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arXiv:cond-mat/0603411v2 [cond-mat.stat-mech] 5 Apr 2006

EPJ manuscript No.

(will be inserted by the editor)

Critical Binder cumulant of two–dimensional Ising models

W. Selke

Institut f¨ ur Theoretische Physik, Technische Hochschule RWTH Aachen, 52056 Aachen, Germany

Received: date / Revised version: date

Abstract. The fourth-order cumulant of the magnetization, the Binder cumulant, is determined at the

phase transition of Ising models on square and triangular lattices, using Monte Carlo techniques. Its value

at criticality depends sensitively on boundary conditions, details of the clusters used in calculating the

cumulant, and symmetry of the interactions or, here, lattice structure. Possibilities to identify generic

critical cumulants are discussed.

PACS. 05.50.+q Ising model, lattice theory – 05.10.Ln Monte Carlo method, statistical theory

1 Introduction

In the field of phase transitions and critical phenomena,

the fourth order cumulant of the order parameter [1], the

Binder cumulant U, plays an important role. Among oth-

ers, the cumulant may be used to compute the critical

exponent of the correlation length, and thence to identify

the universality class of the transition, characterised , e.g.,

by the values of the bulk critical exponents [2].

The value of the Binder cumulant at the transition

temperature in the thermodynamic limit, U∗, the critical

Binder cumulant, has received much attention as well [3],

being a measure of the deviation of the corresponding dis-

tribution function of the order parameter from a Gaussian

function. However, there seem to be conflicting statements

about its ’universality’. For concreteness and simplicity,

let us consider here and in the following results within the

universality class of the two–dimensionalIsing model, with

the magnetization as the order parameter. In particular,

in the case of the isotropic spin-1/2 Ising model with ferro-

magnetic nearest–neighbour couplings on a square lattice

with L2spins, the critical cumulant has been determined

very accurately in numerical work, applying Monte Carlo

techniques [4] and transfer-matrix methods [5] augmented

by finite–size extrapolations to the thermodynamic limit,

L −→ ∞. The resulting value, employing full periodic

boundary conditions, is U∗= 0.61069... [5]. For other re-

lated two-dimensional models on square lattices, including

the nearest-neighbour XY-model with an easy axis and the

spin-1 Ising model, estimates of U∗have been reported

which seem to be consistent with this value [4,5,6,7,8,

9,10,11,12]. Actually, the quoted value for U∗has been

sometimes believed to be ’universal’, i.e. to be generic for

the two–dimensional Ising universality class.

On the other hand, the possible dependence of the crit-

ical cumulant, for instance, on boundary conditions has

been noted already by Binder in his pioneering work [1].

Indeed, different values of U∗have been obtained when

considering various boundary conditions, lattice structures

(or anisotropic interactions) as well as aspect ratios, stay-

ing in the universality class of the two–dimensional Ising

model [3,5,13,14,15,16]. Some of the results can be re-

lated to each other by suitable transformations. For in-

stance, applying periodic boundary conditions, the critical

cumulant of the nearest–neighbourIsing model with differ-

ent vertical and horizontal couplings may be mapped onto

that of the isotropic model on a rectangular lattice with

aspect ratio r [5,16]. Such a scale transformation, keeping

rectangular symmetry and employing periodic boundary

conditions, does not exist, however, for Ising models with

nearest neighbour and anisotropic next–nearest neighbour

interactions on a square lattice (with the triangular lat-

tice being a special case of that anisotropy [16,17]). This

fact has been demonstrated by Chen and Dohm [17] using

renormalization group arguments, and it has been con-

firmed in Monte Carlo simulations [16,18]. It shows a vi-

olation of the two–scale factor universality for finite–size

effects [19], in general, and, specifically, of the universality

of the critical Binder cumulant.

The aim of this paper is, to study spin-1/2 Ising models

with nearest neighbour interactions on square and triangu-

lar lattices in order to analyse in a systematic way possible

dependences of the critical Binder cumulant on boundary

conditions, clusters used in calculating the cumulants, and

lattice structure (or anisotropy of the interactions).

The paper is organized as follows: In the next sec-

tion, the model and the method are introduced, and the

Binder cumulant is defined. Then, simulational results will

be presented, arranged according to boundary conditions.

Finally, the findings will be summarized briefly.

Page 2

2 W. Selke: Critical Binder cumulant of two–dimensional Ising models

2 Model and method

We consider spin-1/2 Ising models on square and triangu-

lar lattices with nearest neighbour ferromagnetic interac-

tions, J. The Hamiltonian reads

H = −J

?

