thesis defense - west virginia university€¦ · 2008-05-12 · thesis defense stefanos...

Thesis Defense

Stefanos Papanikolaou

Adviser: Eduardo Fradkin

May 12, 2008

University of Illinois, Urbana - Champaign

Thanks to collaborators(...and sometimes mentors):

M. Bauer, R. M. Fernandez, W. Krauth, E. Luijten, P. Phillips, K. S. Raman, J. Schmalian, R. Sknepnek

Solved PhD research problems

Mechanism for commensurate-incommensurate

transitions in Mott insulators

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Observables for topological phases with finite correlation length

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Universality & Mott transitions at finite temperatures

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phasecolumnar

phasecoexistence

multicritical point

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dimer-hole liquid

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Statistical mechanics of topological defects in Mott insulators

Universality of Mott Transition Liquid-Gas Critical Points at finite temperatures

Reference : Phys. Rev. Lett. 100, 026408 (2008) Collaborators : R.M. Fernandes, E. Fradkin, P. Phillips, J. Schmalian, R. Sknepnek

• Mott Transitions and Motterials

• Experiments on liquid-gas type of transitions in Mott materials

• Main Fact: The Mott Transition is in the Ising Universality class!

• Conductivity not the best quantity to study the nature of a metal-insulator transition

• Predictions

Outline

Usual liquid-gas transitions

• At a regular (e.g. water) liquid-gas critical point:

1. No symmetry change between the phases

2. Scalar order parameter ( ρ - ρ )

What does this tell us?

Ising universality class

• Is there anything else we could expect ?

Exotic multicritical point of some type (e.g. in frustrated systems)

L G

• What is a Mott material?

‣ Insulating systems with odd number of electrons per unit cell which “should” conduct

Traditional viewpoint Two parameters

(a) U : Coulomb interatomic exchange integral

(b) W: predicted badwidth

• U/W<<1: Fermi-liquid metallic system

• U/W>>1: Mott insulating state (no hopping)

• As U/W is tuned, a variety of phenomena could emerge (phase separation, spin liquids etc.)

Introduction to Motterials

• Experimentally, life sometimes is too simple!

‣ Direct first-order transition from a conventional metallic state to a Mott insulating state with or without magnetic ordering:

Motterials and direct Mott transitions

• Examples:

1. (3D material)

2. quasi-2D organic salts of the κ - ET family,

3. under hydrostatic pressure and (3D materials)

4. more materials...

Universality of liquid-gas Mott transitions at finite temperatures

Stefanos Papanikolaou,1 Rafael M. Fernandes,2, 3, 4 Eduardo Fradkin,1

Philip W. Phillips,1 Joerg Schmalian,2 and Rastko Sknepnek2

1Department of Physics, University of Illinois at Urbana-Champaign, 1110 W. Green St., Urbana, IL 61801-30802Ames Laboratory and Department of Physics and Astronomy, Iowa State University Ames, IA 50011, USA

3Instituto de Fısica ”Gleb Wataghin”, Universidade Estadual de Campinas4Laboratorio Nacional de Luz Sıncrotron,Campinas, SP, Brazil

(Dated: January 3, 2008)

We explain in a consistent manner the set of seemingly conflicting experiments on the finitetemperature Mott critical point, and demonstrate that the Mott transition is in the Ising universalityclass. We show that, even though the thermodynamic behavior of the system near such critical pointis described by an Ising order parameter, the global conductivity can depend on other singularobservables and, in particular, on the energy density. Finally, we show that in the presence of weakdisorder the dimensionality of the system has crucial e!ects on the size of the critical region that isprobed experimentally.

PACS numbers:

Although band theory predicts that a system of elec-trons in a solid with one electron per site (unit cell)should be metallic, such a system ultimately insulates [1,2] once the local electron repulsive interactions exceeds acritical value. The onset of the insulating state, the Motttransition, arises from the relative energy cost of the on-site Coulomb repulsion U between two electrons on thesame lattice site, and the kinetic energy, represented bythe band width W . Then, the transition is governedsolely by the ratio of U/W . At T = 0, it is often the casethat symmetries of the microscopic system, associatedwith charge, orbital or spin order, may be broken in theMott insulating state. However, at su!ciently high tem-peratures T , or in strongly frustrated systems, no sym-metry is broken at the finite-T Mott transition. Then,the transition is characterized by paramagnetic insulat-ing and metallic phases, whose coexistence terminates ata second-order critical point, depicted in Fig. 1(a). In thispaper, we are concerned with the universal properties ofthis classical critical point [3], as revealed by a series ofapparently conflicting experiments on (Cr1!xVx)2O3 [4]and organic salts of the ! ! ET family [5].

Since no symmetry is broken at the finite-T Mott tran-sition, in a strict sense there is no order parameter.Nonetheless, experimental [4, 5], as well as theoreticalevidence [6, 7] suggest that the transition is in the Isinguniversality class, similar to the liquid-vapor transition.For example, Castellani et al. [6] constructed an e"ectiveHamiltonian for this problem, and proposed that doubleoccupancy should play the role of an order parameterfor the Mott transition. On the insulating side, doublyoccupied sites are e"ectively localized, but in the metal,they proliferate. A Landau-Ginzburg analysis [7] pro-vided further evidence for a non-analyticity in the dou-ble occupancy at a critical value of U/W that defines aMott transition. Ising universality follows immediatelybecause double occupancy, "ni"ni##, is a scalar local den-

sity field.Experimentally, the universality of the Mott critical

point is typically probed by some external parameter,such as pressure, which can tune the ratio W/U . Mea-surements of the conductivity, #, on (Cr1!xVx)2O3 [4]found that away from the critical point, the exponentsdefined through

$# (t, h = 0) = #(t, h = 0) ! #c $ |t|!! ,

$# (t = 0, h) $ |h|1/"! ,

"#(t, h)/"h|h=0 $ |t|!#! , (1)

have mean-field Ising values, #$ % 1/2, $$ % 1 and%$ % 3. Here, t = (T !Tc)/Tc and h = (P !Pc)/Pc, with(#c, Tc, Pc) denoting the corresponding values at the crit-ical endpoint. Close to the critical region, Limelette etal. [4] observed a drift to the critical exponents of the 3DIsing universality class. Mean field behavior is also seenin NiS2 [8].

