entropy-vanishing transition and glassy dynamics in frustrated spins

VOLUME 86, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 5 MARCH 2001

2058

Entropy-Vanishing Transition and Glassy Dynamics in Frustrated Spins

Hui Yin and Bulbul ChakrabortyMartin Fisher School of Physics, Brandeis University, Waltham, Massachusetts 02454

(Received 7 June 2000)

In an effort to understand the glass transition, the dynamics of a nonrandomly frustrated spin modelhas been analyzed. The phenomenology of the spin model is similar to that of a supercooled liquidundergoing the glass transition. The slow dynamics can be associated with the presence of extendedstringlike structures which demarcate regions of fast spin flips. An entropy-vanishing transition, with thestring density as the order parameter, is related to the observed glass transition in the spin model.

DOI: 10.1103/PhysRevLett.86.2058 PACS numbers: 64.70.Pf, 61.20.Lc, 64.60.My

The glass transition in supercooled liquids is heraldedby anomalously slow relaxations with a time scalediverging as the liquid freezes into the glassy state [1].In recent years, there have been careful experimentaland theoretical studies aimed at understanding thestructural aspects of this transition. The presence andnature of dynamical heterogeneities near the glass tran-sition have been a dominant underlying theme of bothsimulations [2,3] and experiments [4,5]. Simulations inLennard-Jones liquids [2] have shown the existence ofstringlike dynamical heterogeneities, and similar struc-tures have been observed directly in colloidal glasses [5].In the Adam-Gibbs scenario, the glass transition is re-lated to a phase transition accompanied by the vanishingof configurational entropy [6,7]. An explicit connectionbetween (a) dynamical heterogeneities, (b) anomalousrelaxations, and (c) the Adam-Gibbs scenario wouldprovide useful insight into the nature of the glass transi-tion. In this paper, we present our analysis of a simplemodel where there are naturally occurring dynami-cal heterogeneities in the form of strings and wherethere is an entropy-vanishing transition involving thesestructures. Monte Carlo simulations of the model showthat there is a glasslike transition with diverging timescales and an anomalously broad relaxation spectrum.Analysis of the simulation results provides strong evi-dence that the entropy-vanishing transition underlies theobserved dynamical behavior.

Model.—One of the simplest nonrandomly frustratedspin models is the triangular-lattice Ising antiferromagnet(TIAFM). The TIAFM has an exponentially large numberof ground states and has a zero-temperature critical point[8–10]. The model studied in this Letter is the compress-ible TIAFM (CTIAFM) in which the coupling of the spinsto the elastic strain fields removes the exponential degen-eracy of the ground state. We solve the CTIAFM exactlywithin the ground-state ensemble of the TIAFM and showthat there is an entropy-vanishing transition which involvesextended structures. We then present results of simulationswhich indicate that the entropy-vanishing transition leadsto glassy dynamics.

The Hamiltonian of the CTIAFM is

0031-9007�01�86(10)�2058(4)$15.00

H � JX

�ij�SiSj 2 eJ

X

a

ea

X

�ij�a

SiSj 1 NE2

X

a

e2a .

(1)

Here J , the strength of the antiferromagnetic coupling,is modulated by the presence of the second term whichdefines a coupling between the spins and the homoge-neous strain fields ea , a � 1, 2, 3, along the three nearest-neighbor directions on the triangular lattice. The last termstabilizes the unstrained lattice. The total number of spinsin the system is given by N . The ground state of theCTIAFM is a threefold degenerate striped phase, where upand down spins alternate between rows and there is a sheardistortion characterized by e1 � e and e2 � e3 � 2e ifthe rows are along the direction of e1 [11,12]. Withinthe manifold of the TIAFM ground states, the competitionbetween energy gained from the lattice distortions and theextensive entropy of the TIAFM ground states leads to anentropy-vanishing transition. The order parameter asso-ciated with this entropy-vanishing transition is the den-sity of extended stringlike structures which characterizethe TIAFM ground states.

