Entropy-vanishing transition and glassy dynamics in frustrated spins

4 Pages·0.16 MB·English

by  Hui Yin

Entropy-vanishing transition and glassy dynamics in frustrated spins
Hui Yin and Bulbul Chakraborty
Martin Fisher School of Physics, Brandeis University, Waltham, MA 02454.
(February 1, 2008)
1
0
0 state. We solvethe CTIAFMexactlywithin the ground-
2 In an effort to understand the glass transition, the dy- state ensemble of the TIAFM and show that there is
namics of a non-randomly frustrated spin model has been
n an entropy-vanishing transition which involves extended
analyzed. Thephenomenologyofthespinmodelissimilarto
a structures. We then present results of simulations which
J that of a supercooled liquid undergoing the glass transition.
indicate that the entropy-vanishing transition leads to
Theslowdynamicscanbeassociatedwiththepresenceofex-
2
glassy dynamics.
1 tended string-like structures which demarcate regions of fast
spin flips. An entropy-vanishing transition, with the string The Hamiltonian of the CTIAFM is:

h] density as the order parameter, is related to the observed H =JX SiSj ǫJX eαX SiSj
glass transition in the spin model. <ij> − α <ij>α
c
E
me +N 2Xαe2α . (1)
- Here J, the strength of the anti-ferromagnetic cou-
at pling, is modulated by the presence of the second term
t which defines a coupling between the spins and the ho-
s The glass transition in supercooled liquids is heralded
t. byanomalouslyslowrelaxationswithatimescalediverg- mogeneous strain fields eα,α=1,2,3, along the three
a nearest-neighbordirectionsonthetriangularlattice. The
m ingastheliquidfreezesintotheglassystate[1]. Inrecent
last term stabilizes the unstrained lattice. The total
years,therehavebeencarefulexperimentalandtheoreti-
- number of spins in the system is given by N. The
d calstudiesaimedatunderstandingthestructuralaspects
ground-state of the CTIAFM is a three-fold degener-
n of this transition. The presence and nature of dynami-
o cal heterogeneities near the glass transition has been a ate striped phase where up and down spins alternate
c between rows and there is a shear distortion charac-
dominantunderlyingthemeofbothsimulations[2,3]and
[ terized by e = e and e = e = e if the rows are
2 e[2x]phearvimeesnhtosw[n4,t5h].eeSximistuelnactieoonfssitnrinLge-nlnikaerdd-yJnoanmesiclaiqluhiedts- along the di1rection of e12[11,123]. W−ithin the manifold
v of the TIAFM ground-states, the competition between
erogeneities and similar structures have been observed
3 energy gained from the lattice distortions and the ex-
8 directly in colloidal glasses [5]. In the Adam-Gibbs sce-
tensive entropy of the TIAFM ground-states leads to an
4 nario, the glass transition is related to a phase transi-
entropy-vanishing transition. The order parameter asso-
4 tionaccompaniedby the vanishingofconfigurationalen-
0 tropy [6,7]. An explicit connection between (a) dynam- ciated with this entropy-vanishing transition is the den-
0 sity of extended string-like structures which characterize
ical heterogeneities, (b) anomalous relaxations and (c)
0 the TIAFM ground states.
the Adam-Gibbs scenario would provide useful insight
/
t ThestringpictureoftheTIAFMgroundstatesderives
a into the nature of the glass transition. In this work,
froma well-knownmappingofthese states to dimer cov-
m we present our analysis of a simple model where there
erings [13,14]. In a ground state, there is one unsatisfied
- are naturally occurring dynamical heterogeneities in the
bond per triangular plaquette and the dimers are the
d form of strings and where there is an entropy vanish-
n filled-in bonds of the dual honeycomb lattice that cross
ing transition involving these structures. Monte Carlo
o the unsatisfied bonds of the triangular lattice, as shown
simulations of the model show that there is a glass-like
c inFig.(1). Superposingadimerconfigurationona“stan-
: transitionwithdivergingtimescalesandananomalously
v dard” dimer configuration where all the dimers are ver-
broadrelaxationspectrum. Analysisofthesimulationre-
i tical [13,14] leads to a string configuration (cf Fig.(1)).