However, similar pressure measurements [5] on thequasi-2D organic salts of the !-ET family appear to chal-lenge the view that the Mott transition is in the Ising uni-versality class. In this material, Kagawa et al. [5] foundthat their data is described by the exponents #$ % 1,$$ % 1, and %$ % 2, which do not seem to be consistentwith the known exponents of the 2D Ising model whoseexponents are [9] # = 1

8 , $ = 74 and % = 15. Since the

exponents obey the scaling law $$ = #$ (%$ ! 1), it wasproposed that the Mott transition is in a new, as yet un-known universality class. The situation is further com-plicated by thermal expansion measurements [10] thatclaim to measure the heat capacity exponent & and find0.8 < & < 0.95. This result is not only in sharp con-trast to the expectation for an Ising transition (where& = 0 for d = 2), it also strongly violates the scaling law& + 2#$ + $$ = 2, if one uses the exponents of Ref. [5].

In this paper, we present a unified phenomenologicaldescription of all of these experimental facts within an








PACS numbers:





$# (t, h = 0) = #(t, h = 0) ! #c $ |t|!! ,

$# (t = 0, h) $ |h|1/"! ,

"#(t, h)/"h|h=0 $ |t|!#! , (1)



8 , $ = 74 and % = 15. Since the



NiS2−xSex

Motterials and direct Mott transitions 2

QCP in clean systems? Does standard behavior of QCPshows up only when the system is disordered? To tacklethese questions experimentally, we attempted to realizea “clean” QCP in NiS2!xSex. The parent compound ofNiS2!xSex, NiS2, is pure and presumably clean. If onecan approach the QCP of pure NiS2 by pressure withoutrelying on Se substitution, a clean analogue of AF QCPin NiS2!xSex can be explored and the impact of disor-der on QCP can be captured. Recent progress of highpressure technique enabled us to do so.

In this Letter, we address the issue of criticality anddisorder by examining the critical behavior of resistivityof pure NiS2 under pressures. The AF QCP of NiS2 wasreached at ! 7 GPa, where the system was found veryclean with a low residual resistivity !0 of ! 0.5 µ!cm.Not only right at the QCP but over an entire range of theparamagnetic phase investigated, the recovery of Fermiliquid T 2 of !(T ) is suppressed substantially to a very lowtemperature below ! 2 K and non Fermi liquid behaviorwith T 3/2 dependence of !(T ) dominated. We demon-strate the drastic influence of disorder on this AF QCPby contrasting the result with previous pressure work onNiS1.3Se0.7 with a residual resistivity of 60 µ!cm [9].

NiS2 sample used in this study was prepared by a va-por transport technique. The resistivity measurementwas performed by a conventional four probe techniqueunder hydrostatic pressure up to ! 10 GPa in a cubicanvil type pressure system down to 3 K and also in amodified Bridgman-type pressure cell down to 180 mK.The results obtained by the two di"erent pressure setupsagreed reasonably in the temperature range of overlap,indicating a very good homogeneity of pressure. Pres-sure was calibrated by measuring the superconductingtransition temperature of Pb [13].

The inset of Fig. 1 demonstrates !(T ) of NiS2 at rel-atively low pressures below 4 GPa. With applying pres-sure, the insulating behavior of !(T ) switches into metal-lic behavior, indicating the occurrence of metal-insulatortransition. In between 2.6-3.4 GPa, we observe a discon-tinuous jump of resistivity as a function of temperature,which corresponds to a first order metal-insulator transi-tion line on the phase diagram in Fig. 1. The discontinu-ous jump appears to terminate around 200 K, indicatingthe presence of a critical end point. In the phase dia-gram of NiS2!xSex solid solution, the first order phaseline terminates at much lower temperature and is hardto identify [14]. This di"erence appears to suggest thestrong influence of disorder on the Mott criticality.

As seen in the inset of Fig. 1, !(T ) of pure NiS2 showedmetallic behavior above P = 2.6 GPa. The residual re-sistivity at the critical point was as low as ! 0.5 µ!cm,demonstrating that the system is indeed very clean. Mag-netic ordering in the AF metal phase manifests itself as avery weak but sharp kink in !(T ) at TN as indicated bythe arrows. The antiferromagnetic transition tempera-ture TN thus determined systematically goes down upon

600

500

400

300

200

100

0

T (K

)

1086420

P (GPa)

10-7

10-5

10-3

10-1

101

!

("cm

)

3002001000

T (K)

0, 2.6, 3.0, 3.2, 3.3, 3.36, 3.4 GPa

P

TMI

TNInsulator

PM

AFM

NiS2

80

60

40

20

0

! (

µ"

cm

)

100500

T (K)

4.0 GPa5.0 GPa6.2 GPa7.5 GPa

P

TN

TN

TN

FIG. 1: The electronic phase diagram of clean NiS2 pyrite asa function of pressure. PM and AFM denote paramagneticmetal and AF metal, respectively. The inset shows the tem-perature dependent resistivity under pressures, P = 0 - 3.4GPa (left) and P = 4.0 - 7.5 GPa (right).

pressure and approaches T = 0 somewhere around 7-7.5GPa. No superconductivity was observed between P =6 and 9.1 GPa down to 180 mK, in spite of the low resid-ual resistivity. This appears to suggest that realizing anAF QCP in clean systems alone is not enough to achieveexotic superconductivity as observed in heavy Fermioncompounds [15, 16, 17, 18] and that additional ingredi-ents such as Kondo physics must be invoked.

The pressure dependence of TN , determined by thekink in !(T ), together with the first order metal insulatortransition, is summarized as a phase diagram in Fig. 1.TN appears to decrease almost linearly in contradictionto (Pc " P )2/3 dependence expected from self consistentrenormalization theory [5]. Unusual linear suppression ofthe magnetic transition temperature was also observedanalogously for a helical magnet MnSi [10] and a weaklyferromagnet ZrZn2 [11] when the sample is very clean. Itmay be interesting to infer that, in these clean system,the magnetic transition as a function of pressure is re-ported to be a first order rather than a second order. Inthe clean NiS2, we cannot rule out the possibility of afirst order transition at this stage, because !(T ) is notvery sensitive to TN near the critical point.

The signature of AF criticality in this clean system wasexplored. The inset of Fig. 2 demonstrates !(T ) below 30K, plotted as ! vs. T 3/2. In the antiferromagnetic phaseat P = 6.2 GPa, ! - T 3/2 curve is linear above TN butshows superlinear behavior below TN . The temperaturedependence below TN is found to be approximately T 2,indicative of the formation of coherent quasi particles. Inthe paramagnetic phase above ! 7 GPa, however, the !- T 3/2 curve shows a linear behavior down to very low

FIG. 4: The generic phase diagram of the organic ! ! BEDT materials as a function of hydrostatic

pressure and temperature.Picture taken from ref.[4].

