Table Of ContentDynamic Approach to the
Fully Frustrated XY Model∗
H.J. Luo, L. Schu¨lke and B. Zheng∗
Universit¨at – GH Siegen, D – 57068 Siegen, Germany
∗Universit¨at Halle, D – 06099 Halle, Germany
8 Using Monte Carlo simulations, we systematically investigate the non-equilibrium dynamics of
9 the chiral degree of freedom in the two-dimensional fully frustrated XY model. The critical initial
9 increase of the staggered chiral magnetization is observed. By means of the short-time dynamics
1 approach, we estimate the second order phase transition temperature Tc and all the dynamic and
static critical exponentsθ, z, β and ν.
n
a
J PACS:64.60.Cn, 75.10.Hk, 64.60.Ht, 75.40.Mg
6
2
Forcriticalphenomena,itistraditionallybelievedthat One of the most prominent properties of the scaling
] universal behaviour exists only in equilibrium or in the forms (1) and (2) is that the exponents β, ν and z take
t long-timeregimeofthedynamicalevolution. Theuniver- the same values as in equilibrium or in the long-time
f
o salscalingbehaviouris describedby a number ofcritical regime of the dynamic evolution. It has been suggested
s exponents. Duetocriticalslowingdown,numericalmea- thatitispossibletodeterminenotonlythedynamicbut
.
t surements of the critical exponents are very difficult. alsoallthestaticexponentsaswellasthecriticaltemper-
a
m Recently,muchprogresshasbeenachievedindynamic aturealreadyintheshort-timeregime[5,9](seealsoRefs.
criticalphenomena. Itwasdiscoveredthat, alreadyafter [11–13,6]). The method may be an alternative way for
-
d amicroscopictimescalet ,universalscalingbehaviour overcomingcriticalslowing down since the measurement
mic
n emerges in the macroscopic short-time regime of the dy- does not enter the long-time regime. It is important to
o namic process [1–6]. To illustrate this, we consider the systematicallyverifythisapplicationingeneralandcom-
c
followingdynamicrelaxationprocess: amagneticsystem plex models.
[
initially in a high-temperature state with a small initial The two-dimensional fully frustrated XY (FFXY)
1 magnetizationm0isquenchedtothecriticaltemperature modelhas been the topic of manyrecentstudies [14–22].
v Tc without an external magnetic field and then released Critical properties of this model are rather unconven-
3
to a dynamic evolution with model A dynamics [7]. At tional. On the square lattice, the model has two kinds
5
theonsetoftheevolution,themagnetizationissubjected of phase transitions, i.e. the Kosterlitz-Thouless phase
2
1 to the scaling form [1,4,5,8,9] transition (XY-like) and the second order phase tran-
0 sition (Ising-like). Numerical simulations of the FFXY
8 M(t,τ,m )∼m tθF(t1/νzτ). (1) model suffer severely from critical slowing down. Due
0 0
9 to the frustration, the standard cluster algorithm does
/
t The exponent θ is a new independent exponent, τ ∼ not apply to the FFXY model. Most of the recent work
a
(T −T )/T is the reducedtemperature, β and ν are the on FFXY models supports that these two phase tran-
m c c
static criticalexponents, andz is the dynamicexponent. sitions take place at two different temperatures, how-
- At the critical temperature, τ = 0, the magnetization ever, their critical properties are still not very clear. For
d
n undergoes a critical initial increase M(t)∼tθ. example, for the second order phase transition the es-
o In fact, the short-time dynamic scaling is very gen- timated values of the exponents β and ν differ in the
c eral. Another important example is the dynamic relax- literature [22,21,18,17] and the critical dynamics has
:
v ationofamagneticsystemstartingfromanorderedstate not been investigated. It is still a matter of contro-
i (m = 1) [10,9,11,6]. The scaling form for this dynamic versy whether the chiral degree of freedom of the FFXY
X 0
process is given by modelisinthesameuniversalityclassastheIsingmodel
r [23,22,21,24,25].
