E T T E R S 13 MAY 2002

Hysteresis in the Random-Field Ising Model and Bootstrap Percolation

Sanjib Sabhapandit,1 Deepak Dhar,1 and Prabodh Shukla2

1Department of Theoretical Physics, Tata Institute of Fundamental Research, Mumbai-400005, India2Physics Department, North Eastern Hill University, Shillong-793 022, India

(Received 23 October 2001; revised manuscript received 11 March 2002; published 25 April 2002)

We study hysteresis in the random-field Ising model with an asymmetric distribution of quenched fields,in the limit of low disorder in two and three dimensions. We relate the spin flip process to bootstrap per-colation, and show that the characteristic length for self-averaging L� increases as exp�exp�J�D�� in 2D,and as exp�exp�exp�J�D�� in 3D, for disorder strength D much less than the exchange coupling J . Forsystem size 1 ø L , L�, the coercive field hcoer varies as 2J 2 D ln lnL for the square lattice, and as2J 2 D ln ln lnL on the cubic lattice. Its limiting value is 0 for L ! ` for both square and cubic lattices.For lattices with coordination number 3, the limiting magnetization shows no jump, and hcoer tends to J .

DOI: 10.1103/PhysRevLett.88.197202 PACS numbers: 75.10.Nr, 64.60.Ak

In recent years, there has been a lot of interest in thestudy of hysteresis in magnetic systems, both theoretically[1] and in experiments [2]. Hysteresis in the random-fieldIsing model (RFIM) model was first discussed by Sethnaet al. [3], who proposed it as a model of return point mem-ory, and of Barkhausen noise [4]. Sethna et al. solvedthe model in the mean-field limit, and showed that, if thestrength D of the quenched random field is large, the aver-age magnetization per site is a continuous function of theexternal field, but, for small D, it shows a discontinuousjump as the external field is increased. Interestingly, thenonequilibrium hysteresis response in the RFIM can bedetermined exactly on a Bethe lattice [5,6], though thecorresponding equilibrium problem has not been solvedthus far, even in zero field. These calculations have beenextended to determining the distribution of sizes of theBarkhausen jumps [7], and the calculation of minor hys-teresis loops [8,9].

In this paper, we study the low disorder limit of the hys-teresis loop in the RFIM on periodic lattices in two andthree dimensions. We find that the behavior of hysteresisloops depends nontrivially on the coordination number z.For z � 3, for continuous unbounded distributions of ran-dom fields, the hysteresis loops show no jump discontinu-ity of magnetization even in the limit of small disorder, butfor higher z they do. This is exactly as found in the exactsolution on the Bethe lattice [6].

The analytical treatment of self-consistent equations onthe Bethe lattice is immediately generalized to the asym-metrical case. However, we find that behavior of hysteresisloops in Euclidean lattices can be quite different from thaton the Bethe lattice, for asymmetrical distributions. On hy-percubical lattices in d dimensions, there is an instabilityrelated to bootstrap percolation, that is absent on the Bethelattice. This reduces the value of the coercive field hcoer

away from the Bethe lattice value O �J� to zero, where J

is the exchange coupling. We note that the limit D ! 0 issomewhat subtle, as the system size L� required for self-averaging diverges very fast for small D, and the finite-size

corrections to the thermodynamic limit tend to zero veryslowly.

In the RFIM, the Ising spins �si� with nearest neigh-bor ferromagnetic interaction J are coupled to the on-sitequenched random magnetic field hi and the external fieldh. The Hamiltonian of the system is given by

H � 2JX�i,j

sisj 2X

i

hisi 2 hX

i

si . (1)

We assume that �hi� are quenched independent identicallydistributed random variables with the probability that thevalue of the random field at site i lies between hi andhi 1 dhi being f�hi�dhi.

The system evolves under the zero-temperature Glaubersingle-spin-flip dynamics [10]: A spin flip is allowed onlyif the process lowers energy. We assume that the rate ofspin flips is much larger than the rate at which h is changed,so that all flippable spins may be said to relax instantly, andany spin si always remains parallel to the net local field �i

at the site:

si � sgn��i� � sgn

µJ

zXj�1

sj 1 hi 1 h

∂. (2)

Under this dynamics, for ferromagnetic coupling (J .0), if we start with any stable configuration, and then in-crease the external field and allow the system to relax, thefinal stable configuration reached is independent of the or-der in which the unstable spins are flipped. Also, in therelaxation process, no spin flips more than once.

For a given distribution f�hi�, we define pm�h� with0 # m # z as the conditional probability that the localfield at any site i will be large enough so that it will flipup, if m of its neighbors are up, when the uniform externalfield is h. Clearly,

pm�h� �

Z `

�z22m�J2h

f�hi� dhi . (3)

For any given value of h, the magnetization depends onthe distribution f�hi� only through pm�h�.