Each dot in fig.3 represents one hole that remained in each dimer(two BEDT-molecules),

due to the absorption of an electron in the insulating layer above the plane. The lat-

tice formed by the dots is topologically equivalent to a triangular lattice, but there is an

anisotropy(look for example at [6, 9]...). The hopping energy between diagonally connected

sites is about 20% smaller than the nearest-neighbor ones. Equivalently, this corresponds

to a higher Coulomb repulsion energy by about 25% along this direction. In the Hubbard

model’s terminology, this means that: t!/t = 0.8 or V !/V = 1.25. The phase diagram of this

material, as studied in several experiments[3] , as a function of temperature and hydrostatic

pressure[4] is shown in fig.4.

The experimental system is regarded to be governed by quasi-2D physics, due to the

inorganic insulating X layers, that are in between the BEDT layers. The quasi-2D nature is

protected by polarized electrostatic interactions between the cation and the anion-layers[11].

II. THE LIQUID-GAS CRITICAL POINT

Kanoda and collaborators[4] studied the critical point which emerges at 40K and they

reached some striking conclusions:

1. The transport properties close to this liquid-gas critical point showed power-law behav-

4

Mott Transition from a Spin Liquid to a Fermi Liquid in the Spin-Frustrated Organic Conductor!-!ET"2Cu2!CN"3

Y. Kurosaki,1 Y. Shimizu,1,2,* K. Miyagawa,1,3 K. Kanoda,1,3 and G. Saito2

1Department of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo, 113-8656, Japan2Division of Chemistry, Kyoto University, Sakyo-ku, Kyoto, 606-8502, Japan

3CREST, Japan Science and Technology Corporation, Kawaguchi 332-0012, Japan(Received 15 October 2004; revised manuscript received 6 April 2005; published 18 October 2005)

The pressure-temperature phase diagram of the organic Mott insulator !-!ET"2Cu2!CN"3, a modelsystem of the spin liquid on triangular lattice, has been investigated by 1H NMR and resistivitymeasurements. The spin-liquid phase is persistent before the Mott transition to the metal or super-conducting phase under pressure. At the Mott transition, the spin fluctuations are rapidly suppressed andthe Fermi-liquid features are observed in the temperature dependence of the spin-lattice relaxation rateand resistivity. The characteristic curvature of the Mott boundary in the phase diagram highlights a crucialeffect of the spin frustration on the Mott transition.

DOI: 10.1103/PhysRevLett.95.177001 PACS numbers: 74.25.Nf, 71.27.+a, 74.70.Kn, 76.60.2k

Magnetic interaction on the verge of the Mott transitionis one of the chief subjects in the physics of stronglycorrelated electrons, because striking phenomena such asunconventional superconductivity emerge from the motherMott insulator with antiferromagnetic (AFM) order.Examples are transition metal oxides such as V2O3 andLa2CuO4, in which localized paramagnetic spins undergothe AFM transition at low temperatures [1]. The groundstate of the Mott insulator is, however, no more trivialwhen the spin frustration works between the localizedspins. Realization of the spin liquid has attracted muchattention since a proposal of the possibility in a triangular-lattice Heisenberg antiferromagnet [2]. Owing to the ex-tensive materials research, some examples of the possiblespin liquid have been found in systems with triangular andkagome lattices, such as the solid 3He layer [3], Cs2CuCl4[4], and !-!ET"2Cu2!CN"3 [5]. Mott transitions betweenmetallic and insulating spin-liquid phases are an interestingnew area of research.

The layered organic conductor !-!ET"2Cu2!CN"3 is theonly spin-liquid system to exhibit the Mott transition, tothe authors’ knowledge [5]. The conduction layer in!-!ET"2Cu2!CN"3 consists of strongly dimerized ET[bis(ethlylenedithio)-tetrathiafulvalene] molecules withone hole per dimer site, so that the on-site Coulombrepulsion inhibits the hole transfer [6]. In fact, it is aMott insulator at ambient pressure and becomes a metalor superconductor under pressure [7]. Taking the dimer as aunit, the network of interdimer transfer integrals forms anearly isotropic triangular lattice, and therefore the systemcan be modeled to a half-filled band system with strongspin frustration on the triangular lattice. At ambient pres-sure, the magnetic susceptibility behaved as the triangular-lattice Heisenberg model with an AFM interaction energyJ# 250 K [5,8]. Moreover, the 1H NMR measurementsprovided no indication of long-range magnetic order downto 32 mK. These results suggested the spin-liquid state at

ambient pressure. Then the Mott transition in!-!ET"2Cu2!CN"3 under pressure may be the unprece-dented one without symmetry breaking, if the magneticorder does not emerge under pressure up to the Mottboundary.

In this Letter, we report on the NMR and resistancestudies of the Mott transition in !-!ET"2Cu2!CN"3 underpressure. The result is summarized by the pressure-temperature (P-T) phase diagram in Fig. 1. The Mott

Superconductor

(Fermi liquid)

Crossover

(Spin liquid) onset TC

R = R0 + AT2

T1T = const.

(dR/dT)max

(1/T1T)max

Mott insulator

Metal

Pressure (10-1GPa)

FIG. 1 (color online). The pressure-temperature phase diagramof !-!ET"2Cu2!CN"3, constructed on the basis of the resistanceand NMR measurements under hydrostatic pressures. The Motttransition or crossover lines were identified as the temperaturewhere 1=T1T and dR=dT show the maximum as described in thetext. The upper limit of the Fermi-liquid region was defined bythe temperatures where 1=T1T and R deviate from the Korringa’srelation and R0 $ AT2, respectively. The onset superconductingtransition temperature was determined from the in-plane resis-tance measurements.

PRL 95, 177001 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending21 OCTOBER 2005

0031-9007=05=95(17)=177001(4)$23.00 177001-1 ! 2005 The American Physical Society

κ− (ET)2Cu)2(CN)3κ− (ET)2X

V2O3NiS2

‣ The order parameter is a density variable (maybe the density of doubly occupied sites (DiCastro, Castellani 1979))

‣ Ising universality class

• classical statistical mechanics relevant for criticality (similar as He - modelling at finite-T)

• At infinite dimensions, DMFT decided that the transition is Ising-like (Kotliar et al. 2000).

Mott liquid-gas critical point Ising universality

• Theoretical Intuition/Expectation:

‣ The order parameter is a density variable (maybe the density of doubly occupied sites (DiCastro, Castellani 1979))

‣ Ising universality class

• classical statistical mechanics relevant for criticality (Emery-Griffiths-like models)

• At infinite dimensions, DMFT decided that the transition is Ising-like (Kotliar et al. 2000).

Mott liquid-gas critical point Ising universality

• Theoretical Intuition/Expectation:

Is it consistent with experiments?

• Consider the conductivity as a local order parameter(like magnetization of the Ising model):

Experiments near the Mott critical point

!"

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"

"


$%&''(%&)*&+,&%-*(%&",.-'&"/0-1%-+"23"!456789499:;<=(>?5=?;<@=A"-%2(B/"*.&"C%0*0C-A"







*.&"C%0*0C-A"&I,2B&B*'#"5"#"##"$;"%"5<#"L#"L;D"

!!"

