ferenc igloi and heiko rieger- random transverse ising spin chain and random walks

8/3/2019 Ferenc Igloi and Heiko Rieger- Random transverse Ising spin chain and random walks

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Random transverse Ising spin chain and random walks

Ferenc IgloiResearch Institute for Solid State Physics, H-1525 Budapest, P.O. Box 49, Hungary

and Institute for Theoretical Physics, Szeged University, H-6720 Szeged, Hungary

Heiko Rieger HLRZ, Forschungszentrum Julich, 52425 Julich, Germany

Received 24 September 1997; revised manuscript received 26 November 1997

We study the critical and off-critical Griffiths-McCoy regions of the random transverse-field Ising spin

chain by analytical and numerical methods and by phenomenological scaling considerations. Here we extend

previous investigations to surface quantities and to the ferromagnetic phase. The surface magnetization of the

model is shown to be related to the surviving probability of an adsorbing walk and several critical exponents

are exactly calculated. Analyzing the structure of low-energy excitations we present a phenomenological

theory which explains both the scaling behavior at the critical point and the nature of Griffiths-McCoy singu-

larities in the off-critical regions. In the numerical part of the work we used the free-fermion representation of

the model and calculated the critical magnetization profiles, which are found to follow very accurately the

conformal predictions for different boundary conditions. In the off-critical regions we demonstrated that the

Griffiths-McCoy singularities are characterized by a single, varying exponent, the value of which is related

through duality in the paramagnetic and ferromagnetic phases. S0163-18299803418-3

I. INTRODUCTION

Magnetic systems with quenched disorder at very low or

even vanishing temperature have attracted a lot of interest

recently. In particular the quantum phase transition occurring

in quantum Ising spin glasses in a transverse field 1 and ran-

dom transverse Ising ferromagnets2 4 turned out to have a

number of surprising features. For instance, the presence of

quenched disorder has more pronounced effects on quantum

phase transitions than on those phase transitions, which are

driven by thermal fluctuations. For example, in the Griffithsphase, which is at the disordered side of the critical point, the

susceptibility has an essential singularity in classical sys-

tems, whereas in a random quantum system the correspond-

ing singularity Griffiths-McCoy singularity5,6 is much

stronger, it is in a power-law form.

Many interesting features of random quantum systems

can already be seen in one-dimensional models. After the

pioneering work by McCoy and Wu7 and later studies by

Shankar and Murphy,8 Fisher2 has recently performed an ex-haustive study of the critical behavior of the randomtransverse-field Ising spin chain. He used a renormalization-group RG approach, which he claims becomes exact at the

critical point. The same type of method has later been usedfor other 1 d random quantum problems9 and recently someexact results are obtained through a mapping of the randomXY model onto a Dirac equation in the continuum limit.10

In the present paper we consider the prototype of randomquantum systems the random transverse-field Ising chain de-fined by the Hamiltonian:

Hl

Jllxl1

x

lh ll

z . 1.1

Here the lx , l

z are Pauli matrices at site l and the Jl ex-

change couplings and the h l transverse fields are randomvariables with distributions (J) and (h), respectively.The Hamiltonian in Eq. 1.1 is closely related to the transfermatrix of a classical two-dimensional layered Ising model,which was introduced and studied by McCoy and Wu. 7

In the following we briefly summarize the existing exact,conjectured, and numerical results on the random transverse-field Ising chain in Eq. 1.1. The quantum control parameterof the model is given by

ln h av ln Jav

var ln h var ln J. 1.2

For 0 the system is in the ordered phase with a nonvan-ishing average magnetization, whereas the region 0 cor-responds to the disordered phase. There is a phase transitionin the system at 0 with rather special properties, whichdiffers in several respects from the usual second-order phasetransitions of pure systems. One of the most striking phe-nomena is that some physical quantities are not self-averaging, which is due to very broad, logarithmic probabil-ity distributions. As a consequence the typical value whichis the value on an event with probability one and the aver-age value of such quantities is different. Thus the criticalbehavior of the system is primarily determined by rareevents, dominating the averaged values of various observ-ables.

The average surface magnetization close to the criticalpoint vanishes as a power law m s

s, where

s1, 1.3

PHYSICAL REVIEW B 1 MAY 1998-IIVOLUME 57, NUMBER 18

570163-1829/98/5718/1140417 /$15.00 11 404 1998 The American Physical Society


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is an exact result by McCoy.6 The average bulk magnetiza-tion is characterized by another exponent , which is con-

jectured by Fisher in his RG treatment:2

2, 1.4

where (15)/2 is the golden mean. The average spin-spin correlation function G( l)i

xilx av involves the

average correlation length , which diverges at the criticalpoint as av. The RG result by Fisher2 is

av2. 1.5

On the other hand, the typical correlations have a faster de-cay, since typ

typ with typ1.8

In a quantum system statistics and dynamics are inher-ently connected. Close to the critical point the relaxationtime tr is related to the correlation length as tr

z, where zis the dynamical exponent. The random transverse-field Isingspin chain is very strongly anisotropic at the critical point,since according to the RG picture2 and to numerical results11

ln tr1/2, 1.6

which corresponds to z. On the other hand, the relax-ation time is related to the inverse of the energy-level spac-ing at the bottom of the spectrum tr(E)

1. Then, as aconsequence of Eq. 1.6 some energylike quantities specificheat, bulk, and surface susceptibilities, etc. have an essentialsingularity at the critical point, while the correlation functionof the critical energy density has a stretched-exponential de-cay, in contrast to the usual power-law behavior.

Leaving the critical point towards the disordered phasethe rare events with strong correlations still play an impor-tant role, until we reach the region G , where all trans-verse fields are bigger than the interactions. In the region 0

G , which is called the Griffiths-McCoy phase themagnetization is a singular function of the uniform longitu-dinal field Hx as m singHx

1/z, where the dynamical expo-nent z varies with . At the two borders of the Griffiths-McCoy phase it behaves as z1/21O()2 as 0 and

z1 as G , respectively.

Some of the above-mentioned results have been numeri-cally checked11 and in addition various probability distribu-tions and scaling functions have been numericallydetermined.12 Our present study, which contains analyticaland numerical investigations extends the previous work inseveral respects. In the case of a limiting random distributionwe obtain exact results on the surface magnetization expo-

nent xs

s / and in particular the thermal exponent through a simple directed walk consideration. The mappingof the problem of calculating various universal quantities forthe random transverse Ising chain onto the calculation ofstatistical properties of appropriately defined random walksis one of the main achievements of the present paper. Withits help we are also able to explain quantitatively many of theexotic features of the Griffiths-McCoy region on both sidesof the transition.

Moreover, we improved the accuracy of the numericalestimates on the bulk magnetization exponent in Eq. 1.4.Furthermore we present results on the magnetization profilesin confined critical systems as well as about the probabilitydistribution of several quantities.

Throughout the paper we use two types of random distri-butions. 1 The binary distribution, in which the couplingscan take two values and 1/ with the probability p andq1p, respectively, while the transverse field is constant:

JpJ qJ1,

h hh 0 . 1.7

The critical point is given by (pq)ln ln h0 and Gp/q for 1. 2 The uniform distribution in which thecouplings and the fields have uniform distributions:

J 1, for 0J10, otherwise

1.8

h h

0

1 , for 0hh0

0, otherwise,

and the critical point is at h 01 and G.The structure of the paper is the following. In Sec. II we

present the free-fermionic description of our model togetherwith the way of calculation of several physical quantities inthis representation. In Sec. III the surface magnetization andseveral critical exponents are calculated exactly using a cor-respondence with an adsorbing walk problem. Phenomeno-logical considerations and numerical estimates about the dis-tribution of low-energy excitations are compared in Sec. IV.Numerical results for different critical and off-critical param-

eters are presented in Secs. V and VI, respectively. Our re-sults are discussed in the final section.