X sultsprovidesstrongevidencethattheentropy-vanishing
Assuming spin-flip dynamics for the moment, the only
r transition underlies the observed dynamical behavior.
a spins that can be changed while the system remains in
a. Model One of the simplest non-randomly frus-
the ground-statemanifold,arethe ones whichhavea co-
trated spin models is the triangular-lattice Ising anti-
ordination of 3-3 (3 satisfied and 3 unsatisfied bonds).
ferromagnet (TIAFM). The TIAFM has an exponen-
These are the fast spins in the system, and, as shown in
tially large number of ground states and has a zero-
Figure (1), are located at isolated kinks on the strings.
temperature critical point [8–10]. The model studied
Thestrings,therefore,playtheroleofdynamicalhetero-
in this letter is the compressible TIAFM (CTIAFM)
geneities in this lattice model.
in which the coupling of the spins to the elastic strain
b. Exact results We can solve the CTIAFM exactly
fields removesthe exponentialdegeneracyofthe ground-
within the restricted spin ensemble of the ground states
1
of the TIAFM. All the states in this ensemble can be the coupling constant is increased, this minimum stays
classifiedaccordingtothestringdensityp=N /Lwhere pinned at 2/3 and a second minimum starts developing
s
L is the linear dimension of the sample and N is the at p = 0. The p = 0 state stays metastable until at
s
number ofstrings[13]. The numberofspinstates(Ω(p)) µ/T = (3/4) γ(2/3) 0.24 there is a first-order tran-
1
∗ ≃
belongingtoaparticularstring-densitysectorphasbeen sition from the p = 2/3 state to the p = 0 state. The
showntobeexponentiallylarge[13]: Ω(p)=exp(Nγ(p)), p=2/3 state loses its metastability at a larger coupling
with N = L L being the total number of spins. As given by µ/T∗ = √3π/12. At T∗, the entropy vanishes
×
showninFig. (2),theentropyγ(p)hasapeakatp=2/3 and the order parameter shows a discontinuous change
[13]. In the CTIAFM (Eq. 1), the strain e couples to from p = 2/3 to p = 0. This transition is reminiscent of
α
S S . Thisaveragecountsthenumberofstringsalong the transitions observed in p-spin spin glasses [7,16].
i j α
h i
the α direction, i.e. the number of strings obtained by
taking an overlap with the standard dimer state with −0.1
all the dimers perpendicular to the α direction. Under (b) T=0.45
T=0.50
periodicboundaryconditions,onlyoneofthethreestring −0.2 T=0.60
T=0.90
densities is independent. T=1.20
−0.3
p)
f(
−0.4
0.4
0.3 (a)
−0.5 γ(p) 00..12
0.0
0.0 0.4 p 0.8
−0.6
0.0 0.4 0.8
p
FIG.2. (a)Entropyasafunctionofthestringdensityfrom
theworkofDharetal.([13])(b)Thedimensionlessfreeenergy
f(p)forµ=0.18. T∗forthisvalueofµis0.397. Temperature
is measured in unitsof 1/kB.
FIG. 1. String representation of a TIAFM ground state. These exact results show that in the CTIAFM, there
Dark bonds of the dual honeycomb lattice (cf text) are the isanentropyvanishingtransitionwhichdefinesthelimit
dimers which divide unsatisfied pairs of spins. Light bonds ofstabilityofthehomogeneous,liquid-likep=2/3state.
define the “standard” dimer configuration. The strings are In the Adam-Gibbs scenario, such a transition underlies
made up of these two types and extend across the system. the glass transition. The exact results are valid for the
The fast spins, the ones with 3−3 coordination, have been CTIAFMactingwithintheground-stateensembleofthe
encircled.