(x,y),(x′,y′)

Sx,ySx′,y′

(1)

where Sx,y= ±1 is the spin at site (x,y). Sums are taken

over all pairs of nearest–neighoursites (x,y),(x′,y′). x and

y refer to symmetry axes of the lattices. Usually, lattices

of linear dimensions L and K = rL will be simulated, r is

the aspect ratio. As indicated above, the triangular case is

isomorphic to an anisotropic Ising model on a square lat-

tice with nearest–neighbour couplings augmented by half

of the next–nearest neighbour couplings, also of strength

J, along one diagonal direction of the lattice [16,17,20].

Our aim is to study the Binder cumulant at the phase

transition temperature Tc. For both lattice structures, the

exact critical temperature is known. For the square lattice,

one gets [21]

kBTc/J = 2/ln(√2 + 1) = 2.26918... (2)

For the triangular lattice, the critical temperature is given

by [22]

kBTc/J = 2/ln(

?

3) = 3.64095..(3)

The fourth order cumulant of the magnetization, i.e.

the Binder cumulant, for a spin cluster C is defined by [1]

U(T,C) = 1− < M4>C/(3 < M2>2

where < M2>C and < M4>C denote the second and

fourth moments of the magnetization in that cluster, tak-

ing thermal averages. In principle, clusters of various sizes

or shapes and systems with different boundary conditions

may be studied. In the Ising case, the cumulant approaches

in the thermodynamic limit (with the cardinal number

|C| −→ ∞) the value 2/3 at temperatures T < Tc, while

it tends to zero, reflecting a Gaussian distribution of the

magnetization histogram, at T > Tc [1]. At Tc, U∗=

U(Tc,|C| −→ ∞) acquires a nontrivial value, the critical

Binder cumulant.

To study systematically the possible dependence of

the critical cumulant for two-dimensional Ising models on

boundary conditions, the choice of clusters C as well as

the lattice type (or, more basically [17], the anisotropy of

interactions), we performed Monte Carlo simulations for

both lattice types at criticality.

Note that simulational data of high accuracy are needed,

to obtain reliable estimates for U∗. We computed sys-

tems of various shapes and sizes, usually with up to about

4 × 103spins. In general, the (moderate) system sizes al-

ready seem to allow for a smooth extrapolation to the

thermodynamic limit. Using the standard Metropolis algo-

rithm (a cluster flip algorithm becomes significantly more

efficient for larger system sizes), Monte Carlo runs with

up to 109Monte Carlo steps per site, for the largest sys-

tems, were performed, averaging then over several, up to

C) (4)

about ten, of these runs to obtain final estimates, and to

determine the statistical error bars shown in the figures.

We computed not only the cumulant, but also other quan-

tities like energy and specific heat, to check the accuracy

of our data. Of course, for sufficiently small lattices ther-

mal averages may be obtained exactly and easily by direct

enumeration.

3 Results

The critical Binder cumulant depends sensitively on the

boundary conditions. In fact, Ising systems with periodic

and free boundary conditions will be analysed here. In

addition, the Ising model on a square lattice with mixed,

free and periodic, boundary conditions will be considered.

3.1 Periodic boundary conditions

Employing full periodic boundary conditions, with the

cluster C comprising the entire system, U∗has been de-

termined accurately before, both for square and triangular

lattices. For the square lattice, one gets U∗

[4,5,16], and for the triangular lattice, one finds a slighty

different, but distinct value, U∗

Less attention has been paid in the past, however, to

different choices of clusters. In his pioneering work [1],

Binder considered square subblocks for the Ising model on

a square lattice. In particular, for systems of L2spins, the

clusters then correspond to subblocks of size L′2, where

L′= bL, with the subblock factor b ≤ 1. The finite–size

dependence of the cumulant at criticality has been dis-

cussed as well. For the two–dimensional Ising model, the

leading correction term to the critical cumulant is argued

[1] to behave like U∗− Uc(Tc,L) ∝ 1/L.

In the original analysis [1], the subblock sizes L′have

been enlargened at fixed L. We pursue a somewhat dif-

ferent strategy in computing cumulants at Tc by fixing

the subblock factor b and then enlargening the linear di-

mension of the lattice with L2spins (applying periodic

boundary conditions). In particular, we set b= 1, 1/2, 1/4,

and 1/8. Some representative data of our simulations are

depicted in Fig. 1. Increasing the system size L, the cu-

mulant at criticality allows for a smooth and reliable ex-

trapolation to the thermodynamic limit, yielding U∗

is observed to decrease with decreasing subblock factor

b, and we estimate U∗

b= 0.5925 ± 0.0005,0.577 ± 0.001,

and 0.568±0.0015 for b= 1/2, 1.4 , and 1/8, respectively.