197202-1 0031-9007�02�88(19)�197202(4)$20.00 © 2002 The American Physical Society 197202-1


Historically, RFIM was first studied in the context ofpossible destruction of long range order by arbitrarily weakquenched disorder in equilibrium systems. Accordingly,the distribution of random field was assumed to be sym-metrical. However, in the hysteresis problem, the symme-try between up and down spin states is already broken bythe specially prepared initial state (all down in our case),and the symmetry of the distribution plays no special role.In the following, we shall assume that the distribution hasan asymmetrical shape, given by

f�hi� �

1

Dexp�2hi�D�u�hi� , (4)

where u is the step function. The mean value of hi can bemade zero by a shift in the value of the external uniformfield. Our treatment is easily extended to other continuousunimodal distributions. The exact form of f�x� is notimportant, and other forms such as exp�2x 2 e2x� whichfall sharply for negative x have the same behavior.

Consider first the case of the two-dimensional hexago-nal lattice with z � 3. For periodic boundary conditions(PBC), if D � 0, starting with a configuration with allspins down, clearly one has hcoer � 3J. For D fi 0, thesite with the largest local field flips first, and then if h . J,p1�h� � 1, this causes neighbors of the flipped spin to flip,and their neighbors, and so on. Thus, so long as thereis at least one flipped spin, all other spins also flip, andthe magnetization is 1. The largest local field in a systemof L2 spins is of order 2D lnL. Once this spin turns up,other spins will flip also up, causing a jump in magneti-zation from a value 2 1 to a value 11 in each sample.Hence, the coercive field (the value of h where magnetiza-tion changes sign) to lowest order in D, is given by

hcoer � 3J 2 2D lnL, for 1 ø lnL ø J�D . (5)

Sample to sample fluctuations in the position of the jumpare of order D. On averaging over disorder, the magneti-zation will become a smooth function of h, with the widthof the transition region being of order D.

For a fixed D ø J, if L is increased to a value nearexp�J�D� � L

�

hex, hcoer decreases to a value near J. Forh J, p1�h� is no longer nearly 1, but p0�h� � 0,

p2�h� � p3�h� � 1. The value of magnetization depends

only on p1�h�, which is a function of eh � �h 2 J��D.

As eh increased from 2`, p1�h� increases continuouslyfrom 0 to 1.

In Fig. 1, curve A shows the result of a simulation onthe hexagonal lattice with L � 4096, and PBC. To avoidthe problem of probability of nucleation being very smallfor h near J, we made the local field at a small fractionof randomly chosen sites very large, so that these spinsare up at any h. The number of such spins is of order L,so that their effect on the average magnetization is neg-ligible. Introduction of these “nucleation centers” makesL� O �

pL � (the average separation between centers),

FIG. 1. Magnetization in the increasing field. The curves forthe two values of D coincide. Curves A and B are for 2D and3D lattice with z � 3.

and hcoer drops to a value near J, so that we can study thelarge L limit with available computers. For L . L

�

hex, thebehavior of hysteresis loops becomes independent of L.

We see that magnetization no longer undergoes a singlelarge jump, but many small jumps. In the figure, we alsoshow the plot of magnetization when the random field ateach site is decreased by a factor of 10. This changes thevalue D from 0.1J to 0.01J. However, plotted as a function

of eh, the magnetization for these two different values (forsmall D) fall on top of each other for the same realization

of disorder (except for the overall scale D). Thus, wecan decrease D further to arbitrarily small values, and thelimit of D ! 0 is straightforward for each realization ofdisorder. Then, averaging over disorder, for a fixed D, wesee that hcoer tends to the value J as D tends to 0. Also,we see that there is no macroscopic jump discontinuity forany nonzero D.

We also show, in Fig. 1 (curve B), the results of simula-tion of a three-dimensional lattice with z � 3 of size 2563

with PBC. The behavior is qualitatively the same as thatin two dimensions. The value of hcoer � J in the limitD ! 0 is the same for symmetrical distribution, and alsois the same as predicted by the Bethe approximation.