II. FREE-FERMION REPRESENTATION

We consider the random transverse-field Ising spin chainin Eq. 1.1 on a finite chain of length L with free or fixedboundary conditions, i.e., with JL0. The Hamiltonian inEq. 1.1 is mapped through a Jordan-Wigner transformationand a following canonical transformation13 into a free-fermion model:

Hq1

L

q qq 12 , 2.1

where q and q are fermion creation and annihilation op-

erators, respectively. The fermion energies q are obtainedvia the solution of an eigenvalue problem, which necessitatesthe diagonalization of a 2L2L tridiagonal matrix T withnonvanishing matrix elements T2i1,2iT2i ,2i1h i , i1,2, . . . ,L and T2i,2i1T2i1,2iJi , i1,2, . . . ,L1. We denote the components of the eigenvectors Vq asVq(2i1)q(i) and Vq(2 i)q( i), i1,2, . . . ,L,i.e.,

57 11 405RANDOM TRANSVERSE ISING SPIN CHAIN AND . . .


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T

0 h1

h1 0 J1

J1 0 h2

h2 0

JL1

JL1 0 hL

hL 0

,

Vq q 1

q 1

q 2

qL1

qL

qL

. 2.2One is confined to the q0 part of the spectrum.

14

A. Magnetization

Technically the calculation is more simple, when theboundary condition bc does not break the symmetry of theHamiltonian, therefore we start with a chain with two freeends. As a consequence, in this case the ground-state expec-

tation value of the local magnetization operator 0lx0 free

is zero for finite chains. Then the scaling behavior of themagnetization at the critical point is obtained from theasymptotic behavior of the imaginary time-time correlationfunction:

G l0lxl

x 0 0

i

ilx02expEiE0 , 2.3

where 0 and i denote the ground state and the ith excitedstate of H in Eq. 2.1, with energies E0 and Ei , respec-tively. In the thermodynamic limit in the ordered phase ofthe system the first excited state is asymptotically degeneratewith the ground state, thus the sum in Eq. 2.3 is dominated

by the first term. In the large limit limG l()m l2 , thus

the local magnetization is given by the off-diagonal matrixelement:

m lfree1l

x0 . 2.4

In the fermion representation the magnetization operator isexpressed as

lxA 1B1A 2B 2 . . . A l1B l1A l 2.5

with

A iq

q i qq, B i

qq i q

q.

2.6

Using 110 the matrix element in Eq. 2.4 is evalu-

ated by Wicks theorem. Since for ij 0A iAj00B iBj00 we obtain for the local magnetization

m lfree

H1 G 11 G 12 . . . G 1l1

H2 G 21 G 22 . . . G2l1

Hl G l1 G l2 . . . G ll1

, 2.7

where

Hj01Aj01j ,

Gjk0BkAj0q

q k qj . 2.8

We note that the off-diagonal magnetization m lfree in Eq.

2.4 can be used to study the scaling behavior of the criticalmagnetization through finite-size scaling.

Next we turn to consider the system with symmetrybreaking bcs, when one of the boundary spins is fixed. Thenone should formally put h10 or hL0. For instance, if we

fix the spin at site L we put hL0, implying that Lx now

commutes with the Hamiltonian and SLx is a good quantum

number. In the fermionic description the twofold degeneracy

of the energy levels, corresponding to SLx1 and SL

x

1, is manifested by a zero energy mode: 10 in Eq.2.1, with an eigenvector which satisfies V1(2 i)1(i)0, i1,2, . . . ,L .15 Then the first excited state 1 is de-generate with the ground state 0 and the matrix element inEq. 2.4 corresponds to the ground-state expectation valueof the magnetization

m lfree

0lx0 , 2.9

which is formally given by the determinant in Eq. 2.7. Thesurface magnetization m sm 11(1) can be computed in astraightforward manner by simply solving the equation TV10 with V1(2 i)1(i)0 for i1, . . . ,L and using thenormalization condition i1( i)

21. This leads to the exact

formula

m s 1l1

L1

j1

l

hjJj

21/2

, 2.10

which has been derived in Ref. 16 in the thermodynamic

limit L. Note that with the boundary condition consid-ered here we proved that Eq. 2.10 is also exact for anyfinite system.

Next we consider those boundary conditions, when bothboundary spins are fixed. Then one should formally put h 10 and hL0. This situation is, however, more complicatedthan the mixed bc, since it describes both parallel () andantiparallel () boundary conditions. To determine themagnetization profiles in these cases we make use of theduality properties of the quantum Ising model. First we de-

fine the dual Pauli operators i1/2x ,i1/2

z as

i1/2z

ixi1

x , izi1/2

x i1/2x , 2.11

11 406 57FERENC IGLOI AND HEIKO RIEGER


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in terms of which the Hamiltonian in Eq. 1.1 is expressedas

Hl

Jll1/2z

l

h ll1/2x l1/2

x . 2.12

In the dual model formally the couplings and fields are in-

terchanged, thus the dual Hamiltonian has zero surfacefields, since J0JL0 in Eq. 1.1. Then it can be easilyshown that the even sector i.e., those states which containeven number of fermions of the free chain corresponds tothe parallel boundary condition of the dual model,whereas the odd sector of the free chain to the anti-parallel boundary condition.

To obtain the magnetization profile for fixed boundaryspin conditions we use the expression

l1/2x

0z1

z . . . lz , 2.13

which can be obtained from Eq. 2.11, then express the

product of zs by fermion operators through the relationi

zA iB i and evaluate the right-hand side rhs of Eq. 2.13

in the corresponding free-chain situation. For the parallelspin boundary condition we take the free-chain vacuum ex-pectation value:

m l0l1/2

x 00A 0B 0A 1B 1 . . . A lB l0free,2.14

which can be expressed through Wicks theorem as

m l

1 0 0 . . . 0

0G

11 G

12. . .

G

1l

0 Gl1 Gl2 . . . Gll. 2.15

Here Gjk is the same as in Eq. 2.8, however it is calculatedwith the dual couplings h lJl .

To obtain the magnetization profile in the antiparallel spinboundary condition one should take the expectationvalue of the rhs of Eq. 2.13 in the lowest state of the oddsector of the free chain, which is the first excited state of the

Hamiltonian: 110. Thus

m l0l1/2x 01A 0B 0A 1B 1 . . . A lB l1 free.2.16

To evaluate this expression by Wicks theorem one should

notice that the state 110 can be considered the

vacuum state of such a system, where 1 and 1 are inter-

changed, which formally means that 1(k)1(k) and11 . Then the expectation values in Eq. 2.8 aremodified as

Gjk1BkAj1q1

q kqj 1 k 1j ,

2.17

which again have to be evaluated with the dual couplings

h lJl . Then the m l profile is given by the determinant in

Eq. 2.15, where the matrix elements Gjk are replaced by

Gjk .

B. Susceptibility and autocorrelations

The local susceptibility l at site l is defined through thelocal magnetization m l as

l limHl0

m l

Hl, 2.18

where Hl is the strength of the local longitudinal field, which

enters in the Hamiltonian in Eq. 1.1 as Hllx . l can be

expressed as

l2

ii l

x02

EiE0, 2.19

which for boundary spins is simply given by

12q

q 1 2

q. 2.20

Next we consider the dynamical correlations of the sys-tem as a function of the imaginary time . First, we note thatthe correlations between surface spins can be obtained di-rectly from Eq. 2.3 as

G 1q

q 1 2expq. 2.21

For bulk spins the matrix element ilx0 in Eq. 2.3 is

more complicated to evaluate, therefore one goes back to thefirst equation of Eq. 2.3 and considers the time evolution inthe Heisenberg picture:

lx

expHlxexpH

A1

B1

. . . Al

1B

l

1A

l.