a
M(k)(t,τ,L)=b−kβ/νM(k)(b−zt,b1/ντ,b−1L), (2) In this letter we present results of systematic Monte
Carlo simulations for the short-time dynamic behaviour
where, for later convenience, a system with finite size L of the second order phase transition in the two-
hasbeenconsidered. Thisfinitesizescalinghasthesame dimensional FFXY model. We confirm the short-time
form as in the long-time regime but is assumed valid in dynamic scaling. For the first time, we confidently de-
the macroscopic short-time regime. termine the dynamic exponents θ and z. Based on the
∗
Work supported in part by theDeutscheForschungsgemeinschaft; DFG Schu95/9-1 and SFB418
1
short-timedynamicscaling,thestaticexponentsβ andν on the lattice, then we randomly choose a number of
as well as the critical temperature T are also extracted plaquettes andorienttheir spins accordingto the config-
c
from the numerical data. urationofthegroundstateshowninFig.1tilltheinitial
The Hamiltonian of the FFXY model on a square lat- magnetization m is achieved. After the initial configu-
0
tice can be written as rationisgenerated,thesystemisreleasedtothedynamic
evolutionwiththeMetropolisalgorithmattemperatures
H =−K cos(θi−θj +Aij), (3) aroundTc. Wehaveperformedoursimulationonlattices
<ij> of size L = 128 and 256, and updated the system up to
X
1000 Monte Carlo steps. The average is taken over in-
In our notation the factor 1/kT has been absorbed in
dependent initial configurations with 40000 samples for
thecouplingK,θ isthe angleofthespin(aunitvector)
i L = 128 and 10000 samples for L = 256. Errors are
located on site i, A determine the frustration and the
ij estimated by dividing the samples into three groups.
sum is over the nearest neighbours. A simple realization
of the FFXY model is by taking A = π on half of the
ij
vertical links and A = 0 on other links. This is shown
ij
in Fig. 1, where the links with A = π are marked by 0.1
ij
dotted lines. The spin configuration of one ground state
of the model is also shown in Fig. 1. The order parame-
ter for the secondorderphase transitionis the staggered
chiral magnetization defined as [19]
M (t)
I
T=0.452
1
M = (−1)rx+ry sgn sin(θ −θ +A )
I *L2 i j ij + T=0.456
Xr <ijX>∈Pr
(4)
where (r ,r ) is the coordinate of the plaquette P .
x y r
0.04
100 t 1000
FIG. 2. The time evolution of the chiral magnetization
MI(t)whenstartingfromadisorderedinitialstate. Thetem-
peratures are T = 0.452, 0.454, and 0.456 from above. The
dotted line represents themagnetization at Tc =0.4547.
In order to locate the critical temperature T , simu-
c
lations have been carried out with three temperatures
T = 0.452, 0.454 and 0.456. The initial magnetization
is set to m = 0.06. In Fig. 2, the time evolution of the
0
magnetizationM (t)atdifferenttemperatures is plotted
I
withsolidlinesinlog-logscaleforthelatticesizeL=128.
Datawithinthemicroscopictimescalet ∼100,which
mic
are dependent on microscopic details, are not included.
Indeed we observe that the magnetization increases at
the macroscopic early time. The magnetization at the
temperature between T = 0.452 and 0.456 can be ob-
tainedbyquadraticinterpolation. Fromthescalingform
FIG. 1. One of the ground states of the FFXY model. (1) and as suggested in Ref. [9], searching for a curve
Dotted lines denotethelinks with Aij =π. MI(t) with the best power law behaviour can yield an
estimate of the critical temperature. In Fig. 2, the dot-
At first, we investigate the short-time critical be- ted line represents such a curve and the corresponding
haviour of MI in the dynamic process starting from an critical temperature Tc = 0.4547(8). From the slope of
initial state with a very high temperature and a small this curve, the exponent θ =0.200(3) is obtained.
magnetizationm . Weprepareaninitialconfigurationin From an analysis of the data for L=256 we have ob-
0
the following way: first we randomly generate all spins served that the finite size effect for L = 128 is already
2
sufficiently small to be neglected. In fact, the finite size 1/νz =0.59(3)isobtainedfrom∂ lnM (t,τ)| . How-
τ I τ=0
effect can easily be controlled in the short-time dynam- ever, this dynamic process starting from a disordered
ics, and this is an advantage of the short-time dynamic state is not the best choice to obtain these exponents or
approach. On the other hand, the critical exponent θ is the critical temperature T . A dynamic process starting
c
defined in the limit m → 0. Therefore, the finite m from an ordered state is preferable, since the fluctuation
0 0
effect should also be considered. However, within sta- is weaker.
tistical errors our data for m = 0.04 and m = 0.06 Forthispurpose,simulationswerealsoperformedwith
0 0
show no difference. Hence the finite m effect will also temperatures T = 0.452, 0.454 and 0.456, starting from
0
be ignored. an ordered initial state. The lattice size chosen was
In principle, other static and dynamic critical expo- L = 256 and the system was updated for 2000 Monte
nents can be obtained from ∂ lnM , the second mo- Carlo steps. The average was taken over 2000 samples.