The string picture of the TIAFM ground states derivesfrom a well-known mapping of these states to dimer cov-erings [13,14]. In a ground state, there is one unsatis-fied bond per triangular plaquette and the dimers are thefilled-in bonds of the dual honeycomb lattice that crossthe unsatisfied bonds of the triangular lattice, as shownin Fig. 1. Superposing a dimer configuration on a “stan-dard” dimer configuration where all the dimers are ver-tical [13,14], leads to a string configuration (cf. Fig. 1).Assuming spin-flip dynamics for the moment, the onlyspins that can be changed while the system remains in theground-state manifold are the ones which have a coordi-nation of 3-3 (3 satisfied and 3 unsatisfied bonds). Theseare the fast spins in the system, and, as shown in Fig. 1,are located at isolated kinks on the strings. The strings,therefore, play the role of dynamical heterogeneities in thislattice model.

Exact results.—We can solve the CTIAFM exactlywithin the restricted spin ensemble of the ground statesof the TIAFM. All the states in this ensemble can be

© 2001 The American Physical Society


FIG. 1. String representation of a TIAFM ground state. Darkbonds of the dual honeycomb lattice (cf. text) are the dimerswhich divide unsatisfied pairs of spins. Light bonds define thestandard dimer configuration. The strings are made up of thesetwo types and extend across the system. The fast spins, the oneswith 3-3 coordination, have been encircled.

classified according to the string density p � Ns�L, whereL is the linear dimension of the sample and Ns is thenumber of strings [13]. The number of spin states�V�p�� belonging to a particular string-density sector phas been shown to be exponentially large [13]: V�p� �exp�Ng�p��, with N � L 3 L being the total number ofspins. As shown in Fig. 2, the entropy g�p� has a peakat p � 2�3 [13]. In the CTIAFM [Eq. (1)], the strain ea

couples to �SiSj�a . This average counts the number ofstrings along the a direction, i.e., the number of stringsobtained by taking an overlap with the standard dimerstate with all the dimers perpendicular to the a direction.Under periodic boundary conditions, only one of the threestring densities is independent.

The strain field appears in the Hamiltonian as a purelyGaussian variable and can be integrated out, and theCTIAFM energy per spin, in the restricted ensemble, canbe written in terms of the one, independent string densityp:

E�p� � 2�m�2� ��1 2 2p�2 1 2�1 2 p�2� . (2)

Here m � e2J2�E. The energy function E�p� distin-guishes different string sectors and is minimized by p � 0.The entropy of the ground states, g�p�, on the other hand,favors the p � 2�3 sector and the competition betweenenergy and entropy leads to the possibility of a phase tran-sition. In the thermodynamic limit, the partition functionZ �

Pp e2Nf� p�, is dominated by the string density which

minimizes f�p� � bE�p� 2 g�p�, where b is the in-verse temperature. The exact free energy corresponds tothis minimum value of f�p� and the only relevant couplingconstant in the problem is bm. For small values of thecoupling constant, the function f�p� shown in Fig. 2(b)has only one minimum at p � 2�3. As the coupling con-stant is increased, this minimum stays pinned at 2�3 anda second minimum starts developing at p � 0. The p �0 state stays metastable until at m�T1 � �3�4�g�2�3� �

0.0 0.4 0.8p0.0

0.1

0.2

0.3

0.4

γ(p)

0.0 0.4 0.8

p

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

f(p)

T=0.45T=0.50T=0.60T=0.90T=1.20

(b)

(a)

FIG. 2. (a) Entropy as a function of the string density from thework of Dhar et al. [13] (b) The dimensionless free energy f�p�for m � 0.18. T � for this value of m is 0.397. Temperature ismeasured in units of 1�kB.

0.24 there is a first-order transition from the p � 2�3 stateto the p � 0 state. The p � 2�3 state loses its metasta-bility at a larger coupling given by m�T� �

p3 p�12. At

T�, the entropy vanishes and the order parameter shows adiscontinuous change from p � 2�3 to p � 0. This tran-sition is reminiscent of the transitions observed in p-spinspin glasses [7,15].