2.22

The general time and position-dependent correlation func-tion

lx

lnx A 1B 1 A lA 1B 1 A ln ,

2.23

can then be evaluated by Wicks theorem as a product oftwo-operator expectation values, which in turn is written intothe compact form as a Pfaffian:



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lx

lnx

A 1B 1 A 1A 2 A1B2 A 1A l A 1A 1 A 1A ln

B 1A 2 B 1B 2 B 1A l B1A1 B 1A ln

A 2B 2 A 2A l A 2A 1 A 2A ln

B ln1A ln

detC

i j1/2, 2.24

where Ci j is an antisymmetric matrix Ci jCji , with theelements of the Pfaffian 2.24 above the diagonal. At zerotemperature the elements of the Pfaffian are the following:

AjA kq

qj q kexpq,

AjB kq

qj q kexpq,

2.25

BjB kq

qj q kexpq ,

BjA kq

qj q kexpq ,

whereas the equal-time contractions are given in Eq. 2.6.For the finite-temperature contractions see Ref. 17.

III. SURFACE MAGNETIZATION AND THE MAPPING

TO ADSORBING WALKS

In this section we analyze the surface magnetization ofthe RTIM random transverse Ising model using a mappingto an adsorbing random-walk problem. In this way we obtainexact results for the critical exponents s and , as well as

for the surface magnetization scaling dimension x ms . These

observations will then be used in the following section toidentify the structure of the strongly coupled domainsSCDs, which are responsible for the low-energy excita-tions in the system.

A. Surface magnetization and correlation length

The surface magnetization in Eq. 2.10 represents per-haps the simplest order parameter of the transverse-field

Ising chain. Note that the scaling behavior of end-to-end cor-relations

CL1xL

x av 3.1

is identical to that of the surface magnetization since CL aswell as m s involve only surface spin operators that haveanomalous dimension x s . So, for instance, CL(0)L2xs and CL()

2s.In the following we determine its average behavior for the

symmetric binary distribution, i.e., with pq1/2 in Eq.1.7. First we consider the system of a large, but finitelength L at the critical point. In the limit 0 the surfacemagnetization in Eq. 2.10, which is expressed as a sum of

products has a simple structure. Considering a random real-ization of the couplings the surface magnetization is zero,whenever a product of the form of i1

lJi2 , l

1,2, . . . ,L is infinite, i.e., the number of couplings ex-ceeds the number of 1 couplings in any of the 1,l inter-vals. On the other hand, if the surface magnetization has afinite value, it could be of the form m s1/(n1)

1/2, wheren0,1,2, . . . measures the number of intervals, which arecharacterized by having the same number of and 1 cou-plings. To have a more transparent picture we represent thedistribution of couplings by directed walks, which start atzero and make the ith steps upwards for a coupling Ji1) or downwards for a coupling Ji). As illustratedin Fig. 1 the surface magnetization corresponding to a ran-dom sequence is nonzero, only if the representing walk doesnot go below the time axis,18 while the correspondingsurface magnetization is given as m s1/(n1)

1/2, wherenow n just counts the number of times the walk has touchedthe t axis for t0.

Then the ratio of walks representing a sample with finitesurface magnetization is just the survival probability of thewalk P surv , which is given by P surv(L)L

1/2, for walkssamples of length L . In the thermodynamic limit this prob-

ability vanishes, thus the t y p i c a l realization of the chain,i.e., the event with probability one has zero surface magne-tization. This is certainly different from the average value,which is dominated by the rare events represented by surviv-ing walks with an m sO(1). Consequently the average sur-face magnetization of a critical chain of length L is given by

m sL,0 avAL1/2OL3/2. 3.2

On the other hand, one knows from finite-size scaling that

m s(L,0) avLx

ms

, where xmss / is the scaling di-

mension of the surface magnetization. Thus from Eq. 3.2we can read the exact result:

FIG. 1. Sketch of the correspondence between a bond configu-

ration and a random walk. The thick segments in the horizontal line

indicate strong (Ji1) bonds, thin segments weak bonds (Ji

). This bond configuration corresponds to a surviving walk as

indicated by the broken line staying completely above the horizon-

tal line, implying a finite surface magnetization. Actually for this

example ms1/31/2 in the case 0, since the random walk

touches the horizontal line three times including the starting point.



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x ms

1

2, 3.3

while the prefactor in Eq. 3.2 is obtained as A0.643 196 1 from a numerical calculation.

In the following we argue that the exponent in Eq. 3.3 isthe same far all values of the binary distribution. First wemention that the finite-size critical surface magnetization

m s(L ,0) av is a monotonically decreasing function of

1: ms

1(L ,0) av m s2(L ,0) av 0 121 for any

value of L . Thus the corresponding exponents also satisfy

x ms (1)xm

s (2). However according to exact results the

value of xms is the same, i.e., 1/2 for the homogeneous

model19 (1) and for the extreme inhomogeneous model(0), thus the relation in Eq. 3.3 should hold for anyparameters of the distribution. We note that this is the first

exact derivation of the x ms exponent, the value of which is in

agreement with previously known exact and conjectured re-sults in Eqs. 1.3 and 1.5.

Next we calculate the exponent from the dependence

of the surface magnetization. In the scaling limit L

1, 1 the surface magnetization can be written as

m s

L,av m s

L ,0 avmsL1/. 3.4

Here the scaling function ms(y), which depends on the ratioof L and the correlation length , can be expanded as

ms(y)1ByO(y2), so that one obtains for the correc-

tion to the surface magnetization:

m sL ,avm sL ,0 avL, 3.5

with 1/xms . This exponent can also be determined in

the 0 limit of the binary distribution. Now, slightly out-

side the critical point the products of l terms in the sum ofEq. 2.10 will contain a factor of (1)2l 12l in lead-ing order of. Then the surface magnetization of a couplingdistribution which is represented by a surviving walk isgiven by

m s 1i1

n

12l i1/2

1n 1/2

i

l i

n1 3/2O2, 3.6

where l i gives the position of the ith touching point of thewalk with the t axis. Next we consider a typical survivingwalk, which has nO(1) return points since the probabilityof n returns decreases exponentially, and these points aresituated at l iO(L

1/2). Consequently for a typical survivingwalk the correction term in Eq. 3.6 is O(L1/2), whichshould be multiplied by the surviving probability ofO(L1/2). Since the surviving walks have a sharp probabilitydistribution we are left with the result: m s(L ,) av m s

(L ,0) avBO(2), where the constant is given

from numerical calculations as: B0.270 563 17/20.Comparing our result with that in Eq. 3.5 we get for thecorrelation length critical exponent

2. 3.7

This is an exact determination of the exponent . We ob-tained it for a particular limit 0 for the binary distribu-tion 1.7, which is a very broad distribution of couplingsand therefore essentially the limit in which Fishers RGanalysis2 becomes exact, too. Cum grano salis one might saythat here we presented a way to perform exact calculation by

using directly the fixed-point distribution occuring in this RGstudy.

B. Relation between surface magnetization

and adsorbing random walks

Here we summarize and extend the mapping between thesurface magnetization of the RTIM and the surviving prob-ability of the corresponding adsorbing random walk. First,we consider again the extreme binary distribution of cou-plings in Eq. 1.7 with h 01, such that the control param-eter in Eq. 1.2 is given by

qp

4pq

1

ln . 3.8

Then, according to the considerations of the previous sec-tion, the corresponding adsorbing walk has an asymmetriccharacter: it makes steps with probabilities p and q1poff and towards the wall, respectively. The correspondingcontrol parameter wqp in Eq. A1 is proportional to in Eq. 3.8. Our basic observation can be summarized as

m s,L P survw ,L , w . 3.9

At the critical pointfrom Eq. A7 P surv(0,L)L and

1/2 is just x ms , the surface magnetization scaling dimen-

sion of the RTIM. In the paramagnetic phase of the RTIM0 and the corresponding walk has a drift towards theadsorbing wall. The surviving probability in Eq. A8

P survw0,L expL/w,

w8pq

pq 2w

2 , 3.10

is characterized by a correlation length, which diverges asw

with 2. We note that the expression for w inEq. 3.10 agrees with the RTIM result by Fisher.2 Finally, inthe ferromagnetic phase 0 the corresponding walk driftsoff the wall and the surviving probability has a finite limit as

L:

P surv0,L pq

pw . 3.11

This expression then corresponds to a finite average surfacemagnetization of the RTIM, which linearly vanishes at thecritical point. Thus the surface magnetization exponent of theRTIM is s1, in agreement with the exact results byMcCoy.6 The finite-size corrections to the surviving prob-ability in Eq. 3.11 are exponential, according to Eq. A10

P surv0,L Psurv0,LexpL/w ,3.12



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with the correlation length w , as in Eq. 3.10. Conse-quently the critical exponent is the same below and abovethe transition point, as it generally should be.