τ I
ment and the fourth moment of the magnetization, the Theorderedinitialstatewastakentobethegroundstate
auto-correlation and other observables. For example, shown in Fig. 1.
1
M (t)
I
T=0.452
T=0.456
0.5
100 t 1000 2000 0.452 0.453 0.454 0.455 0.456
FIG.3. (a) The time evolutions of the chiral magnetizations MI(t) when starting from an ordered state. From above, the
solid lines represent the MI(t) at T=0.452, 0.454, and 0.456. The dotted line is themagnetization at thecritical temperature
Tc =0.4545. (b) Deviation of MI(t) from the power law behaviourwithin thetime interval [200,1000].
The estimation of T can now be performed again. [9]. In this paper, we choose to measure the deviation
c
From the scaling form (2) and for sufficiently large L as the error by fitting M (t) directly to a power law in
I
we can easily deduce the scaling behaviour for the mag- the time interval [200,2000]. Furthermore we perform
netization (k =1) the fitting in log scale, i.e. less weight is given to the
data in the longer time regime. In Fig. 3 (b), the devi-
M (t,τ)=t−β/νzG(t1/νzτ). (5) ation of M from the power law is plotted as a function
I I
of the temperature. The clear minimum confidently in-
Atthecriticaltemperature,τ =0,M undergoesapower dicates the critical temperature T . The resulting value
I c
lawdecay. Whenτ 6=0thispowerlawismodifiedbythe T = 0.4545(2) is consistent with T = 0.4547(8) ob-
c c
scaling function G(t1/νzτ). As pointed out earlier, the tainedfromFig.2 andverycloseto thosevalues ranging
temperature for which the magnetization has the best from T = 0.451 to 0.454 reported in most of the recent
c
power law behaviour is the critical temperature T . In references[17,18,21,22]. Ourstatisticalerror,however,is
c
Fig. 3 (a), the time evolution of M (t) at T = 0.452, smaller. Thecorrespondingmagnetizationisalsoplotted
I
0.454 and 0.456 is plotted in log-log scale. M (t) at in Fig. 3 (a) with a dotted line. The slope of this curve
I
other temperatures in the interval [0.452,0.456] can be yields the criticalexponent β/νz =0.0602(2). The qual-
estimated by a quadratic interpolation. To avoid the ef- ity of this measurement is very good. With β/ν given,
fect of t , measurements are performed in the time in- oncanobtainarigorousz orviceversa[10,26,11,9,6]. As
mic
terval [200,2000]. The deviation from the power law can compared to simulations with a disordered initial state,
be estimated in different ways, e.g. as described in Ref. the measurements here carry much less fluctuations.
3
10 10−2
10−3
1 ∂ lnM U(t)
τ I
10−4
100 t 1000 2000 100 t 1000 2000
FIG.4. (a) Thederivative∂τlnMI(t,τ)|τ=0 plottedversustimein log-log scale. (b) Thetime-dependentBindercumulant
U.
To extract the critical exponent 1/νz, differentiation cumulant U(t,L)=M(2)/M2−1. Due to the short spa-
I I
of Eq. (5) leads to tial correlationlength in the short-time regime, a simple
finite size scaling analysis shows at the critical tempera-
∂τ lnMI(t,τ)|τ=0 =t1/νz∂τ′lnG(τ′)|τ′=0. (6) ture
Therefore,∂τ lnMI(t,τ)|τ=0 shouldalsopresenta power U(t,L)∼td/z. (7)
law behaviour in the beginning of the time evolution. In
Fig. 4(a), ∂ lnM (t,τ) atT =0.4545is plottedin log- In Fig. 4 (b), the curve for U(t,L)in log-logscale shows
τ I c
log scale. The power law behaviour is clearly seen. The a nice power law behaviour. The slope gives the critical
slope yields the critical exponent 1/νz =0.57(1). exponent d/z =0.92(2). Table I summarizes the results.
The final step is to seek the dynamical critical expo- The errorsin the measurements from m =1 are clearly
0
nent z. For this, we introduce a time-dependent Binder smaller than those from m =0.06.
0
Tc θ β/νz 1/νz d/z
m0=1 0.4545(2) 0.0602(2) 0.57(1) 0.92(2)
m0=0.06 0.4547(8) 0.200(3) 0.59(3)
TABLEI. Thecritical temperatureandthecritical exponentsmeasured bysimulations starting from an ordered as wellas
a disordered state.