These exact results show that, in the CTIAFM, thereis an entropy-vanishing transition which defines the limitof stability of the homogeneous, liquidlike p � 2�3 state.In the Adam-Gibbs scenario, such a transition underliesthe glass transition. The exact results are valid for theCTIAFM acting within the ground-state ensemble of theTIAFM. Defects [16], which correspond to triangles withall three bonds unsatisfied, can take the system out of theground-state manifold. For low defect densities, however,it is possible that the entropy-vanishing transition survivesin some form and leads to slow glassy dynamics. Wehave investigated this scenario by performing Monte Carlosimulations of the CTIAFM.

Simulations.—The parameters of the model were cho-sen to be J � 1, e � 0.6, and E � 2. These values yield asmall value of m �� 0.18� and ensures that at the transition,T� �� 0.397�, the defect density is low. The average defectnumber density was measured to be �0.04% at T � 0.45.We used Monte Carlo simulations to study the dynamicsof the supercooled state following instantaneous quenchesto temperatures below T1 � 0.75. Spin-exchange kineticswas extended to include moves which attempted changesof the strain fields ea . Details of the simulation algorithmwere published earlier [12]. System sizes ranged from

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48 3 48 up to 120 3 120. Unless otherwise stated, theresults presented in this letter were obtained from 96 3 96systems.

As the glass transition is approached, global quantitiessuch as the energy per spin should exhibit anomalouslyslow relaxation processes. Figure 3 shows the energyautocorrelation function CE�t, t0� � �E�t0�E�t 1 t0��.For quench temperatures between T � 0.6 and T � 0.47,CE�t, t0� is independent of the time origin t0 and has astretched exponential form; e2�t�tE�b

. The stretching expo-nent b decreases from 0.45 to 0.35 over this temperaturerange and the time scale tE increases rapidly (cf. Fig. 3).Below T � 0.47, the energy autocorrelation functiondepends on the time origin t0, indicating that the equi-libration times have become longer than our observationtimes. To illustrate the dependence on the waiting timet0, we have shown the autocorrelation function averagedover three different ranges of t0 at T � 0.45. The systemis seen to relax more slowly for longer waiting times t0.This behavior of the energy autocorrelation function issimilar to that of supercooled liquids, and the temperatureT � 0.47 is analogous to the laboratory glass transitiontemperature at which the equilibration time becomeslonger than the observation time. In the CTIAFM, the

0 200 400 600 800 1000time(MCS)

0.0

0.2

0.4

0.6

0.8

1.0

CE(t

)

0 200 400 600 800 10000.0

0.2

0.4

0.6

CE(t

)

T=0.48T=0.50T=0.52

T=0.45

FIG. 3. Top frame shows CE�t, t0� for three different tempera-tures where CE�t, t0� does not depend on t0. The solid linesare stretched exponential fits. The bottom frame shows thewaiting-time dependence of the energy autocorrelation func-tion at T � 0.45. The curves were obtained by averagingCE�t, t0� over three different ranges of t0. From bottom to top,these ranges are 0 , t0 , 25 000, 18 000 , t0 , 48 000, and50 000 , t0 , 80 000.

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proximity of this transition to T� suggests that the glassydynamics is related to the entropy-vanishing transition.

The microscopic picture of the entropy-vanishingtransition is one where the string density vanishes. Wefind that the string-density autocorrelation functions arewell described by exponential relaxations, in contrastto the energy autocorrelation functions. Figure 4 showsthe results of our simulations for the relaxation timesand fits to a power law and a Vogel-Fulcher form [17].Both fits yield a time-scale divergence at a temperatureT � T�. The tE obtained from the stretched exponentialfits to CE�t, t0� has a temperature dependence whichtracks that of the string relaxation time. This observationsuggests that the slow, nonexponential relaxations are aconsequence of the freezing of the string-density relax-ation which, in turn, is related to the entropy-vanishingtransition. The static susceptibility associated with thestring density changes only by a factor �2 over the tem-perature range in which the time scales change by a factor�40, indicating that the entropy-vanishing transitionsuffers from anomalously strong critical slowing.