"

"

Figure 4"| Scaling plot of the whole data set of GT(P) in the metallic region."#$%"&'()"(*"

!+h,",-h!-#".%/010"h-|"|#$"23)$")$%"&/%0%4)".5'1%06"+#6"$,"%"+76"!,6"*5''0"34)(")2("085'349"81/.%0"

8(//%0&(4:349")(")$%"085'349"*148)3(4"f&+x,"(.%/"5"23:%"/549%;"<!6"h6""="34")$%"'5>%'0"(*")$%"5?%0"

8(//%0&(4:0")("<GT+P,"'"GT+P8/(00+T,,6"P"'"P8/(00+T,6"@T"'"T8@="*(/"T"A"T8"54:"<GT+P,"'"G86"P"'"

PB+T,6"@T"'"T8@="*(/"T"C"T8"+0%%"D%)$(:0,;"

∆Σ ≡ Σ(T,P)− Σ(Tc,Pc)

• Assumed behavior near the critical point:

•

•

•

where ,

• Also, scaling relation holds:

∆Σ��(a)∼ |t|βσ

∆Σ��(b)∼ |h|1/δσ

∂∆Σ/∂h��(c)∼ |t|−γσ

t ∼ T− Tc h ∼ P− Pc

βσ + γσ = βσδσ

MaterialCritical exponents identified

Theorists’ initial comments on the exper. facts

Limelette et al.(2003)

(β, γ, δ) = (1/2, 1, 3) and crossover to Ising

Expected(3D Ising), Mean-Field Ising, Narrow critical region

Kagawa et al.(2005) κ-ET organic salts under pressure

(β, γ, δ) = (1, 1, 2) Not expected!Exotic, maybe due to 2D physics...

Takeshita et al.(2007)

under pressure (β, γ, δ) = (1/2, 1, 3)

Expected(3D Ising), Mean-Field Ising, Narrow critical region

M. de Souza et al. (2006)

κ-ET organic salts under pressure

Not expected...... if connected with specific heat exp. (?)

Experiments near the Mott critical point

l−1∂l/∂T ∼ t−0.85








PACS numbers:





$# (t, h = 0) = #(t, h = 0) ! #c $ |t|!! ,

$# (t = 0, h) $ |h|1/"! ,

"#(t, h)/"h|h=0 $ |t|!#! , (1)



8 , $ = 74 and % = 15. Since the










PACS numbers:





$# (t, h = 0) = #(t, h = 0) ! #c $ |t|!! ,

$# (t = 0, h) $ |h|1/"! ,

"#(t, h)/"h|h=0 $ |t|!#! , (1)



8 , $ = 74 and % = 15. Since the



• Which critical exponent corresponds to the thermal expansion coefficient in the organics ?

• M. de Souza et al. assumed specific heat exponent

Thermal Expansion measurements...

α ∼ (∂E/∂T )|p ∼ Cp

But, is this correct?



• M. de Souza et al. assumed specific heat exponentα ∼ (∂E/∂T )|p ∼ Cp





T

P

TISING

PISING

α ∼ (∂E/∂T )|p ∼ Cp

∂E

∂T

��p∼ a

∂E

∂TISING+ b

∂E

∂PISING∼ aCIsing

p + b∂m

∂TISING





T

P

TISING

PISING

α ∼ (∂E/∂T )|p ∼ Cp

∂E

∂T

��p∼ a

∂E

∂TISING+ b

∂E


p + b∂m

∂TISING

weak logarithmic divergence

∼ t−0.875





T

P

TISING

PISING

α ∼ (∂E/∂T )|p ∼ Cp

∂E

∂T

��p∼ a

∂E

∂TISING+ b

∂E


p + b∂m

∂TISING

weak logarithmic divergence





T

P

TISING

PISING

α ∼ (∂E/∂T )|p ∼ Cp

∂E

∂T

��p∼ a

∂E

∂TISING+ b

∂E


p + b∂m

∂TISING

weak logarithmic divergence ∼ t−0.875





• From another point of view:

• Assume an analytic equation of state

• To first order (no symmetry constraints): ∆l ∼ ∆ncar ∼ m

α ∼ ∂m

∂T∼ t−0.875

T

P

TISING

PISING

α ∼ (∂E/∂T )|p ∼ Cp

∂E

∂T

��p∼ a

∂E

∂TISING+ b

∂E


p + b∂m

∂TISING






• From another point of view:

• Assume an analytic equation of state

• To first order (no symmetry constraints): ∆l ∼ ∆ncar ∼ m

α ∼ ∂m

∂T∼ t−0.875

T

P

TISING

PISING

α ∼ (∂E/∂T )|p ∼ Cp

∂E

∂T

��p∼ a

∂E

∂TISING+ b

∂E


p + b∂m

∂TISING


found: 0.8 - 0.95

✓critical Ising

What is not consistent for the conductivity...

• Implications:

1. Large critical region for the κ-ET materials

2. Conductivity’s critical exponents cannot be the correct order parameter’s exponents


• Implications:


2. Conductivity’s critical exponents cannot be the correct order parameter’s exponents

• Candidate culprits:

1. definition of the relevant fields (P h, T T)

2. Regarding the conductivity as a local order parameter


• Implications:


2. Critical exponents cannot be the correct order parameter’s exponents

• Candidate culprits:

1. definition of the relevant fields (P h, T T)

2. Regarding the conductivity as a local order parameter

I. THE MATERIAL

The structure of the ! ! (BEDT ! TTF)2X family of materials is distinguished from

other families by the presense of the letter !. The BEDT molecule is shown in figure 1:

FIG. 1: The complicated organic molecule which is responsible for these materials. Its name is

written on the figure...[11]

The studied compound that we are going to discuss in this paper is the one where

X = Cu [N(CN)2] Cl. The structure of the material[11] is shown in the figures2,3.

FIG. 2: The 3D structure of the ! ! BEDT materials.The BEDT molecules are packed between

layers of insulating inorganic molecules(black dots.)

2

I. THE MATERIAL

The structure of the ! ! (BEDT ! TTF)2X family of materials is distinguished from

other families by the presense of the letter !. The BEDT molecule is shown in figure 1:

FIG. 1: The complicated organic molecule which is responsible for these materials. Its name is

written on the figure...[11]

The studied compound that we are going to discuss in this paper is the one where

X = Cu [N(CN)2] Cl. The structure of the material[11] is shown in the figures2,3.

FIG. 2: The 3D structure of the ! ! BEDT materials.The BEDT molecules are packed between

layers of insulating inorganic molecules(black dots.)