Plotting now U∗

bagainst b, we obtain, in the limit b −→ 0,

the critical cumulant, U∗

b=0= 0.560±0.002. This estimate

may be checked by fixing L′and increasing L to estimate

U∗

L′, see Fig. 2. U∗

on L′, L′≥ 4. Taking into account estimates for L′= 4,

8, and 16, we arrive at a value for U∗

nicely with the one quoted above. Note that the critical

cumulant in the limit b −→ 0 refers to arbitrarily large

clusters or subblocks being eventually embedded in their

indefinitely larger ’natural’ heat bath. In that sense, the

clusters themselves are subject to a ’heat bath boundary

s= 0.61069...

t= 0.61182... [5,16].

b. U∗

b

L′ is found to depend only rather weakly

b=0, which agrees

Page 3

W. Selke: Critical Binder cumulant of two–dimensional Ising models3

0

0.025 0.05 0.075

0.1

0.125

1/L

0.57

0.58

0.59

0.6

0.61

0.62

U(Tc,L)

Fig. 1. Binder cumulant U(Tc,L) vs. 1/L using subblocks of

linear size L′= bL for b= 1 (squares), 1/2 (diamonds), and 1/4

(circles) in the Ising model with L2sites on a square lattice,

employing full periodic boundary conditions. The square with

a cross, at 1/L = 0, refers to the result of Ref. 4.

0

0.05

0.1

0.15

0.2

0.25

1/L

0.4

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

0.62

0.64

U(Tc,L)

Fig. 2. Fourth–order cumulant U(Tc,L), using subblocks of

linear size L′= 4 (squares) and 8 (diamonds) as clusters, ver-

sus 1/L for square lattices of linear dimension L, employing

periodic (broken lines) and free (solid lines) boundary condi-

tions.

condition’. This boundary condition is expected to per-

turb the intrinsic bulk fluctuations of the magnetization

of the cluster only very mildly. Therefore, U∗

candidate for a critical Binder cumulant, which is robust

against various modifications of the model. Indeed, within

the accuracy of the simulations, one obtains the same esti-

mate for the critical cumulant, b −→ 0, when one replaces

the periodic boundary conditions by free boundary condi-

tions, as will be discussed below.

The critical cumulant for the heat bath boundary con-

dition of the clusters, b = 0, is found to depend for rect-

angular subblocks, L′× rCL′, on their aspect ratio rC,

tending to decrease significantly, as rCdeviates more and

more from one.

Moreover, when going from square to triangular lat-

tices, using periodic boundary conditions and subblocks

b=0may be a

of L′2spins, the critical cumulant U∗

very close to that for the square lattice. The possible dif-

ference occurs, perhaps, in the third digit. But already for

b = 1, the difference between U∗

Here, the heat bath boundary condition for the clusters,

b = 0, tends to reduce such differences furthermore, and

one has to be careful in drawing definite conclusions. In-

deed, on physical grounds I tend to believe that also under

heat bath boundary conditions for the subblocks, there is

a difference in the value of the critical cumulant for the tri-

angular and the square lattice, unless one uses subblocks

of special shapes, as will be discussed below.

b=0is observed to be

sand U∗

tis quite small.

3.2 Mixed boundary conditions

We study the Ising model on a square lattice consisting

of L lines, running from left to right, having L sites or

spins in each line. At the bottom and top, free boundary

conditions are employed, while the left and right hand

sides are connected by periodic boundary conditions. The

Hamiltonian, eq. (1), is slightly extended by still assuming

ferromagnetic nearest neighbour interactions, which now

may be different in the two surface lines at the top and

bottom, Js, as compared to those, Jb, in the bulk, i.e.

when at least one spin of the nearest neighbour pair of

spins is not in a surface line, as usually assumed [23,24,

25,26].

Let us first consider Js= Jb. To compute the critical

cumulant, U∗

thermally excitable spins. Results for various system sizes

are depicted in Fig. 3. Again, the data may be smoothly

extrapolated to the thermodynamic limit, L −→ ∞, lead-

ing to the estimate U∗

mixed= 0.514± 0.001.