TIAFM.Defects[15],whichcorrespondtotriangleswith
all three bonds unsatisfied, can take the system out of
ThestrainfieldappearsintheHamiltonianasapurely
theground-statemanifold. Forlowdefectdensities,how-
Gaussian variable and can be integrated out, and the
ever, it is possible that the entropy-vanishing transition
CTIAFMenergyperspin,in the restrictedensemble,can
survivesinsomeformandleadstoslowglassydynamics.
bewrittenintermsoftheone,independentstringdensity
We have investigated this scenario by performing Monte
p:
Carlo simulations of the CTIAFM.
c. Simulations The parameters of the model were
E(p)= (µ/2)((1 2p)2+2(1 p)2)) . (2)
− − − chosen to be J = 1, ǫ = 0.6 and E = 2. These val-
Here µ = ǫ2J2/E. The energy function E(p) distin- ues yield a small value of µ (= 0.18) and ensures that
∗
at the transition, T (= 0.397), the defect density is
guishesdifferentstringsectorsandisminimizedbyp=0.
low. The average defect number-density was measured
Theentropyofthegroundstates,γ(p),ontheotherhand
to be 0.04% at T = 0.45. We used Monte Carlo sim-
favors the p = 2/3 sector and the competition between
≃
ulations to study the dynamics of the supercooled state
energy and entropy leads to the possibility of a phase
following instantaneous quenches to temperatures below
transition. In the thermodynamic limit, the partition
function Z = exp−Nf(p), is dominated by the string T1 = 0.75. Spin-exchange kinetics was extended to in-
Pp
cludemoveswhichattemptedchangesofthe strainfields
density which minimizes f(p) = βE(p) γ(p), where β
− e . Details of the simulation algorithm have been pub-
is the inverse temperature. The exact free energy corre- α
lished earlier[12]. System sizes rangedfrom 48x48up to
sponds to this minimum value of f(p) and the only rel-
120x120. Unless otherwise stated, the results presented
evant coupling constant in the problem is βµ. For small
in this letter were obtained from 96x96 systems.
values of the coupling constant, the function f(p) shown
Astheglasstransitionisapproached,globalquantities
in Fig (2(b)), has only one minimum at p = 2/3. As
2
such as the energy per spin should exhibit anomalously the glass transition. We find that the string-density au-
slow relaxation processes. Fig (3) shows the energy au- tocorrelation functions are well described by exponen-
tocorrelation function C (t,t ) =< E(t )E(t + t ) >. tial relaxations, in contrast to the energy autocorrela-
E 0 0 0
For quench temperatures between T = 0.6 and T=0.47, tion functions. Fig. (4) shows the results of our simu-
C (t,t ) is independent of the time origin t and has a lations for the relaxation times and fits to a power-law
E 0 0
stretchedexponentialform; exp−(t/τE)β. The stretching- and a Vogel-Fulcher form [17]. Both fits yield a time-
∗
exponentβ decreasesfrom0.45to0.35overthistemper- scale divergence at a temperature T T . The τE ob-
≃
ature range and the time scale τE increases rapidly as is tained from the stretched exponential fits to CE(t,t0)
evident from the top panel of Fig (3). Below T 0.47, has a temperature dependence which tracks that of the
≃
the energy autocorrelationfunction depends on the time string relaxation time. This observation suggests that
origint ,indicatingthattheequilibrationtimeshavebe- the slow, non-exponential relaxations are a consequence
0
come longer than our observation times. To illustrate of the freezing of the string-density relaxation which, in
the dependence on the waiting time t , we have shown turn, is related to the entropy vanishing transition. The
0
the autocorrelation function averaged over three differ- static susceptibility associated with the string density
ent ranges of t at T =0.45. The system is seen to relax changesonly by a factor 2 overthe temperature range
0 ≃
more slowly for longer waiting times t . This behavior in which the time scales change by a factor 40, indi-
0 ≃
of the energy autocorrelation function is similar to that cating that the entropy-vanishing transition suffers from
of supercooled liquids and the temperature T = 0.47 is anomalously strong critical slowing down.