2

FIG. 3: Looking the previous picture from above. We easily see the anisotropic triangular structure

formed. The molecules are very closely packed together, allowing for overlaps between the orbitals.

Each two molecules contribute one electron to the insulating anion-layer forming a half-filled 2D-

hole gas. The blue dots are supposed to represent the ’sites’ of the holes. Obviously, these ’sites’

have a thickness equal to the length of the molecule.

3

Organic materials in the κ - ET family

• aligned organic molecules, separated by insulating layers of salts(quasi-2D conduction)

• Two molecules form a dimer and contribute one electron

• Positions of the dimers form an anisotropic triangular lattice

Define a system at an Ising critical point

• label grains of high-low carrier density.

• grain size: decoherence length

• Effective Ising Hamiltonian:

2

P

T

AFI

PI

PM

(a) (b)

FIG. 1: (a) Typical phase diagram of Mott transitions as afunction of pressure P and temperature T . At low T and P , aNeel antiferromagnetic insulator (AFI) appears; it becomes aparamagnetic insulator (PI) if T increases, or a paramagneticmetal (PM) if P increases. Dashed line: first-order transi-tion ending at a liquid-gas critical point. Full lines: contin-uous phase transition to the ordered state. Colored regions:critical (dark) and mean-field (light) regimes of the criticalpoint. (b) An Ising configuration on the triangular lattice.Up (down) spins correspond to conducting (insulating) grainsof linear size of the order of the system’s dephasing length.

of these experiments requires to take into account thatthe conductivity depends on all possible singular observ-ables of the associated critical system, and not just onthe thermodynamic order parameter associated with thephase transition. Similar considerations were made inmagnetic systems near the Curie temperature [11–13], toexplain the critical exponent of the conductivity alongthe coexistence curve. In that case, a symmetry of themicroscopic definition of the conductance prevented anycoupling of the global conductivity to odd moments ofthe order parameter, along the coexistence line. Eventhough similar in spirit, the situation here is much di!er-ent. Starting from an e!ective microscopic model nearan Ising critical point, we show that: 1) the conductivitytypically depends on all possible singular thermodynamicobservables of the system, namely the order parameterand energy density of the Ising model; 2) when the cou-pling to the energy density dominates, there exists a largeregime around the critical point, where the critical expo-nents for the conductivity are (!!, "!, #!) = (1, 7

8 , 158 ),

that agree (within the error bars) with the findings ofKagawa et al. [5], and the corresponding mean-field ex-ponents are (!MF

! , "MF! , #MF

! ) = (1, 12 , 3

2 ); 3) a crossoverto Ising exponents is obtained in the order parameterdominated regime as seen in Refs. [4, 8]; 4) in the pres-ence of disorder the Mott critical point ultimately belongsto the random-field Ising model universality class, andtherefore the dimensionality of the system under study iseven more important for specifying its critical properties.

In order to resolve the discrepancies raised by theseexperiments, we consider the behavior of the conductiv-ity of the system near the Mott critical point, assumingthat it belongs to the 2D Ising universality class. Ratherthan starting from a microscopic picture, e.g. a Hub-bard model, we consider a coarse-grained model with thecorrect symmetries in which the physics of the relevant

transport degrees of freedom is captured. In this picture,one defines coarse-grained regions, of linear size of theorder of the dephasing length l" of the system, which areeither insulating or conducting. Along these lines, weconsider an Ising model on a 2D lattice (cf. Fig. 1(b)).Near the critical point, where the correlation length fordensity fluctuations $ diverges, it is expected that therelevant degrees of freedom behave classically. The Isingvariables Si on each lattice site represent the fluctuatingdensity of mobile carriers on microscopic “grains” of lin-ear size of the order of the dephasing length l", whichare conducting (Si = +1), or insulating (Si = !1). TheHamiltonian is

!H = !1

T

!

!ij"

SiSj +h

T

!

i

Si , (2)

where T is the temperature, P and Pc are the pressureand the critical pressure, respectively, and h " P ! Pc

plays the role of the Ising magnetic field. This model isexpected to describe the physics near the critical point,where $ # l". In this limit, all other interactions beyondnearest-neighbor are irrelevant. Near the critical point,the most singular e!ect of the pressure is described by acoupling to the order parameter.

To relate the order parameter fluctuations to the trans-port properties we will define an associated resistor net-work for this model, an approach that has been success-fully used in other strongly correlated systems [14, 15].Let %C and %I be the local conductivities of the conduct-ing and insulating regions, respectively. We define thebond conductance of the network model simply by addingthese two conductivities in series. The bond conductancehas three possible values, depending on the state of eachgrain, which can both be conducting, both insulating,one conducting and the other insulating. Thus, the con-ductance of the bond (i, j) has the form

%ij = %0 (1 + gm(Si + Sj) + g#SiSj) . (3)

Even in this toy model, the microscopic conductivity, %ij ,couples both to the order parameter, Si, and to the en-ergy density, SiSj, of the Ising model with naturally largecouplings, gm and g#, defined in Eq.(3). More specifically,we find that %0 = 1

4 (%C +%I)+ !C!I

!C+!I, gm = !C#!I

4!0, and

g# = (!C#!I )2

4!0(!C+!I ) . At high contrast, %C # %I , we get

gm $ g# $ 1, whereas, at low contrast, |%I ! %C | % %C ,we get g# < gm & 0.

The conductivity of the 2D Ising model we described isa non-trivial quantity to compute. As it was shown in thesimpler case of the random resistor network (RRN) [16],networks of bonds with conductance %C (%I) chosen ran-domly with probability p and 1!p, the global conductiv-ity becomes non-zero as soon as an infinite percolatingand conducting cluster emerges in the system. When%I = 0, the critical exponent !! of the conductivity is

Si = ±1

lφ

2

P

T

AFI

PI

PM

(a) (b)



8 , 158 ),


! , "MF! , #MF

! ) = (1, 12 , 3




!H = !1

T

!

!ij"

SiSj +h

T

!

i

Si , (2)




%ij = %0 (1 + gm(Si + Sj) + g#SiSj) . (3)


4 (%C +%I)+ !C!I

!C+!I, gm = !C#!I

4!0, and

g# = (!C#!I )2




Each Ising configuration: a resistor network

σinsul

σconduct

• Incoherent hopping between grains classical resistor

• What is crucial?

(A) geometric distribution of resistors (problem on its own)

(B) distinguishability or contrast:

• Solving the network:

‣ analytically, only at small contrast,

‣ numerically, solve exactly the Kirchoff equations(Franck-Lobb and others) for every Ising configuration.

g =σconduct − σinsul

σconduct

Full contrast limit: Looking at the fractal

, :

• percolating clusters

• conductivity diffusion of an “ant” in the incipient infinite Ising cluster.