These results obtained for the extreme binary distributioncan be generalized for other random distributions, too. It isenough to notice that the surface magnetization in a samplewith nonsurviving walk character is exponentially vanishingwith the size of the system. Therefore the basic relation in

Eq. 3.9 remains valid. Then at the critical point theP surv(0,L)L

1/2 is a consequence of the Gaussian na-

ture of the random walk. Similarly, the relations ww2 ,

w0 and P surv(w0,L)w , w0 follow from thescaling behavior of the random walks.

IV. DISTRIBUTION OF THE LOW-ENERGY

EXCITATIONS

In the previous section we saw that the surface order isconnected to a coupling distribution, which can be repre-sented by a surviving walk. Since local order and smallvanishing excitation energies are always connected, we can

thus identify the local distribution of couplings, which resultsin a strongly coupled domain SCD. Note that a SCD is notsimply a domain of strong bonds, but it generally has a muchlarger spatial extent.20

To estimate the excitation energy of an SCD we makeuse the exact result for the lowest gap 1(l) of an Isingquantum chain of l spins with free bc:21 Since we are inter-ested in a bond and field configuration that gives rise to anexponentially small gap 1 we can neglect the rhs of theeigenvalue equation

TV11V1 , 4.1

cf. Eq. 2.2 and derive approximate expressions for the

eigenfunctions 1 and 1 . With these one arrives at

1 l m smsh li1

l1h i

Ji. 4.2

Here m s and ms denote the finite-size surface magnetizations

at both ends of the chain, as defined in Eq. 2.10 for mssimply replace hj /Jj by hLj /JLj in this equation. If thesample has a low-energy excitation, then both end-surfacemagnetizations are of O(1), consequently the coupling dis-tribution follows a surviving walk picture. The correspond-ing gap estimated from Eq. 4.2 is given by

1i1

l1

h iJi

expl trlnJ/h , 4.3

where l tr measures the size of transverse fluctuations of a

surviving walk of length l and ln(J/h) is an average coupling.In the following we assume that the excitation energy ofsurface SCDs are of the same order of magnitude as thoselocalized in the bulk of the system and have the same type ofcoupling distributions. Thus we identify the SCDs, both atthe surface and in the volume of the system, as a realizationof couplings and fields with surviving walk character andhaving an excitation energy given in Eq. 4.3. With thisprerequisite we are now ready to apply our theory for thecritical and off-critical regions of the RTIM of L sites.

At the bulk critical point the characteristic length l ofsurviving regions is of the order of the size of the system L,thus the SCD extends over the volume of the system. Thetransverse fluctuations of the couplings in the SCD are fromEq. A13 as l trL

1/2, thus we obtain for the scaling relationof the energy gap at the critical point:

0,L expconstL 1/2, 4.4

in accordance with the existing numerical results.11 At thispoint it is useful to point out the origin of the exponent 1/2accompanying the length scale L in Eq. 4.4: it is the factthat the sequence of h i /Ji is random and uncorrelated, for ageneral sequence one would have l trL

with being thewandering exponent the scaling dimension of the transversefluctuations of the particular sequence under consideration.For instance one could also consider relevant aperiodic se-quences generated in a deterministic fashion, which haveeither the same or different wandering exponents, leadingeither to the same or a different scaling behavior of the en-ergy scale at the critical point as the random chain studiedhere.22

In the paramagnetic phase the probability of finding aSCD of size l , which is localized at a given point is propor-tional to exp(l/), cf. Eq. 3.10. Since the SCD can belocated at any point of the chain, the actual probability isproportional to the length of the chain, thus PL(l)Lexp(l/). The characteristic size of SCDs obtainedfrom the condition PL(l)O(1) is given by

l ln L , 0, 4.5

which grows very slowly with the linear size of the system.The characteristic transverse fluctuations of such a walk isgivenaccording to Eq. A16as l tr(qp) l, with

O(1). Setting this expression into Eq. 4.3 we obtain thescaling relation

0,L Lz, 4.6

where the dynamical exponent z()2/ is a continuousfunction of the control parameter . Our estimate qualita-tively agrees with Fishers result,2 that close to the criticalpoint z()1/2.

The dynamical exponent z() is conveniently measuredfrom the scaling behavior of the probability distributionPL(l) PLln()L . For a given large L the scaling com-bination from Eq. 4.6 is L1/z, thus

P

1

1/z. 4.7

The distribution function of the gap in Eq. 4.7 has alreadybeen studied in Ref. 11 for periodic boundary conditions.Here we considered free chains and investigated the accumu-lated probability distribution

L ln

ln

dy PLy . 4.8

As seen in Fig. 2 the accumulated probability distribution forlow energies is approximately a straight line on a log-logplot and from the slope one can estimate 1/z() quite accu-rately.



8/17

In the ferromagnetic phase of the RTIM the size of theSCD is of the order of the sample, lL and also the trans-verse fluctuations of the couplings are l trL . Consequentlythe energy of the first excitations scale exponentially with thesize of the system: exp(constL). Here, however, oneshould take into account thatdue to the duality relation inEq. 2.12in a strongly coupled environment there are

always weakly coupled domains WCDs, which are the

counterparts of the SCDs in the paramagnetic phase. Thecharacteristic size of a WCD is O(ln L), and their presencewill reduce the size of the SCDs, such that one expectslogarithmic corrections to the size of transverse fluctuationsl tr . Indeed the numerical results on the accumulated gap dis-tribution function in Fig. 3 can be interpreted with the pres-ence of such corrections.

In the ferromagnetic phase many physical quantities con-nected autocorrelation function, susceptibility, etc. are con-nected with the distribution of the second gap. Unfortu-nately, we cannot make an estimate for 2 on the base of ourpresent approach. However, our model is self-dual, the dis-tributions of the couplings and the fields transform to each

other in Eq. 2.12 for . Therefore, we assume thatthe scaling behavior of1 in the paramagnetic phase and thatof 2 in the ferromagnetic phase are also related throughduality, thus

20,L Lz. 4.9

Indeed, as seen in Fig. 4 the scaling relation in Eq. 4.9 issatisfied, however with strong, logarithmic corrections.

V. CRITICAL PROPERTIES

A. Surface magnetizationcanonical vs microcanonical

ensemble

The surface magnetization of the RTIM has already beenstudied in Sec. III. Here we revisit this problem in order toanswer the question, whether the values of the average quan-tities and the corresponding critical exponents depend or noton the ensemble used in the calculations. Our present studyis motivated by a recent work23 in which finite-size scalingmethods and their predictions for critical exponents24 havebeen scrutinized for random systems.