This work Ref. [22] Ref. [21] Ref. [18] Ref. [17] Ising
(1996) (1995) (1994) (1993)
Tc 0.4545(2) 0.451(1) 0.452(1) 0.454(2) 0.454(3)
ν 0.81(2) 0.898(3) 1 0.813(5) 0.80(5) 1
2β/ν 0.261(5) 0.22(2) 0.38(2) 0.25
z 2.17(4) 2.165(10)
θ 0.202(3) 0.191(3)
TABLEII. Critical exponentsobtained in thiswork and valuesreported in somerecent references. Reference[21] doesnot
provide an estimate of the error on ν = 1. For the Ising model, exponents ν and 2β/ν are exact values and θ is taken from
Refs. [8,27]. The exponent z in theliterature ranges from 2.155 to 2.172 [6,26,8,27]. Herean ‘average’ value is given.
4
The interesting and important property of the Binder
cumulantis thatone canestimate independently the dy-
namicexponentz. Withz inhand,wecalculatethecrit-
icalexponents2β/ν andν fromβ/νz and1/νz. TableII
listsallcriticalexponentsalongwiththeresultsreported
[1] H.K.Janssen,B.SchaubandB.Schmittmann,Z.Phys.
intherecentliterature. Nowθ =0.202(3)ismeasuredat
B 73 (1989) 539.
T =0.4545,whichshows a smalldifference from that at
c [2] D. A. Huse,Phys. Rev. B 40 (1989) 304.
T =0.4547. Ourshort-timedynamicmeasurementssup-
[3] K.HumayunandA.J.Bray,J.Phys.A24(1991) 1915.
portthosefromRef.[18]andprovideextranewresultsfor
[4] Z.B. Li, U. Ritschel and B. Zheng, J. Phys. A: Math.
the dynamic exponents z and θ. The exponent ν of the Gen. 27 (1994) L837.
FFXY modelis differentfromthatofthe Isingmodel by [5] L.Schu¨lkeandB.Zheng,Phys.Lett.A 204(1995) 295.
nearly 20 per cent. This indicates that the chiral degree
[6] B. Zheng, Monte Carlo Simulations of Short-time Crit-
of freedom of the FFXY model is in a new universality ical Dynamics, Halle Univ., 1998, review article, to be
class. Other exponents of the FFXY model do not differ published in Int.J. Mod. Phys. B.
muchfromthoseofthe Isingmodel. Fromournumerical [7] P.C. Hohenberg and B.I. Halperin, Rev. Mod. Phys. 49
data,weobservethatthevaluesofthecriticalexponents (1977) 435.
are rather sensitive to the assumed or measured critical [8] P. Grassberger, Physica A 214 (1995) 547.
temperature T . This should be one of the main reasons [9] L. Schu¨lke and B. Zheng, Phys. Lett A 215 (1996) 81.
c
for the different values of the exponents reported in the [10] D. Stauffer, Physica A 186 (1992) 197.
literature. Figure 2 and, in particular, Fig. 3, give us [11] D.StaufferandR.Knecht,Int.J.Mod.Phys.C7(1996)
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c
the data in the close neighbourhood of T , in contrast 2940.
c
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In conclusions, using Monte Carlo methods, we have
[15] D.B. Nicolaides, J. Phys. A24 (1991) L231.
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[16] G. Ramirez-Santiago and J. Jos´e, Phys. Rev. Lett. 68
short-time dynamic behaviour of the second order phase
(1992) 1224.
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[17] E.GranatoandM.P.Nightingal,Phys.Rev.B48(1993)
model. Basedontheshort-timedynamicscalingform,all
7438.
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(1994) 9567.
critical temperature Tc is very precise. The estimated [20] S.J.Lee,J.R.LeeandB.Kim,Phys.Rev.E 51(1995)
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Isingmodel. Ourinvestigationofthechiraldegreeoffree- [21] P. Olsson, Phys.Rev. Lett. 75 (1995) 2758.
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culties associated with large correlation times since our [25] G.S.Jeon,S.Y.ParkandM.Y.Choi,Phys.Rev.B 55
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615.
ponents have been obtained [28,29]. However, owing to
[29] H.J.Luo,L.Schu¨lkeandB.Zheng,Universal Short-time
the absence of symmetry breaking,a clear signalsuchas
BehaviouroftheDynamicFullyFrustrated XYModel,to
in Fig. 3 does not exist for the Kosterlitz-Thoulesstran-
be published in Phys.Rev.E, cond-mat/9705222, 1997.
sition temperature T . The determination of T and
KT KT
of the exponent ν is very difficult and requires extensive
simulations and a careful analysis.
5