• critical exponents controlled by fractal properties

• Connection between conductivity exponents and thermodynamic exponents not known!

σinsul = 0

10 20 50 100 200L

1.5

2

2.5

ΣTc

10 100

gm= gε= 0.999999

ΣTc=3.9 L− 0.20

g = 1

Zero contrast limit: Almost local conductivity :

Possible to calculate the conductivity in a cluster perturbation theory, in powers of the contrast parameter g

Consider generally:

Solve the Kirchoff laws perturbatively (Blackman 1976)

(...) Result:

σinsul → σconduct, g → 0

2

P

T

AFI

PI

PM

(a) (b)



8 , 158 ),


! , "MF! , #MF

! ) = (1, 12 , 3




!H = !1

T

!

!ij"

SiSj +h

T

!

i

Si , (2)




%ij = %0 (1 + gm(Si + Sj) + g#SiSj) . (3)


4 (%C +%I)+ !C!I

!C+!I, gm = !C#!I

4!0, and

g# = (!C#!I )2




Σσ0

= 1 + gm�S� + (g� + g2mΓαβ)�(SS)α� + g�gmΓαβ�(SS)αSβ� +

g2� Γαβ�(SS)α(SS)β� + O(g3)

Σσ0





Consider generally:


(...) Result:


Lattice functions, decaying exponentially with distance between links α, β

2

P

T

AFI

PI

PM

(a) (b)



8 , 158 ),


! , "MF! , #MF

! ) = (1, 12 , 3




!H = !1

T

!

!ij"

SiSj +h

T

!

i

Si , (2)




%ij = %0 (1 + gm(Si + Sj) + g#SiSj) . (3)


4 (%C +%I)+ !C!I

!C+!I, gm = !C#!I

4!0, and

g# = (!C#!I )2




Σσ0





Consider generally:


(...) Result:


when Ising critical:

∼ m + · · ·

2

P

T

AFI

PI

PM

(a) (b)



8 , 158 ),


! , "MF! , #MF

! ) = (1, 12 , 3




!H = !1

T

!

!ij"

SiSj +h

T

!

i

Si , (2)




%ij = %0 (1 + gm(Si + Sj) + g#SiSj) . (3)


4 (%C +%I)+ !C!I

!C+!I, gm = !C#!I

4!0, and

g# = (!C#!I )2




Σσ0





Consider generally:


(...) Result:


when Ising critical:

∼ � + · · ·

2

P

T

AFI

PI

PM

(a) (b)



8 , 158 ),


! , "MF! , #MF

! ) = (1, 12 , 3




!H = !1

T

!

!ij"

SiSj +h

T

!

i

Si , (2)




%ij = %0 (1 + gm(Si + Sj) + g#SiSj) . (3)


4 (%C +%I)+ !C!I

!C+!I, gm = !C#!I

4!0, and

g# = (!C#!I )2




Zero contrast limit: Almost local conductivity

• At the critical point:

• local correlation functions have the singular behavior of the leading local observables.

• At an Ising critical point, these are the magnetization and the energy density:

3

non-trivially related to the fractal properties of the incip-ient infinite conducting cluster. This exponent is largerthan unity for random uncorrelated networks and largerthan the exponent of the order parameter, because dan-gling bonds of the infinite cluster do not contribute tothe conductivity. On the other hand, it becomes muchless than unity for correlated networks, and typically veryclose to the exponent of the order parameter, since theinfinite cluster is connected with few dangling bonds.

On the other hand, when !I > 0, a conducting clus-ter is less distinguishable from that of an insulating one,and the complex e!ects coming from the fractal clusterboundaries are smeared out. In the context of RRN, thepercolation transition is not seen in the behavior of theconductivity, which seems to show just a crossover. Ifthe contrast is low, !I ! !C , the actual conductivity ofa single bond between sites i, j, ", should depend onlyon local observables, and we can formally expand it inpowers of gm and g! [17],

" = !0 + gm"(Si + Sj)# + g!"SiSj# + . . . , (4)

where the ellipsis represents more complex products oflocal spin operators (weighed by rapidly decaying func-tions) [17]. Near the Ising critical point, the most singu-lar contribution of the expectation values of multi-spinoperators in Eq. 4 is given by the expectation value ofthe most singular, “primary”, operators of the Ising crit-ical point, the order parameter m and the energy density". Thus, the most singular term of multi-spin operatorswith odd (even) number of spins is proportional to theorder parameter (energy density). Therefore, within therange of convergence of this expansion,

" = "0(gm, g!) + fm(gm, g!)"m# + f!(gm, g!)""# , (5)

where "0, fm, f! are non-universal regular polynomialsin gm and g!. Provided that the critical behavior isstill controlled by the fixed point theory of the Isingmodel, the total conductivity should have the structureof Eq. (5). Thus, at finite contrast, Eq. (5) predictsthat the actual conductivity is the sum of even and oddcomponents, under the Ising symmetry transformation," = "0 + "even + "odd, and it should exhibit a crossoverfrom an energy density dominated behavior at short dis-tances to an order parameter dominated behavior at longdistances. The crossover scale is controlled by the relativesize of the functions fm and f! (cf. Fig. 2). This behaviorbreaks down at high contrast where there is multi-fractalbehavior (cf. Inset in Fig.2 and Ref. [18]).

We can understand the experiments of Refs. [4, 5, 8], ifwe assume that Eq. (5) applies. The results of Refs. [4, 8]follow by assuming that fm(gm, g!) > f!(gm, g!), and theconductivity scales as the order parameter. Conversely,the results of Ref. [5] follow if f!(gm, g!) $ fm(gm, g!),and the conductivity, for an extended regime near thecritical point, scales as the energy density of the Ising

10 20 50 100 200L

1.5

2

2.5

!T

c

10 100

gm

= g"= 0.999999

!T

c

=3.9 L# 0.20

10 20 50 100 300

L

0.013

0.015

!T

c# !

0

10 100

gm

= 0.001, g"= 0.01

!T

c

# !0= 0.02 L

# 1.0+ 0.012

!T

c

# !0= 0.003 L

# 0.125+ 0.011

FIG. 2: Crossover behavior of the conductivity at finite con-trast: the energy density (order parameter) dominates atshort (long) length scales. Inset: Fractal scaling at large con-trast.

model. In this case #" % |m|" holds, where # =(1 & $) /%. Then, it follows that %# = #%, &# = &/#and '# = ' + % (1 & #). The resulting critical exponentsare (%#, '#, &#) = (1, 7

8 , 158 ), very close to the experimen-

tal values. These exponents obey '# = %# (&# & 1), if' = % (& & 1), i.e., the conductivity exponents obey ascaling relation identical to that for the the Ising expo-nents, in agreement with the experimental verification ofthis scaling in Ref. [5]. In addition, the scaling obtainedby Kagawa et al. [5] only depends on %& = %#&#, as inour theory.