In our approach in Sec. III the bond and field configura-tions were taken completely random according to the corre-sponding distribution. We call this the canonical ensemble,since only the ensemble average of ln Ji and ln hi is heldfixed. One can also confine oneself on a subset of this en-

FIG. 2. The integrated gap probability distribution L(ln 1) in

the disordered phase (h1) for different values of h . The dynami-

cal exponent z(h) is extracted from the expected asymptotic form

ln L(ln 1)1/z(h)(ln 1)const which is a straight line when us-

ing a logarithmic scale on the y axes. Thus 1/z(h)0.82, 0.62 and0.40 for h4.0, 2.0, and 1.5, respectively. The data for the uniform

distribution averaged over 50 000 samples. Note that for large h ,

i.e., far away from the critical point, there is essentially no system

size dependence, whereas closer to the critical point the asymptotic

slope is reached only for large enough system sizes.

FIG. 3. The same as in the Fig. 2 in the ordered phase (h

1) . The data do not scale with ln 1/L, there are strong logarith-

mic corrections. A scaling with ln 1/L/ln(L) is also poor as can

be seen in the figure, most probably higher powers of ln(L) are

involved.



9/17

semble, in which we fix the value of the product of all bondsin the chain and similarly the product of all fields. Then the

critical point of the system is exactly given by i1L1 ln Ji

i1L1 ln hi , note that we study the surface magnetization

in a chain with one fixed boundary condition such that thereare exactly as many bonds as fields. This we call the micro-canonical ensemble. The motivation for the introduction ofthis ensemble can be found in Ref. 23: essentially it is a morerestrictive way of fulfilling the criterium ln Jav ln hav forbeing at the critical point.

The critical exponents of the canonical ensemble Eqs.3.3, 1.5, which agree with other exact and RG results, arethen the canonical ones.

We start to analyze the behavior of the surface magneti-zation in the paramagnetic phase (0). In the microca-

nonical ensemble the product of the couplings and that of thefields are fixed, therefore the last term in the sum of Eq.2.10 is of j(hj /Jj)

2exp(2L), for each realizations. Asa consequence the surface magnetization of the samples con-tains a prefactor exp(AL), which is also present in theaverage value. It is easy to see that the leading finite-sizedependence of the surface magnetization of typical samplesis related to the above term and given by

m sL ,0 typ mcexpAL . 5.1

Thus in the microcanonical ensemble there are no rare eventswith O(1) surface magnetizations and therefore the scalingbehavior of the average and typical values are identical andfor 0 we have the scaling combination L .

In the canonical distribution, due to fluctuations in theproduct of the couplings, there are rare events with O(1 )surface magnetizations and their fraction exp(A2L) inEq. 3.10 governs the finite-size scaling behavior of the av-erage surface magnetization, yielding av2. On the otherhand, the typical behavior in the canonical ensemble is thesame as in the microcanonical ensemble, see Eq. 5.1, bywhich typ1.

In the ferromagnetic phase (0) the fraction of realiza-tions with finite surface magnetization can be estimated asfollows in the microcanonical ensemble. After lL steps thewalk has an average drift of l drwl, which exceeds the

size of transverse fluctuations, l trl1/2, if ll cw

2 .

Therefore a walk which has been survived up to lc steps with

a probability ofP survlc1/2w will not be adsorbed with

probability of O(1 ) in the following steps lclL. Fromthis argument the microcanonical surface magnetization hasa linear dependence close to the critical point, therefore thesurface magnetization exponent

s mc1, 5.2

is the same as in the canonical distribution Eq. 1.3.We can easily estimate the finite-size corrections to the

surface magnetization in the ferromagnetic phase by noticingthat the corrections to the surviving probability of the corre-sponding walk are proportional to the probability that thewalk has a transverse fluctuation of the size of the drift l trwL , which is given by exp(Bl tr

2/L2)exp(BLw2 ).

Consequently the finite-size corrections to the surface mag-netization are given by exp(L/), with 2. Thus thecorrelation length exponent in the ordered phase is

mc

2,

0, 5.3which agrees with the canonical result in Eq. 3.7. We notethat the above arguments about the surviving probability ofrandom walks for 0 essentially hold for the canonicaldistribution, as a consequence the ferromagnetic phase of theRTIM has the same type of description in the two ensembles.

At the critical point of the model we use again the sym-metric binary distribution in Eq. 1.7, such that the sampleshave the same number of and 1 couplings. In the ex-treme limit 0, as in the canonical case, the critical pointsurface magnetization can be determined exactly, throughstudying the surviving probability of the corresponding ad-sorbing random walk. To determine the microcanonical sur-

viving probability first we note that from the canonical walksonly a fraction of O(L1/2) is microcanonical. Second, themicrocanonical surviving walks have their end at the startingpoint. Such returning surviving walks are of a fraction ofO(L3/2) among the canonical walks. Thus the survivingprobability of microcanonical walks is P surv(mc) L

1,therefore the microcanonical surface magnetization satisfiesthe scaling relation:

m sL ,0 av mcL1. 5.4

The scaling combination between L and 0 is againobtained by analyzing the expression in Eq. 3.6. The typi-cal number of return points of the surviving walks is againnO(1), but now l iO(L), since the endpoint of the walkis a return point. Consequently, the correction term in Eq.3.6 for surviving walks is O(L), what should be multipliedby the surviving probability to obtain the average of O(1),from which the scaling combination L, 0 follows, inagreement with the previous determination below Eq. 5.1.

To summarize, the average surface magnetization of theRTIM has anomalous scaling behavior in the microcanonicalensemble: in Eq. 5.1 there is an exponentionally vanishingprefactor exp(AL), which governs the scaling behaviorof the surface magnetization in the paramagnetic phase. Wenote that in scaling theory the different scaling behavior inthe low- and high-temperature phases is generally attributed

FIG. 4. The same as in Fig. 3 for the second lowest excitation,

i.e., L(ln 2). One observes that asymptotically ln L(ln 2)

1/z(h)ln 2const, with z(h0.5)0.62z(h2.0) as one

would expect from duality, by which z(h)z(1/h).



10/17


11/17

B. Profiles of observables

A real system is always geometrically constrained anddue to modified surface couplings its properties in the sur-face region are generally different from those in the bulk.Close to the critical point this surface region, which has acharacteristic size of the correlation length, intrudes far intothe system. At the very critical point the appropriate way todescribe the position-dependent physical quantities is to usedensity profiles rather than bulk and surface observables. Fora number of universality classes much is known about thespatially inhomogeneous behavior, in particular in two di-mensions, where conformal invariance provides a powerfultool to study various geometries.27

In a critical system confined between two parallel plates,which are at a large, but finite distance L apart, the localdensities (r) such as the order parameter magnetizationor energy density vary with the distance l from one of theplates as a smooth function of l/L . According to the scalingtheory by Fisher and de Gennes28

l abLxFab l/L , 5.8

where x is the bulk scaling dimension of the operator ,while ab denotes the boundary conditions at the two plates.In a d-dimensional system the scaling function in Eq. 5.8has the asymptotic behavior:

Fab l/L A 1Bab lL d

. . . lL1. 5.9

Here the amplitude of the first correction term is universal,

the corresponding exponent is just xsd the surface scaling

dimension of .In two dimensions conformal invariance gives further pre-

dictions on the profile:

l abL sin l

Lx

G ab l/L , 5.10

where the scaling function G ab ( l/L) depends on the univer-sality class of the model and on the type of the boundarycondition. With symmetric boundary conditions the scalingfunction is constant G aaA . For conformally invariant, non-symmetric boundary conditions the scaling function has beenpredicted for several models. For the Ising model the mag-netization profiles with free-fixed (f) and boundaryconditions are predicted as

GfA sinl

2L x

ms

, 5.11

and

GA cosl

L, 5.12

respectively.In two dimensions conformal invariance can also be used

to predict the critical off-diagonal matrix-element profiles

( l)0, where denotes the lowest excited stateleading to a nonvanishing matrix element see Eq. 2.4.These off-diagonal profiles give information about the sur-face and bulk critical behavior via finite-size scaling, while

avoiding the contribution of regular terms. With symmetricboundary conditions one obtains for the profile29

l 0L

x sin l

L

xsx

, 5.13

which involves both the bulk and surface scaling dimensions.The numerically calculated average diagonal and off-

diagonal magnetization profiles see Sec. II for the RTIMare presented in Figs. 7 and 8 for the uniform distribution inRef. 30 some data for the binary distribution have been pre-

sented. Here we do not use xm

and xm

s as fit parameters, butfix them to the theoretically predicted values in Eqs. 1.4,3.3. The only fit parameter is the nonuniversal prefactor A ,which is found remarkably constant for the different bound-ary conditions. As one can see on Fig. 7 the data for differentlength L collapses to scaling curves, which are very welldescribed by the scaling functions predicted by conformalinvariance. Thus we can conclude that not only the scalingprediction by Fisher and de Gennes28 in Eq. 5.8 is verywell satisfied for the RTIM, but the corrections to the appro-priate conformal results are also very small, practically neg-ligible. This is an unexpected result, since the RTIM is notconformally invariant, due to anisotropic scaling at the criti-cal point Eq. 1.6.

