The crossover from first to second-order finite-size scaling: a numerical study

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The crossover from first to second-order finite-size scaling: a numerical study

Christian Borgs, P. Rakow, Stefan Kappler

To cite this version:

Christian Borgs, P. Rakow, Stefan Kappler. The crossover from first to second-order finite-size scaling: a numerical study. Journal de Physique I, EDP Sciences, 1994, 4 (7), pp.1027-1048.

�10.1051/jp1:1994182�. �jpa-00246962�

Classification Puysics Abstracts

05.50 05.70F 75.10H

The _crossover ifom first _to second-order finite-size

scaling:

a numerical

study

Christian Borgs (~), P-E-L- Rakow (~) ^{and Stefan} Kappler (~)

(~) Institut für Theoretische Physik, Freie Universitàt Berlin, Amimallee 14, D-14195 Berlin, Germany

(~) Institut für Physik, Universitàt Mainz, Staudinger Weg 7, D-55128 Mainz, Germany

(Received 4 November 1993, received in final form 8 Marcll 19g4, accepted 21 Marcu 1994)

Abstract, We consider

a particular _case of the two dimensional Blume-Emery-Grilfiths mortel to study the finite-size scaling for _a field driven first-order phase transition with two _co-

existing phases not related by _a symmetry. For low temperatures we venfy trie asymptotic (large volume) predictions of trie ngorous theory of Borgs and Koteckf. Near the critical temperature

we show that ail data fit onto _a unique curve, even when the correlation length ( ^becomes _com- parable to _or forger thon the size of the system, provided trie hnear dimension L of trie system

is rescaled by (.

1. Introduction.

First-order phase transitions play _an important rote in physics. Examples _range from well known transitions _m solid state physics (like ordinary melting) to phase transitions in the early universe. For _many mortels describing ^such _a transition, exact solutions _{are not} available,

and systematic expansions such _as "weak coupling" _or "high temperature series" _are of limited value. In this situation, the numerical simulation of physically mteresting mortels has gained

more and _more popularity. With the increasing speed of computers, the statistical _errors of Monte Carlo simulations have been drastically reduced, but _a systematic limitation of this method is the limite _size of the simulated system. ^{A clear} understanding of finite-size elfects is

therefore _more and _more important in order to interpret ^the resulting numerical data.

The theory of finite-size scaling (FSS) ^for second-order phase transitions has _a rich and

longstanding literature, starting with the pioneering work of Fisher [1] and Fisher and Barber [2]. ^{In the last} ten years, the theory of FSS has been extended to first-order transitions.

For systems ^with two coexisting phases, three main approaches _can be distinguished in the literature: the phenomenological renormalisation _group approach to FSS of Berker/Fisher

[3] and Blôte/Nightingale _[4], the thermodynamic fluctuation theory for the order parameter

distribution Pv(s) of Binder and cc-workers [5-7] and the transfer matrix method of Privman and Fisher [8], see references [9, 10] ^{for reviews.}

Recently, the theory of FSS _near first order transitions has been put _{on a} rigorous basis [11, 12] ^{within the} context of the Pirogov-Sinai theory _[13] which applies whenever it is possible

to describe the configurations of the system in terms of energetically unfavourable contours.

Typical mortels to which these methods _can be applied _are field-driven transitions in low temperature spin systems with discrete spins, temperature-driven transitions for Potts mortels, lattice field theories with double well potential, Higgs/confinement transitions in large N lattice gauge theories, and certain 2d continuum field theories, such _as P(#)2 field theories in the broken phase and supersymmetric Wess-Zumino mortels.

In the context of the field-driven transitions with two coexisting phases considered in this

paper, the main result of reference [1ii _con be summanzed by _a formula of the form

Z(V, h) ₌ (e~~f+(~>~)~ ⁺ e~~f-

(~,~)~j ^{(1 +} O(Ve~~/~°_{)) (1.1)}

for the partition ^function m a cubic volume V

=

Ld _with periodic boundary conditions [14].