When varying Js/Jb, the cumulant appears to depend

strongly on the ratio of the surface to the bulk coupling,

considering systems of fairly small sizes, see Fig. 3 for

Js/Jb= 0.1. 1.0, and 2.0. However, in the thermodynamic

limit, the critical cumulant seems to approach a unique

value, independent of Js/Jb, as may be inferred also from

that figure.

mixed, we take clusters C consisting of all, L2,

It is well known that the critical behaviour of the bulk

is distinct from that of the surface [23,24,25,26]. In par-

ticular, in the two–dimensional case, the vanishing of the

surface magnetization, on approach to Tc, is described by a

power law with an exponent 1/2, while the exponent of the

bulk magnetization is 1/8 [26,27,28]. Thence, it may be

interesting to restrict the clusters C to the surface lines at

the bottom and top of the lattice, in analogy to what has

been done before for Ising films in three dimensions [29].

Here, we find that the critical cumulant tends to vanish in

the thermodynamic limit, reflecting a Gaussian distribu-

tion of the histograms for the surface magnetization. The

vanishing may be explained by the fact that the surface is

one–dimensional in our case.

Page 4

4 W. Selke: Critical Binder cumulant of two–dimensional Ising models

0

0.05

0.1

0.15

0.2

0.25

1/L

0.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

U(Tc,L)

Fig. 3. Binder cumulant U(Tc,L) for the Ising model with

mixed boundary conditions and L2spins on a square lattice as

a function of 1/L, varying the ratio Js/Jb= 0.1 (squares), 1.0

(diamonds), and 2.0 (circles).

3.3 Free boundary conditions

Free boundary conditions allow one to study arbitrary

shapes of the lattice. Moreover, when compared to pe-

riodic boundary conditions, they are more realistic.

Let us first consider square and triangular lattices with

L2spins, i.e. with aspect ratio r= 1, applying free bound-

ary conditions at the four sides of the system. The clusters

C are comprising all spins, b= 1. As may be inferred from

Fig. 4, the critical cumulant, in the thermodynamic limit,

may be estimated from a smooth extrapolation of the sim-

ulational data. The resulting values deviate appreciably

for the two different lattice structures: For the square lat-

tice, we find U∗

fbc,s= 0.396±0.002, while for the triangular

lattice, we obtain U∗

fbc,t= 0.379±0.001. Accordingly, free

boundary conditions are very useful to show the relevant

influence of the lattice type (or anisotropy of the interac-

tions) on the critical Binder cumulant. In comparison, the

difference in the critical cumulant for the Ising model on

square and triangular lattices is rather small in the case

of periodic boundary conditions, see above.

The critical cumulant is expected to depend on the as-

pect ratio, as we confirmed by considering square lattices

with the aspect ratio r = 1/2. The critical cumulant U∗

is estimated to be 0.349± 0.002.

We also computed the cumulant, for the square lattice,

with the clusters C being square subblocks of fixed linear

dimension L′, to study the effect of heat bath boundary

conditions, with the subblock factor b −→ 0. As illustrated

in Fig. 2, by increasing L, the resulting critical cumulant

U∗

fbc(L′) tends to approach the same value as in the case

of periodic boundary conditions using the same subblocks

L′. The finite–size correction term of the cumulant has

opposite sign for the two boundary conditions, see Fig. 2.

We conclude that there is strong evidence that the criti-

cal cumulant acquires the same value for free and periodic

boundary conditions, when the clusters are embedded in

their natural heat bath. We tend to suggest, that the very

same value holds for other boundary conditions as well.

0

0.025 0.050.075

0.1

0.125

1/L(R)

0.37

0.38

0.39

0.4

0.41

0.42

U(Tc,L(R))

Fig. 4. Binder cumulant U(Tc,L(R)) at criticality for square

(squares) and triangular (triangles) lattices with free boundary

conditions along two symmetry axes of the lattices (broken

lines) and along discretized circles (solid lines) as a function of

1/L and 1/LR, respectively.

On the other hand, as discussed above, the critical cumu-

lant U∗, in the limit b −→ 0, still depends on the shape

of the clusters and, presumably, on the lattice type (or

anisotropy of interaction).

The (’non–universal’)dependences of U∗may be partly

explained by the fact, that the shape of the cluster C

does not fit to the spatial structure of the spin corre-

lation function. Indeed, we propose that an appropriate

cluster shape follows from the Wulff construction at criti-

cality [30], which determines equilibrium shapes and pre-

serves the intrinsic symmetry of the correlations. In the

case of square and triangular lattices, this consideration

leads to free boundary conditions of circular shape. In-

terpreting the Ising model on the triangular lattice as a

model with anisotropic next–nearest neighbour interac-

tions on a square lattice, the circle would transform into

an ellipse, rotated with respect to the principal axes, on

the square lattice. Obviously, for finite radii, the circular

shape may be approximated by a discretization. More con-

cretely, we define a radius R from the center of the square

or triangular lattice, and we keep all spins, NR, within

this radius as active, thermally excitable spins, while the

remaining spins are set to be equal to zero. From that con-

struction, an effective linear dimension LRmay be defined

by LR=√NR(being proportional to an effective radius

of the cluster). In the thermodynamic limit, R −→ ∞,

one arrives at a perfect circle. Certainly, an analogous ap-

proach is feasible for clusters with heat bath boundary

conditions, b = 0, being, however, more cumbersome, be-

cause one had to take, at each given radius, the thermo-

dynamic limit L −→ ∞.