FIG. 7. Scaling plots of the magnetization profiles for nonsym-

metric boundary conditions ( ll0.5). Top: plus-minus ( )

bc, the broken line is a fit to the form 5.10 and 5.12 with A as a

fit parameter. Bottom: free-fixed f bc., the broken line is a fit to

the form 5.10 and 5.11 with A as a fit parameter.



12/17

To close this section we present numerical results for thebulk magnetization scaling dimension x m and compare itwith Fishers perhaps most striking prediction in Eq. 1.4.Here we have made effort to increase the numerical accu-racy, therefore we worked with the binary distribution in Eq.1.7 on chains with both ends fixed with length L24 andperformed the exact average of the local magnetization onthe central spin. From the finite-lattice magnetizations, whichscales as m(L ,0) avL

xm we have determined x m by atwo-point fit, comparing systems with sizes L and L2.From the finite-size exponents, presented in Table I for 2, 3, and 4 one concludes that they are in agreement with

Fishers result: x m(35)/40.191. Unfortunately the

numerical data in Table I show log-periodic oscillations,which are a consequence of the energy scale introduced bythe binary distribution. Therefore one cannot use the accuratesequence-extrapolation methods to analyze the limiting be-havior of the series. Instead, from a simple linear fit one canobtain the estimate

x m0.1900.003, 5.14

improving the accuracy of previous MC estimates.11

C. Dynamical correlations

The general time- and position-dependent correlations inEq. 2.23 have a complicated structure at the critical point.

Therefore we consider dynamical correlations on the samespin, which has a simpler asymptotic behavior. First we con-sider the bulk autocorrelation function

GL/2x L/2x av 5.15

and recapitulate the scaling argument in Ref. 31.The autocorrelation function, like the local magnetiza-

tion, is not self-averaging at the critical point: its averagevalue is determined by the rare events, which occur with aprobability P r , and P r vanishes in the thermodynamic limit.In the random quantum systems the disorder is strictly cor-related along the time axis, consequently in the rare eventswith a local order, i.e., with a finite magnetization also theautocorrelations are nonvanishing. Under a scaling transfor-mation, when lengths are rescaled as ll/b , with b1 the

probability of the rare events transforms as P rbxm, like

the local magnetization. As we said above the same is truefor the autocorrelation function

G ln bxmG ln /b 1/2 0, 5.16

where we have made use of the relation between relevanttime tr and length at the critical point in Eq. 1.6. Takingnow the length scale as b(ln )2 we obtain

G ln 2xm 0. 5.17

For surface spins in Eqs. 5.16, 5.17 the surface magneti-

zation scaling dimension xms appears.

In Fig. 9 we present the numerical results for the criticalbulk autocorrelation function obtained via evaluating the

Pfaffian Eq. 2.24. Note that we have chosen L to be odd, sothat L /2 denotes the central spin, representing the bulk be-havior in a system with free bc. A plot of G()1/2xm, withxm as in Eq. 5.14, versus ln or on a logarithmic scaleshould yield a straight line in the infinite system size limitaccording to Eq. 5.17. As can be seen in Fig. 9 the dataagree well with this prediction. For the surface autocorrela-

tions G 1()1x ()1

xav , evaluated according to Eq.2.21, which is much less involved than the computation ofa Pfaffian, a similar plot with the bulk magnetization expo-

nent xm replaced by the surface magnetization exponent xms

gives also an excellent agreement with the prediction

G 1(ln )(ln )2x

ms

.

FIG. 8. Top: scaling plot of the magnetization profile for sym-

metric here: fixed boundary conditions. The broken line is a fit to

the form 5.10 and GaaA with A as a fit parameter. Bottom: the

profile of the off-diagonal matrix element with free bc. The broken

line is a fit to the form 5.13.

TABLE I. Numerical estimates for the bulk magnetization ex-

ponent xm(L) for the binary distribution for various values of .

x m(L)

L 2 3 4

6 0.127071 0.162136 0.181770

8 0.142310 0.161044 0.169656

10 0.157063 0.179177 0.18981512 0.167197 0.195090 0.207268

14 0.173605 0.197072 0.206820

16 0.176458 0.196602 0.204265

18 0.178444 0.195288 0.201673

20 0.179836 0.194391 0.199992

22 0.181044 0.194279 0.199270

24 0.182175



13/17

To complete the results on critical dynamics we mentionthe scaling behavior of the autocorrelation function

lz

()lz

av . We mention that lz

represents one part ofthe local energy operator, the other part of which

lxl1

x is related to it through duality. As shown in Ref. 31this quantity at the critical point can be characterized by apower-law asymptotic decay with novel critical exponents,which are different in the bulk and at the surface of thesystem.

VI. OFF-CRITICAL PROPERTIES

A surprising property of random quantum systems is theexistence of Griffiths-McCoy singularities in the paramag-netic side of the critical point. In the corresponding Griffiths-

McCoy region the autocorrelation function decays as apower G()1/z(), where the dynamical exponent z()characterizes also the distribution of low-energy excitationsin Eq. 4.7. As a consequence, the free energy is a nonana-lytic function of the magnetic field and the susceptibilitydiverges in the whole region.

According to the phenomenological theory11 in theGriffiths-McCoy region the singularities of all physical quan-tities are entirely characterized by the dynamical exponentz(). Numerical calculations11,12 give support to this as-sumption, although there are discrepancies between the val-ues of z() obtained from different quantities.

Here we extend previous investigations in several re-

spects. First, we consider also the surface properties, such asthe surface autocorrelation function and the surface suscep-tibility. Second, we investigate also the ferromagnetic side ofthe critical point. In the neighborhood of the critical pointFisher2 has already obtained some RG results in the ferro-magnetic phase. Here we are going to check these resultsnumerically and to extend them for finite 0.

A. Phenomenological scaling considerations

As already shown in Sec. IV the dynamical exponent z()is conveniently measured from the probability distribution ofthe energy gap in the ferromagnetic phase one considers thesecond gap in a finite system, which does not vanish expo-

nentially. For large systems the gap distribution is given byEq. 4.7 and with this the average autocorrelation function isgiven by

G0

Pexpd1/z. 6.1

In a finite system of size L for long enough time, such that

Lz(), the decay in Eq. 6.1 will change to a G()1/ form, which is characteristic for isolated spins. Itmeans that in the above limit the system can be considered asan effective single spin.

In the following we present a simple scaling theory whichexplains the form of the asymptotic decay in Eq. 6.1. Herein the Griffiths-McCoy phase we modify the scaling relationin Eq. 5.16 by two respects. First, the scaling combinationis changed to /bz, since the dynamical exponent z() isfinite in the off-critical region. Second, the rare events,which are responsible for the Griffiths-McCoy singularitiesare now samples with very low-energy gaps and their num-ber is practically independent of the size of the system. Con-

sequently the rescaling prefactor is b1

and the scaling rela-tion is given by

G,1/L b1G/bz,b/L 0, 6.2

where the inverse size of the system 1/L is also included as ascaling field. Now taking b1/z we obtain

G,1/L 1/zG1/z/L 0, 6.3

thus in the thermodynamic limit we recover the power-law

decay in Eq. 6.1. The scaling function G(y ) in Eq. 6.3

should behave as G(y )y 1z for large y , in this way onerecovers the limiting 1/ decay, as argued below Eq. 6.1.