analogousto the laboratoryglasstransitiontemperature
at which the equilibration time becomes longer than the 20000
observationtime. In the CTIAFM, the proximity of this
transition to T∗ suggests that the glassy dynamics is re- 15000 e0.1/(T−0.44)
lated to the entropy vanishing transition. (T−0.41)(−3.3)
S)
C 10000
M
τ(
0.6
5000
T=0.48
T=0.50
0.4 T=0.52
C(t) E 00.45 0.50 T(01.5/5k ) 0.60 0.65
0.2 B
FIG.4. Temperature dependence of the string-density re-
laxation time. The simulation results are shown with error
0.00 200 400 600 800 1000 bars. Thesolid lineisafittotheVogel-Fulcherform andthe
dashed line to a power-law.
1.0
T=0.45 In order to investigate the nature of the string-
0.8
relaxations further, we measured the distribution,
0.6
C(t)E 0.4 Pstr(∆inpg),deonfs∆ityp =in pt(hte+titm0)e−inpt(etr0v)a,ltthe(Fdiegv.iat(i5o)n).ofTthhee
distributions were generated by choosing different time
0.2
originst . Themoststrikingfeatureofthedistributions,
0
0.0 observed at temperatures close to T = T∗, is its non-
0 200 400 600 800 1000
time(MCS) Gaussiannatureatintermediatetimes. AtT =0.55,the
non-Gaussian feature is most pronounced at t = 4000.
FIG.3. Top frame shows CE(t,t0) for three different tem-
peratures where CE(t,t0) does not depend on t0. The solid BeyondthistimethedistributionrelaxestowardsaGaus-
lines are stretched exponential fits. Bottom frame shows the sian and for t 8000, the distribution is time indepen-
≥
waiting-time dependence of the energy autocorrelation func- dent. At T = 0.47, a time-independent behavior is not
tion at T = 0.45. The curves were obtained by averaging observed for times as long as 30,000 and the distribu-
CE(t,t0) over three different ranges of t0. From bottom to tionsarenon-Gaussianatallintermediatetimes. Inusual
top, these ranges are 0 < t0 < 25000, 18000 < t0 < 48000 critical-point dynamics [18], one would expect to find a
and 50000<t0<80000. distribution of ∆p which takes longer to reach its time-
independent formas the criticalpointis approachedand
Themicroscopicpictureoftheentropy-vanishingtran-
tofindnon-Gaussianbehavior(withinthelimitsoffinite-
sition is one where the string density vanishes. An anal-
size cutoffs) in the stationary distribution. In contrast,
ysis of the string-density relaxation can, therefore pro-
we observe the most pronounced non-Gaussian features
vide some insight into the nature of the dynamics at
at intermediate times. Drawing an analogy with critical
3
phenomena, this observation leads us to speculate that volving extended structures forced in by frustration.
there exists a time-dependent length scale, ξ(τ) has a The work of BC was supported in part by NSF grant
∗
peak at a time τ (T). As T T , the time scale τ and number DMR-9815986 and the work of HY was sup-
0 0
→
theheightofthepeak,ξ(τ )appeartodiverge. Theexact portedbyDOEgrantDE-FG02-ER45495. Wewouldlike
0
nature of the divergence is difficult to extract from the to thank R. K. P. Zia, W. Klein, H. Gould, S. R. Nagel
current data. These apparent divergences indicate that and J. Kondev for many helpful discussions.
the thermodynamic transition present in the zero-defect
sector has been replaced by a dynamical transition [19].
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ity would be a direct test of the connection between the
glass-transition and an entropy-vanishing transition in-
4