Simulational data for both lattices with discretized cir-

cular free boundary conditions are depicted in Fig. 4. In

contrast to the case of free boundary conditions along sym-

metry axes of the lattices, the critical cumulants for both

lattices now tend to approach close-by, if not identical val-

ues, U∗

circle= 0.406 ± 0.001. Of course, further numerical

Page 5

W. Selke: Critical Binder cumulant of two–dimensional Ising models5

Table 1. Selected critical Binder cumulants of the two–

dimensional Ising model with periodic (pbc), free (fbc), mixed,

and heat bath (hbbc), b = 0, boundary conditions on the

square or triangular (tri) lattice, considering various system

shapes, see text.

boundarylatticeshapeU∗

pbc

pbc

fbc

fbc

mixed

hbbc

fbc

square

tri

square

tri

square

square

square, tri

square

rhombus

square

rhombus

square

square

circle

0.61069... [5]

0.61182... [5]

0.396 ± 0.002

0.379 ± 0.001

0.514 ± 0.001

0.560 ± 0.002

0.406 ± 0.001

as well as analytical work will be very useful to clarify this

interesting aspect.

4 Summary

In this article, we estimated, using Monte Carlo tech-

niques, the critical Binder cumulant U∗for Ising models

with nearest neighbour interactions on square and triangu-

lar lattices, employing various boundary conditions, types

of clusters, and aspect ratios. Selected examples are listed

in Tab. 1.

In particular, in the case of periodic boundary condi-

tions we considered square clusters with decreasing sub-

block factor b. In the limit b = 0, we estimate, for the

square lattice, U∗

b=0= 0.560±0.002. The critical cumulant

is observed, when studying clusters of rectangular shapes,

to depend on their aspect ratio. The, presumably, rather

weak dependence of U∗

b=0on the lattice structure for these

’heat bath boundary conditions’ for the clusters is not re-

solved in our simulations.

For the Ising model with mixed boundary conditions,

analysing square lattices with the aspect ratio r = 1 and

clusters comprising all spins, the strength of the surface

coupling is found to be irrelevant for the critical cumulant.

For clusters containing only the surface spins, the fluctua-

tions of the surface magnetization seem to be of Gaussian

form with vanishing U∗. This behaviour reflects the fact

that the surface is a one-dimensional object here.

Applying free boundary conditions, the critical Binder

cumulants U∗for systems with the aspect ratio r = 1

and clusters including all spins, b = 1, are cleary different

for square and triangular lattices (U∗

U∗

fbc,t= 0.379± 0.001). They differ significantly from the

known corresponding values for periodic boundary condi-

tions. In the limit b = 0, we obtain for the square lattice

an estimate for U∗

b=0which agrees, within the error bars,

with the one for periodic boundary conditions. Perhaps

most interestingly, employing free boundary conditions for

clusters of circular form, we find numerical evidence for

a unique value, both for square and triangular lattices,

U∗

circle= 0.406 ± 0.001. In general, we suggest that the

dependence of the critical cumulant on the anisotropy of

fbc,s= 0.396± 0.002,

interactions or the lattice structure may be overcome by

using cluster shapes obtained from the Wulff construction

at criticality.

Certainly, previous standard analyses of the critical

cumulant, using especially periodic boundary conditions

with the subblock factor b = 1, are not invalidated by our

study, when they are interpreted properly. In particular,

when comparing critical cumulants on different models,

one has to make sure that the models satisfy the same

symmetries determined by, for instance, the interactions

and/or lattice structure. In other words, for such analy-

ses universality of the critical cumulant holds in a rather

restricted sense, when compared to universality of critical

exponents. In any event, care is needed in applying the

critical Binder cumulant when one tries to identify uni-

versality classes or the location of the phase transition.

In general, a finite-size scaling theory including bound-

ary conditions, system shapes and anisotropy of interac-

tions would be desirable, extending previous descriptions

[19,3,31,17].

Useful discussions with V. Dohm, D. Stauffer, L.N.

Shchur, and W. Janke are gratefully acknowledged.

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