On the square lattice also, the value of hcoer is deter-mined by the need to create a nucleation event. Arguingas before, we see that hcoer to lowest order in D is givenby hcoer 4J 2 2D lnL, for 1 ø lnL ø J�D. Addinga small number of nucleation sites suppresses this slowtransient, and lowers hcoer from 4J to a value near 2J.However, in this case, even after adding the nucleationcenters, the system shows a large single jump in magneti-zation, indicating the existence of another instability. Weobserved in the simulation that at low D, as h is increased,the domains of up spins grow in rectangular clusters (seeFig. 2) and, at a critical value of hcoer, one of them sud-denly fills the entire lattice. This value hcoer fluctuates a bitfrom sample to sample. In Fig. 3 we have plotted the dis-

tribution of the scaled variable ehc � �hcoer 2 2J��D fordifferent system sizes L, for D � 0.001J. The number of

197202-2 197202-2


FIG. 2. A snapshot of the up spins just before the jump (h �

1.998 243J). The lattice size is 200 3 200 and D � 0.001J .Initial configuration is prepared with 0.05% up spins.

different realizations varies from 104 (for the largest L) to105 (for the smallest L). Note that the distribution shiftsto the left with the increasing system size, and becomesnarrower.

This instability can be understood as follows: On asquare lattice, for the asymmetric distribution [Eq. (4)] forh . 0, pm � 1 for m $ 2, and any spins with more thanone up neighbor flips up. Therefore, stable clusters of upspins are rectangular in shape. The growth of domains ofup spins is the same as in the bootstrap percolation processBPm with m � 2 [11–13]. In the process BPm, the initialconfiguration is prepared by occupying lattice sites inde-pendently with a probability p and the resulting configu-ration is evolved by the rules: The occupied sites remainoccupied forever, while an unoccupied site having at leastm occupied neighbors, becomes occupied. For m � 2, ona square lattice, in the final configuration, the sites whichare occupied form disjoint rectangles, such as the clusterof up spins in Fig. 2. It has been proven that, in the ther-modynamic limit of large L, for any initial concentration

FIG. 3. Distribution of the scaled coercive field on a squarelattice for different lattice size L2.

p . 0, in the final configuration all sites are occupied withprobability 1 [12].

Now consider a rectangular cluster of up spins, of lengthl and width m. Let P�l, m� be the probability that, ifthis rectangle is put in a randomly prepared background ofdensity p1�h�, this rectangle will grow by the BP2 processto fill the entire space. The probability that the randomfields at any sites neighboring this rectangle will be largeenough to cause it to flip up is p1�h�. The probability thatthere is at least one such site along each of two adjacentsides of length l and m of the rectangle is �1 2 ql� �1 2qm�, where q � 1 2 p1�h�. Once these spins flip up, thisinduces all the other spins along the boundary side to flipup and the size of the rectangle grows to �l 1 1� 3 �m 11�. Therefore,

P�l, m� $ �1 2 ql� �1 2 qm�P�l 1 1, m 1 1� . (6)

Thus, the probability of occurrence of a nucleation whichfinally grows to fill the entire lattice is

Pnuc $ p0�h�Yj�1

�1 2 qj�2. (7)

The right-hand side can be shown to vary as p0�h� 3

exp�2p2

3p1�h� � for small p1�h�. The condition to determinehcoer is that, for this value of h, Pnuc becomes of order1�L2, so that we get

p0�hcoer� exp

µ2

p2

3p1�hcoer�

∂

1

L2. (8)

This equation can be solved for hcoer for any given L. Forthe distribution given by Eq. (4), this becomes

exp

µhcoer 2 4J

D

∂exp

∑2

2p2

3exp

µ2hcoer 1 2J

D

∂∏

1

L2.

(9)

It is easy to see from this equation that, for 1 ø lnL øJ�D, the leading L dependence of hcoer, to lowest order inD is given by

hcoer 4J 2 2D lnL , (10)

and for J�D ø lnL ø exp�2J�D�,

FIG. 4. p1�hcoer� vs 1� lnL for square lattice.

197202-3 197202-3


hcoer 2J 2 D ln

∑3

p2�lnL 2 J�D�

∏. (11)

This agrees with our observation that the scaled critical

field ehc shifts to the left with increasing system size.To test the validity of Eq. (8) in simulations, we put

p0�h� � 0.005 independent of h. Equation (8) then sim-plifies to

p1�hcoer� p2

6 lnL. (12)

In Fig. 4, we have plotted p1 for the mean hcoer from Fig. 3versus 1� lnL. The graph is approximately a straight line,which agrees with Eq. (12). The slope of the line is lessthan in Eq. (12), which gives only an upper bound to hcoer.

If h . 0, we will have p2 � 1, and bootstrapping en-sures that, as long as p0 . 0, we will have all spins upin the limit of large L. This implies that hcoer � 0 in thislimit.

If there are sites with large negative quenched fields,the bootstrap growth stops at such sites. Hence, the boot-strap instability cannot be seen for symmetric distributions.Even if the quenched fields are only positive, the instabil-ity does not occur on lattices with z � 3. On such lattices,if the unoccupied sites percolate, there are infinitely ex-tended lines of unoccupied sites in the lattice. These can-not become occupied by bootstrapping under BP2. Thus,the critical threshold for BP2 on such lattices is not 0.