Then the finite-size scaling behavior of the autocorrelationfunction is of the form of Lz1, and after integratingG(,1/L) by the same scaling behavior will appear in thelocal susceptibility:

iL Lz1, 0. 6.4

B. Numerical calculation of the dynamical exponent

The phenomenological description of the Griffiths phasesuggests that all Griffiths-McCoy singularities emerging intemperature, energy, time- or frequency-dependent quantitiesshould be parametrizable by a single dynamical expontentz(). In this subsection we present the results on our numeri-cal estimates for z() resulting from the calculation of thefollowing quantities:

i distribution of low-energy excitations,

ii autocorrelation function on bulk and surface spins,

iii distribution of surface susceptibilities.

The distribution functions for the energy gaps have alreadybeen presented in Sec. IV. The same quantity for the surfacesusceptibility in Eq. 2.20 has a similar form as the inversegap, as seen in Fig. 10. The only difference that for thesusceptibility the matrix element in the denominator of Eq.2.20 select one special position of the SCD. As a conse-quence the corresponding probability distribution has no L

FIG. 9. Bulk spin-spin autocorrelation function G() Eq. 5.15

for various system sizes and the uniform distribution. The straight

line is the prediction according to Eq. 5.17.



14/17

dependence, as already discussed in Ref. 11 and can also beseen by comparing Eqs. 4.6 with 6.4. The z() exponentscalculated from the surface susceptibility distribution agreewell with those obtained from the gap distribution.

The average autocorrelation function is measured at twosites of the chain: on the central spin, giving an estimate forthe bulk correlation function and on the surface spin. Theaverage bulk autocorrelation functions are drawn on a log-log plot in Fig. 11 for several values of0. One can easilynotice an extended region of the curves, which are well ap-proximated by straight lines, the slope of which is connectedto the dynamical exponent through Eq. 6.1. Similar behav-ior can be seen in Fig. 12, where the average surface auto-correlation functions are drawn. Our investigation on the dy-

namical exponent is completed by studying the connectedsurface autocorrelation function in the ferromagnetic phase.As seen in Fig. 13 the scaling form in Eq. 6.1 is wellsatisfied for this function, too.

The behavior of the dynamical exponents calculated bydifferent methods are summarized in Fig. 14. First we notethat the numerical estimates are very close to each other. Theonly exception is the data obtained from the bulk autocorre-lations. To explain the possible origin of this discrepancy weturn to Sec. VII. The z() values well satisfy the two theo-retical limits: limz()1 and lim01/2

2. Fur-thermore the dynamical exponents show the duality relation:z()z().

VII. DISCUSSION

In this paper the critical and off-critical properties of therandom transverse-field Ising spin chain are studied by ana-lytical and numerical methods and by phenomenologicalscaling theory. The previously known exact,7 RG,2 and nu-merical results11,30,31,12 about the model have been extendedand completed here in several directions. The scaling behav-ior of the surface magnetization is obtained through a map-ping to an adsorbing random walk and the critical exponents

s , , and xms are calculated exactly. We have also shown

that the scaling behavior in the microcanonical ensemble isanomalous.

Using the correspondence between the surface magnetiza-tion and the adsorbing walks we have identified stronglycoupled domains in the system, where the couplings have asurviving walk character, and estimated the distribution oflow-energy excitations both in the critical and off-criticalregions. This provides a comprehensive explanation of themicroscopic origin of the Griffiths-McCoy singularities. Itturns out that most of the astonishing features of the criticalas well as the off-critical Griffiths-McCoy properties can besimply explained via random-walk analogies. However, oneprediction by Fisher,2 namely, the exact value of the bulkmagnetization exponent and its surprising relation to the

golden mean, still lacks a simple explanation in terms ofuniversal properties of random walks.In the numerical part of our work we have treated rela-

tively large (L128) finite systems. At the critical point wehave calculated the magnetization profiles for differentboundary conditions, which are found to follow accuratelythe conformal predictions, although the system is not confor-mally invariant. We have also increased the numerical accu-racy in the calculation of the bulk magnetization scaling di-mension. In the off-critical regions we have determined thedynamical exponent z() from different physical quantities.The obtained results give support to the scaling predictionthat the Griffiths-McCoy singularities are characterized bythe single parameter z(). Here we note that the numerical

FIG. 10. Integrated probability distribution of the zero-

frequency surface susceptibility for different system sizes in the

disordered phase at h2.0. Note that z(h), as determined from the

slope of the straight line, turns out to be within the error margin of

z(h) determined via the gap distribution see Fig. 2.

FIG. 11. The bulk autocorrelation function L/2x ()L/2

x (0 ) avin imaginary time in the disordered phase ( h1), calculated at a

central spin iL /2 with Eq. 2.23 via the Pfaffian method pre-

sented in Sec. II. The straight lines are fits to the expected power-

law decay 1/z(h).



15/17

data show systematic differences, when z() is calculatedfrom bulk or from surface quantities. A similar observationhas been made in Ref. 12, too. The possible origin of thediscrepancies is, that the SCDs, which are responsible forthe Griffiths-McCoy singularities, have different environ-ments at the surface and in the volume of the system. Then,from the argument leading to Eqs. 4.5, 4.6 one can obtain

logarithmic corrections between the dynamical exponents.This fact can then explain the differences in the finite-sizedata. We have numerically studied the Griffiths-McCoy sin-gularities in the ferromagnetic phase, too. In this region thesecond gap of the Hamiltonian and the connected autocorre-lation function scale with the dynamical exponent, which,according to numerical results, satisfies the duality relation.

ACKNOWLEDGMENTS

F.I.s work has been supported by the Hungarian NationalResearch Fund under Grants Nos. OTKA TO12830, OTKATO23642, OTKA TO17485, and OTKA TO15786, and bythe Ministery of Education under Grant No. FKFP 0765/1997. We are indebted to L. Turban for a critical reading ofthe manuscript. Useful discussions with T. Nieuwenhuizen,F. Pazmandi, G. Zimanyi, and A. P. Young are gratefullyacknowledged. H.R.s work was supported by the DeutscheForschungsgemeinschaft DFG and he thanks the AspenCenter for Physics, the International Center of TheoreticalPhysics in Trieste, and the Research Institute for Solid State

Physics, Budapest, where part of this work has been com-pleted, for kind hospitality.

APPENDIX: ADSORBING RANDOM WALKS

Here we summarize the basic properties of one-dimensional random walks in the presence of an adsorbingwall. For simplicity, first we consider a walker, which makessteps of unit lengths with probabilities p and q1p to thepositive and to the negative directions, respectively. Startingat a distance s0 from an adsorbing wall we are interestedin the surviving probability P surv(w ,L) after L steps. Here

wqp A1

FIG. 12. The surface autocorrelation function 1x ()1

x (0 ) avin imaginary time in the disordered phase (h1) , calculated via

Eq. 2.21. The straight lines are fits to the expected power-law

decay 1/z(h).

FIG. 13. The connected part of the surface autocorrelation func-

tion 1x()1

x (0 ) av1x ()1

x (0 )ms2av in imaginary

time in the ordered phase ( h1) , calculated via Eq. 2.21, but

now substracting 1(1 ) 2. The straight lines are fits to the ex-

pected power-law decay 1/z( h).