In order to further investigate the nature of the string re-laxations, we measured the distribution P�Dp�, of Dp �p�t 1 t0� 2 p�t0�, the deviation of the string density inthe time interval t (Fig. 5). The most striking feature ofthe distributions, observed at temperatures close to T �T�, is its non-Gaussian nature at intermediate times. AtT � 0.55, the non-Gaussian feature is most pronouncedat t � 4000. Beyond this time the distribution relaxes to-wards a Gaussian and, for t $ 8000, the distribution is

0.45 0.50 0.55 0.60 0.65

T(1/kB)

0

5000

10000

15000

20000

τ(MC

S)

e0.1/(T−0.44)

(T−0.41)(−3.3)

FIG. 4. Temperature dependence of the string-density relax-ation time. The simulation results are shown with error bars.The solid line is a fit to the Vogel-Fulcher form and the dashedline is a fit to a power law.


−0.05 −0.03 −0.01 0.01 0.03 0.0510

−4

10−3

10−2

P(a

rbit

rary

uni

ts)

1000200040008000

−0.05 −0.03 −0.01 0.01 0.03 0.05

∆p10

−4

10−3

10−2

200080001200030000

T=0.55

T=0.47

t:

t:

FIG. 5. Distribution of string-density deviation Dp for differ-ent time intervals, t, shown at T � 0.55 and T � 0.47. Theareas under the curves have been normalized to unity.

time independent. At T � 0.47, a time-independent be-havior is not observed for times as long as 30 000 and thedistributions are non-Gaussian at all intermediate times.In usual critical-point dynamics [18], one would expect tofind a distribution of Dp, which takes longer to reach itstime-independent form as the critical point is approachedand to find non-Gaussian behavior (within the limits offinite-size cutoffs) in the stationary distribution. In con-trast, we observe the most pronounced non-Gaussian fea-tures at intermediate times. Drawing an analogy withcritical phenomena, this observation leads us to speculatethat there exists a time-dependent length scale, j�t�, whichhas a peak at a time t0�T �. As T ! T�, the time scale t0and the height of the peak j�t0� appear to diverge. The ex-act nature of the divergence is difficult to extract from thecurrent data. These apparent divergences indicate that thethermodynamic transition present in the zero-defect sectorhas been replaced by a dynamical transition [19].

Connection to real glasses.— In conclusion, our studyof the CTIAFM provides strong indication that the slow,glassy dynamics in this model is associated with anentropy-vanishing transition involving extended, stringlikestructures. These structures are a manifestation of thefrustration embodied in the nearest-neighbor, antiferro-magnetic interactions, and the entropy-vanishing transitionis a consequence of coupling to another degree of freedom,the lattice strain, which tends to remove the frustration in

the system. The strings are naturally occurring dynamicalheterogeneities since they demarcate regions of fast spinflips. If the relation between frustration and dynamicalheterogeneities is a generic feature of glass formers,then our observations would suggest that the clusters ofmobile particles observed in Lennard-Jones simulations[2] and in colloidal systems [5] should consist of the mostfrustrated particles in the system. In a Lennard-Jonesmixture, these should be the particles with the leastnumber of unlike bonds (if unlike bonds are energeticallypreferred), and, in a colloidal system, these should be theparticles which have coordinations that are farthest frombeing icosahedral. An experimental verification of thiscorrelation between geometry and mobility would be adirect test of the connection between the glass transitionand an entropy-vanishing transition involving extendedstructures forced in by frustration.

The work of B. C. was supported in part by NSF GrantNo. DMR-9815986, and the work of H. Y. was supportedby DOE Grant No. DE-FG02-ER45495. We would like tothank R. K. P. Zia, W. Klein, H. Gould, S. R. Nagel, andJ. Kondev for many helpful discussions.

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entropy-vanishing transition and glassy dynamics in frustrated spins

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