The above analysis is easily extended to higher dimen-sions. In d � 3, if h . 0, then pm�h� � 1 for m $ 3;therefore the spin flip process is similar to the spanningprocess of three-dimensional BP3 [14]. But in this case, itis known that for any initial nonzero density, in the thermo-dynamical limit, the final configuration has all sites occu-pied with probability 1. The clusters of up spins grow ascuboids, and at each surface of the cluster, the nucleationprocess is similar to that in two dimensions. Let e be theprobability that a nucleation occurs at a given point of asurface of the clusters of up spins which sweeps the entiretwo-dimensional plane at h.

e p1�h� exp

µ2

p2

3p2�h�

∂. (13)

The probability that there exists at least one nucleationwhich sweeps the entire plane of size l 3 l is 1 2 �1 2

e�l2

. Therefore, the probability Pnuc that a nucleationsweeps the entire three-dimensional lattice at h satisfies

Pnuc $ p0�h�Yl�1

�1 2 �1 2 e�l2

�3. (14)

For small e, the infinite product can be shown to vary as

exp�2A�p

e �, with A �

3

2

pp z �3�2�.

hcoer is determined by the condition that Pnuc must beof the order of 1�L3:

p0�hcoer� exp

∑2

App1�hcoer�

exp

µp2

6p2�hcoer�

∂∏ 1�L3.

(15)

The leading L dependence of hcoer is different in dif-ferent ranges of hcoer, depending on whether the strongestdependence of the left-hand side comes from variation ofp0�h�, p1�h�, or p2�h�. We find that hcoer 6J 2 3D lnL,for 4J , hcoer , 6J. It is 4J 2 2D ln�lnL 2 �2J��3D��, for 2J , hcoer , 4J; and 2J 2 D ln ln�lnL 2�2J��3D��, for D ø hcoer , 2J. It is straightforward todetermine the corresponding ranges of L for the validityof these equations.

In the limit L ¿ L�

cub � exp�exp�exp�2J�D��, theloop becomes independent of L, with hcoer ! 0. We havealso verified the existence of jump in numerical simu-lation for z � 4 (diamond lattice) in three dimensions.

In brief, we have shown that the hysteresis loops on lat-tices with coordination number three are qualitatively dif-ferent from those with z . 3. For the square and cubiclattices, hcoer decreases to 0 very slowly for large L. Ingeneral, it is true for lattices where the corresponding boot-strap percolation problem has an instability.

We thank M. Barma and N. Trivedi for critically readingthe manuscript.

[1] B. K. Chakrabarti and M. Acharyya, Rev. Mod. Phys.71, 847 (1999); I. F. Lyuksyutov, T. Nattermann, andV. Pokrovsky, Phys. Rev. B 59, 4260 (1999).

[2] J.-S. Suen and J. L. Erskine, Phys. Rev. Lett. 78, 3567(1997), and references therein.

[3] J. P. Sethna, K. A. Dahmen, S. Kartha, J. A. Krumhans,B. W. Roberts, and J. D. Shore, Phys. Rev. Lett. 70, 3347(1993).

[4] E. Puppin, Phys. Rev. Lett. 84, 5415 (2000), and referencestherein; see also L. B. Sipahi, J. Appl. Phys. 75, 6978(1994).

[5] P. Shukla, Physica (Amsterdam) 233A, 235 (1996); Phys.Rev. E 62, 4725 (2000).

[6] D. Dhar, P. Shukla, and J. P. Sethna, J. Phys. A 30, 5259(1997).

[7] S. Sabhapandit, P. Shukla, and D. Dhar, J. Stat. Phys. 98,103 (2000).

[8] P. Shukla, Phys. Rev. E 63, 027102 (2001).[9] L. Dante, G. Durin, A. Magni, and S. Zapperi, Phys. Rev. B

65, 144441 (2002).[10] See, for example, K. Kawasaki, Phase Transition and Criti-

cal Phenomena, edited by C. Domb and M. S. Green (Aca-demic Press, London, 1972), Vol. 2.

[11] J. Chalupa, P. L. Leath, and G. R. Reich, J. Phys. C 12, L31(1981); P. M. Kogut and P. L. Leath, J. Phys. C 14, 3187(1981).

[12] M. Aizenman and J. L. Lebowitz, J. Phys. A 21, 3801(1988).

[13] For a review, see J. Adler, Physica (Amsterdam) 171A, 453(1991).

[14] R. Cerf and E. N. M. Cirillo, Ann. Probab. 27, 1833 (1999).

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