FIG. 14. Summary of all estimates of the dynamical exponent

z(h) as a function of the distance from the critical point

ln h /2. The open circles are the estimates from the gap distribu-tion, as exemplified in Fig. 2, open triangles: surface susceptibility

distribution cf. Fig. 10, full circles: surface autocorrelation func-

tion cf. Fig. 12, crosses: surface autocorrelation function in the

ordered phase cf. Fig. 13, open squares: bulk autocorrelation func-

tion cf. Fig. 11. Note that whereas all former estimates agreewithin the error margin which is roughly the size of the symbols

the latter estimate, namely the one obtained via the bulk autocorre-

lation function, differs significantly from all others.



16/17

measures the average drift of the walk in one step: for w0 (w0) the walk has a drift towards off the wall.

The probability WL(l), that the walker after L steps is ata position lL, can be easily obtained by the mirrormethod:32

WL l pLl /2q Ll /2 L!

Ll /2!Ll 2!

L!

Ll /2s!Ll /2s! . A2

In the following we take s1 and in the limit L1, l1 weuse the central limit theorem to write in the continuum ap-proximation lx and WL(l)PL(x) as

PLx 1

p

x

L

1

2L2exp xx2

2L2 A3

with

xpq LwL , 24pq . A4

The surviving probability is then given by

P survw ,L 0

PLx dx 22

p 2Lexpy2

12

y

2expy2 1y ,

A5

where

y

x

2L2

wL

22, A6

and (z)2/ 0z exp(t2)dt is the error function.33

In the following we evaluate Psurv(w ,L) in Eq. A5 inthe different limits. In the symmetric case w0, p1/2:

Psurvw0,L 1

8LL1/2. A7

For w0, when pq and the walk has a drift towards thewall the surviving probability has an exponential decay as

y:

Psurvw0,L 22

p2Lexpy

2 14y2

L1/2expL/ww /L , A8

with a correlation length

w22

w2

. A9

Finally, for w0, when pq and the walk is driftedfrom the wall the surviving probability has a finite limit:

Psurv

w0,L

22

p2Ly

22

p 2Lexpy2

1

4y

2

w

p 22

p 2LexpL/ww /L ,

A10

which is approached exponentially.The size of transverse fluctuations of the adsorbing walk

is given by

l trw ,L 0

PLx xdx/Psurvw ,L , A11

where

0

PLx xdx22

pexpy2 y

2

4 2y21 expy2

1y . A12In the symmetric limit w0:

l trw0,L 8LL1/2. A13

For w0 the transverse fluctuations in leading order areindependent of L:

l trw0,L 22

w, A14

while for w0, when there is a drift of the walk from thewall the transverse fluctuations grow linearly with L:

l trw0,L wL . A15

The maximal value of the transverse fluctuations ltrmax(w ,L)

for 0 are in the same order of magnitude as their averagevalues in Eqs. A13 and A15. However for 0 the maxi-mal value is generally larger than the average one in Eq.

A14. In this case l trmax(w0,L) is determined by a rare

event, in which a large fluctuation of positive steps is fol-lowed by a drift process towards the average behavior. If thenumber of steps in the drift process is L , where 01,then

l trmax

w0,L Lw . A16

1 See, H. Rieger and A. P. Young, in Complex Behavior of Glassy

Systems, edited by M. Rubi and C. Perez-Vicente, Lecture Notes

in Physics Vol. 492 Springer-Verlag, Heidelberg, 1997, p. 256,

for a review.2 D. S. Fisher, Phys. Rev. Lett. 69, 534 1992; Phys. Rev. B 51,

6411 1995.

3 S. Sachdev and T. Senthil, Phys. Rev. Lett. 77, 5292 1996.4 H. Rieger and N. Kawashima unpublished; C. Pich and A. P.

Young unpublished; T. Ikegami, S. Miyashita, and H. Rieger,

J. Phys. Soc. Jap. to be published.5 R. B. Griffiths, Phys. Rev. Lett. 23, 17 1969.6 B. McCoy, Phys. Rev. Lett. 23, 383 1969.



17/17

7 B. M. McCoy and T. T. Wu, Phys. Rev. 176, 631 1968; 188,

982 1969; B. M. McCoy, ibid. 188, 1014 1969.8 R. Shankar and G. Murthy, Phys. Rev. B 36, 536 1987.9 R. A. Hyman, K. Yang, R. N. Bhatt, and S. M. Girvin, Phys. Rev.

Lett. 76, 839 1996.10 R. H. McKenzie, Phys. Rev. Lett. 77, 4804 1996.11 A. P. Young and H. Rieger, Phys. Rev. B 53, 8486 1996.12 A. P. Young, Phys. Rev. B 56, 11 691 1997.

13 E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. N.Y. 16, 4071961; S. Katsura, Phys. Rev. 127, 1508 1962; P. Pfeuty, Ann.

Phys. Paris 57, 79 1970.14 F. Igloi and L. Turban, Phys. Rev. Lett. 77, 1206 1996.15 In the eigenvalue problem in Eq. 2.2 the 10 eigenvalue is

twofold degenerate, which corresponds to the fact that the h10, hL0 and the h 10, hL0 boundary conditions are re-

lated through duality in Eq. 2.11, and the second eigenvector is

given by V2(i)2L,i . In the fermionic problem in Eq. 2.1 we

have just one zero mode, which is characterized by the first type

of eigenvector above Eq. 2.9 for hL0. If one takes h10

and hL0), then the zero-energy eigenvector is of the second

type, i.e., V1(i)1,i , and the expectational value of the first

spin from Eq. 2.7 is 01x

0

1, as it should be.16 I. Peschel, Phys. Rev. B 30, 6783 1984.17 J. Stolze, A. Noppert, and G. Muller, Phys. Rev. B 52, 4319

1995.18 The condition that the random walk does not go below the

time axis corrresponds to the condition that two end spins of

an effective spin cluster not be decimated in Fishers RG treat-

met Ref. 2.19 B. M. McCoy and T. T. Wu, Phys. Rev. 162, 436 1967.

20 The strongly coupled domains here are similar to the effec-

tive spin clusters in Fishers RG treatment Ref. 2.21 F. Igloi, L. Turban, D. Karevski, and F. Szalma, Phys. Rev. B 56,

11031 1997.22 F. Igloi, D. Karevski, and H. Rieger, Eur. Phys. J. B 1, 513

1998.23 F. Pazmandi, R. T. Scalettar, and G. T. Zimanyi, Phys. Rev. Lett.

79, 5130 1997.

24 J. T. Chayes, L. Chayes, D. S. Fisher, and T. Spencer, Phys. Rev.Lett. 57, 2999 1986.

25 M. E. Fisher, in Renormalization Group in Critical Phenomena

and Quantum Field Theory, edited by J. D. Gunton and M. S.

Green Temple University, Philadelphia, 1974, p. 65; F. Igloi

and L. Turban, Phys. Rev. B 47, 3404 1993.26 C. de Calan, J. M. Luck, T. M. Nieuwenhuizen, and D. Petritis, J.

Phys. A 18, 501 1985; T. M. Nieuwenhuizen and M. C. W.

van Rossum, Phys. Lett. A 160, 461 1991. Note that in these

works the distribution of a random variable z1x1x 1x2x 1x2x 3 with random uncorrelated x i has been studied,

which is identical to our variable ms2 .

27 F. Igloi, I. Peschel, and L. Turban, Adv. Phys. 42, 683 1993.28

M. E. Fisher and P.-G. De Gennes, C. R. Seances Acad. Sci., Ser.B 287, 207 1978.

29 L. Turban and F. Igloi, J. Phys. A 30, L105 1997.30 F. Igloi and H. Rieger, Phys. Rev. Lett. 78, 2473 1997.31 H. Rieger and F. Igloi, Europhys. Lett. 39, 135 1997.32 See, e.g., R. K. Pathria, Statistical Mechanics Pergamon, Oxford,

1972.33 I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and

Products Academic, London, 1980.


ferenc igloi and heiko rieger- random transverse ising spin chain and random walks

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