In order to verify the theory presented above, we per-formed Monte Carlo simulations of the 2D Ising modelon square and triangular lattices, using the Wol! clusteralgorithm [19]. For the calculation of the conductivity,for each Ising configuration we used the Franck - Lobbalgorithm [20], or explicitly solved Kirchho! equations.As expected, we found that at the Ising critical point,for gm, g! ' 1, the even component of the conductivity"even scales as the energy density, while the odd compo-nent "odd scales as the order parameter (cf. Fig. 3). Asgm, g! approach unity, a slow crossover exists to a fractalregime of the Ising clusters, which is crucial for specify-ing the critical exponent of the conductivity, consistentwith the results of Ref. [18] (cf. Inset in Fig.2.)

Refs. [4, 8] report 3D mean-field Ising behavior anda small critical region in (Cr1!xVx)2O3 and NiS2 underpressure respectively, in contrast to the extended criticalregion with 2D Ising exponents of Ref. [5]. We can un-derstand these experiments by considering the e!ects ofquenched disorder on an Ising critical point. The di!er-ence between a quasi-2D and a 3D material is a stronglyanisotropic Ising interaction along the direction perpen-dicular to the planes. Disorder that locally favors thelocalized over the delocalized state or vice versa, corre-sponds to a random magnetic field, and couples to theorder parameter of the Ising transition. This induces den-sity fluctuations. The relevant model for this discussion

polynomial functionsdependent on microscopics

accurate formula in the range of convergence of the expansion

Simulations at small contrast

• At the Ising critical point:

• Show that the conductivity has odd and even parts which scale like the magnetization and energy density, respectively

10 20 50 100 300L

0.012

0.013

σeven

Tc

10 100

gm = gε = 0.01

σeven

Tc = 0.016 L − 1.007 + 0.011

10 20 50 100 300L

0.01

0.013

σodd

Tc

10 100

0.01

gm = gε = 0.01

σodd

Tc= 0.017 L− 0.125

Simulations at small contrast

10 20 50 100 300L

0.013

0.015Σ

T c− Σ

0

10 100

gm= 0.001, gε= 0.01

ΣTc− Σ0= 0.02 L− 1.0+ 0.012

ΣTc− Σ

0= 0.003 L− 0.125+ 0.011

• Show that there is a crossover length scale, depending on the ratio of the contrasts g

Relation to the experiments on the organics

• Consider the possibility that :

• Then, energy-controlled true-critical exponents ( )

• very close to experimental finding ~ ( 1, 1, 2 )

• Crossover between the energy dominated regime (smaller length scales) and the order parameter dominated regime (larger length scales).

• But, why so small:

‣ Proliferation of domain walls

gm � g�

(βσ, γσ, δσ) = (1, 7/8, 15/8)

gm

∆Σ ∼ ��

Relation to the experiments on the organics

• A lesson from microscopics:

• becomes large when there is strong interface scattering (relevant to inhomogeneous states)

• , to be determined by DMFT-type calculations on inhomogeneous (e.g. R. W. Helmes et al., arXiv: 0805.0566)

gm =σmet−met − σins−ins

2σ0

g� gm

g�

g� =σmet−ins − σ0

σ0

Looking at the full picture of the Mott transition

• , are in the limit and the experiments are in the mean-field regime.

• κ - ET materials are in the and the experiments are in the energy-dominated true critical regime.

• prediction 1: Conductivity jump has different exponent (Ising) than the bare conductivity

• prediction 2: All thermodynamic variables should show Ising criticality

• prediction 3: Multiple crossover regimes, away from criticality








PACS numbers:





$# (t, h = 0) = #(t, h = 0) ! #c $ |t|!! ,

$# (t = 0, h) $ |h|1/"! ,

"#(t, h)/"h|h=0 $ |t|!#! , (1)



8 , $ = 74 and % = 15. Since the










PACS numbers:





$# (t, h = 0) = #(t, h = 0) ! #c $ |t|!! ,

$# (t = 0, h) $ |h|1/"! ,

"#(t, h)/"h|h=0 $ |t|!#! , (1)



8 , $ = 74 and % = 15. Since the



gm > g�

gm � g�

Looking at the full picture of the Mott transition

Why the experiments “see” only true critical?

Disorder and Dimensionality

• , are in the limit and the experiments are in the mean-field regime.

• κ - ET materials are in the and the experiments are in the energy-dominated true critical regime.

• prediction 1: Conductivity jump has different exponent (Ising) than the bare conductivity

• prediction 2: All thermodynamic variables should show Ising criticality

• prediction 3: Multiple crossover regimes, away from criticality








PACS numbers:





$# (t, h = 0) = #(t, h = 0) ! #c $ |t|!! ,

$# (t = 0, h) $ |h|1/"! ,

"#(t, h)/"h|h=0 $ |t|!#! , (1)



8 , $ = 74 and % = 15. Since the










PACS numbers:





$# (t, h = 0) = #(t, h = 0) ! #c $ |t|!! ,

$# (t = 0, h) $ |h|1/"! ,

"#(t, h)/"h|h=0 $ |t|!#! , (1)



8 , $ = 74 and % = 15. Since the



gm > g�

gm � g�

Full problem: Ising model in a random field

• In the organics, disorder comes from orientational flexibility of the ethylene molecules which host the carriers: like a random Ising field

• Correct model: strongly anisotropic random-field Ising model

!xy > 1

!z, !xy < 1

!z > 1

1

1/2

Jz/Jxy

Jxy

!

P

F

00

J

!

!!!!!

3D

C

4

10 20 50 100 300

L

0.01

0.013

!odd

Tc

10 100

0.01

gm

= g" = 0.01

!odd

Tc

= 0.017 L# 0.125

10 20 50 100 300

L

0.012

0.013

!even

Tc

10 100

gm

= g" = 0.01

!even

Tc

= 0.016 L# 1.007

+ 0.011

FIG. 3: Monte Carlo data which verify the expected behaviorof the conductivity when gm, g! ! 1. !even scales as theenergy density (see text). Inset: !odd scales as the orderparameter.

is the anisotropic 3D random-field Ising model (RFIM),

H = !Jxy

!

{ij}xy

SiSj ! Jz

!

{kl}z

SkSl +!

i

hiSi , (6)

where hi is a random field with variance !. For d = 3,there is a continuous phase transition in the 3D random-field Ising model (3DRFIM) universality class [21] forany anisotropy Jxy/Jz, an irrelevant operator at the 3DRFIM fixed point. However, for large anisotropy andweak disorder, relevant to the quasi-2D organics, thereis a large dimensional crossover regime from 2D RFIMbehavior, with an exponentially long correlation length,to the narrower 3D RFIM criticality [22]. What changesbetween the 3D isotropic materials and the quasi-2D or-ganics is not the universality class, but where the planarcorrelations become critical. For weak disorder ! " Jxy

and strong anisotropy Jz/Jxy " 1, the planes are es-sentially decoupled and 2D-RFIM behavior holds with!xy # 1 in a large region away from the transition point.When Jz $ Jxy, the critical region is narrow, and con-trolled by the 3D RFIM fixed point.

With regards to the thermal expansion measurementsthat claim to measure the heat capacity exponent " [10],we argue that the volume change is proportional to theIsing order parameter of the Mott transition, i.e. !l %m, yielding l!1dl/dT % t!!1. The thermal expansiondiverges with exponent 1 ! # = 0.875, consistent withRef. [10] who find it in the range 0.8 ! 0.95.

Some major predictions can be drawn from our pic-ture. Firstly, all thermodynamic observables near theMott critical point should have Ising critical exponents.Secondly, regarding the critical behavior in quasi-2D or-ganic salts [5], in the clean system, the conductivityalong the coexistence line should have the same crit-ical exponent (#" = 1) in both mean-field and true-critical regimes. This means that the conductivity jump!"J & "(T, h = 0+) ! "(T, h = 0!) along the coex-istence line, which should be proportional to the orderparameter, should have distinct mean-field and critical

regimes, where #!"J = 1/2 and #!"J = 1/8 respec-tively. Finally, the first-order Mott transition shouldbe broadened by disorder, thereby rounding the conduc-tance jump to a continuous transition [23, 24], with ef-fects of glassiness and spatial inhomogeneity [15, 25, 26].

Acknowledgements We thank K. Kanoda for discus-sions. This work was supported in part by the NSFgrants DMR 0442537 (EF) and DMR 0605769 (PP) atUIUC, by DOE, Division of Materials Sciences underAward DE-FG02-07ER46453 (EF), through the Freder-ick Seitz Materials Research Laboratory at UIUC, andthe Ames Laboratory, operated for DOE by Iowa StateUniversity under Contract No. DE-AC02-07CH11358(JS), and by CAPES and CNPq (Brazil) (RF).

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(1970).[3] This should not be confused with the scenario of M.

Imada, Phys. Rev. B, 72, 075113 (2005).[4] P. Limelette et al., Science 302, 89 (2003).[5] F. Kagawa, K. Miyagawa, and K. Kanoda, Nature 436,

534 (2005).[6] C. Castellani et al., Phys. Rev. Lett. 43, 1957 (1979).[7] G. Kotliar, E. Lange, and M. J. Rozenberg, Phys. Rev.

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private communication.[9] B. M. McCoy and T. T. Wu, The Two-Dimensional Ising

Model (Harvard University Press, Cambridge, 1973).[10] M. de Souza et al., arXiv:cond-mat/0610576v1 (2006).[11] P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49,

435 (1977).[12] M. E. Fisher and J. S. Langer, Phys. Rev. Lett. 20, 665

(1968).[13] I. Mannari, Phys. Lett. A 26, 134 (1968).[14] E. W. Carlson et al., Phys. Rev. Lett. 96, 097003 (2006).[15] E. Dagotto, T. Hotta, and A. Moreo, Phys. Rep. 344, 1

(2001).[16] D. Stau"er and A. Aharony, Introduction to Percolation

Theory (CRC Press, 1991).[17] J. A. Blackman, J. Phys. C: Solid State Phys. 9, 2049

(1976).[18] P. J. M. Bastiaansen and H. J. F. Knops, J. Phys. A:

Math. Gen. 30, 1791 (1997).[19] U. Wol", Phys. Rev. Lett. 62, 361 (1989).[20] D. J. Franck and C. J. Lobb, Phys. Rev. B 37, 302 (1988).[21] T. Nattermann, in Spin Glasses and Random Fields,

edited by A. P. Young (World Scientific, Singapore,1998).

[22] O. Zachar and I. Zaliznyak, Phys. Rev. Lett. 91, 036401(2003).

[23] Y. Imry and M. Wortis, Phys. Rev. Lett. 19, 3580 (1979).[24] M. Aizenman and J. Wehr, Phys. Rev. Lett. 62, 2503

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• Dimensional crossover of the critical point (Zachar, Zalinyak 2003)


• Disorder mainly from orientational flexibility of the ethylene molecules which host the carriers: like a random Ising field



!xy > 1

!z, !xy < 1

!z > 1

1

1/2

Jz/Jxy

Jxy

!

P

F

00

J

!

!!!!!

3D

C

4

10 20 50 100 300

L

0.01

0.013

!odd

Tc

10 100

0.01

gm

= g" = 0.01

!odd

Tc

= 0.017 L# 0.125

10 20 50 100 300

L

0.012

0.013

!even

Tc

10 100

gm

= g" = 0.01

!even

Tc

= 0.016 L# 1.007

+ 0.011



H = !Jxy

!

{ij}xy

SiSj ! Jz

!

{kl}z

SkSl +!

i

hiSi , (6)

























3D Materials small critical region, RFIM universality


• Disorder mainly from orientational flexibility of the ethylene molecules which host the carriers: like a random Ising field



!xy > 1

!z, !xy < 1

!z > 1

1

1/2

Jz/Jxy

Jxy

!

P

F

00

J

!

!!!!!

3D

C

4

10 20 50 100 300

L

0.01

0.013

!odd

Tc

10 100

0.01

gm

= g" = 0.01

!odd

Tc

= 0.017 L# 0.125

10 20 50 100 300

L

0.012

0.013

!even

Tc

10 100

gm

= g" = 0.01

!even

Tc

= 0.016 L# 1.007

+ 0.011



H = !Jxy

!

{ij}xy

SiSj ! Jz

!

{kl}z

SkSl +!

i

hiSi , (6)

























3D Materials small critical region, RFIM universality

quasi-2D materials wide 2D Ising critical region, still RFIM universality

Conclusions

• The Mott critical point is in the Ising universality class

• All relevant experiments can be explained within the framework of the Ising universality

• Domain-wall effects dominate the conductivity in the organic κ-ET materials

• The conductivity, intrinsically non-local, in the limit of low contrast, depends on all possible local singular observables of the critical system (magnetization, energy density)

Conclusions

• The Mott critical point is in the Ising universality class

• All relevant experiments can be explained within the framework of the Ising universality

• Domain-wall effects dominate the conductivity in the organic κ-ET materials

• The conductivity, intrinsically non-local, in the limit of low contrast, depends on all possible local singular observables of the critical system (magnetization, energy density)

!ank y" !

thesis defense - west virginia university€¦ · 2008-05-12 · thesis defense stefanos...

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