Here Lo is _a constant of the order of the infinite volume correlation length, and fm(fl,h) is

some sort of metastable free energy for the phase _m. It is equal to the free energy f(fl, h)

if _m is stable, strictly Iarger than f(fl, h) if _m is unstable, and may be chosen _{as a} smooth function of h. The finite size scaling of the partition function and its derivatives _are obtained

by expanding fm(fl, h) around the transition point ht

It is interesting to discuss the consequences of (1.1) for the order pararneter distribution Pv (s). Within the framework of the thermodynamic fluctuation theory ^for the order parameter, this distribution is described _{as a} superposition ^of two Gaussians. Depending _on the _version of the theory, different formulae for the FS S of the order parameter _are obtained assuming ^different

ansàtze for the relative normalization of the two Gaussians [6, ii. If the normalisation is chosen

m such _{a way} that both Gaussians have equal weight at the infinite volume transition point ht the resulting formula for the magnetisation in _a cube with periodic boundary conditions

leads _to the prediction _[6] that the shift of the susceptibility maximum with respect to ht is

proportional to L~~d, while _a normalisation with equal peak height at ht leads to the prediction iii that this shift is proportional to L~d

In reference [11] ^it was shown that the predictions of equation (1.1) _agree with those of _the

equal weight prescription. Unfortunately, equation (1.1) is only _prouen for _very Iow tempera-

tures (for Ising type systems ⁱⁿ d

= 2, ^this restricts the temperature to about half T~), while most numerical simulations _are performed _near T~, where it is unclear which _ansatz is best (or

indeed whether finite size effects _can be described by _any simple ansatz). It therefore seemed desirable to test the predictions followmg from (1,1) in _an actual numerical simulation.

This was clone _m [16]. ^{We considered} a spm mortel with spins _s~ E (-1, 0,1) and Hamilto- nian

H =

~j

£

s~

,

(1.2a)

<~y> ~

where the first _{sum goes over} the _set of dLd _nearest neighbours _m V and

OE(8~) =

~~ ~~ " ~l

+1 if _8~

= Ù, +1 (1.2b)

It is defined in such _{a way} that boundaries between -1 and 0 _or +1 _are energetically _un- favourabIe, while boundaries between 0 and +1 do not cost energy. As _a consequence, the Iow

temperature phases of the mortel consist of _a phase "minus" corresponding to spin configura-

tions where most spins _{s~ are} equal to -1 with small islands of 0 _or +1, ^and a phase "plus"

corresponding to spin configurations which _{are a} mixture of 0 and +1 with small islands of -1.

The phase minus is stable _as long _as h is smaller than _a certain critical value ht _" ht(fl), while the plus phase is stable for h _> ht. Because in the "plus" phase spins _can be flipped from 0 to +1 without creating _any energetically unfavourable boundaries _we expect ^the magnetic susceptibility of this phase to be larger than that of the "minus" phase. A large susceptibility

difference between the phases is desirable because it Ieads to _a clear difference between the

predictions of the equal-weight and equal-height mortels of first order finite _size effects.

This mortel is _a special _case of the Blume-Emery-Grifliths (BEG) mortel [17], ^{which is the}

most general mortel with _a three-valued spin and _nearest neighbour interactions. The general

BEG mortel has _a complicated phase _structure, with _{up to} three coexisting phases. Here _we have chosen the coupling constants in such _{a way} that the mortel has _two coexisting phases at the phase transition point ht, with _a large susceptibility difference.

In [16], we simulated the mortel (1.2) in d

= 2 and determined the shift of the susceptibility

maximum with respect to ht For fl ₌ o-s, ^and lattices with linear dimensions L from 4 to

30, we found evidence for the equal weight prescription, as to be expected from the rigorous results of reference [11]. ^{We summarize the main results of reference} [16], as well _as certain

extensions (in particular to fl ₌ 0.45 and fl ₌ 0.47) and refinements in section 3.

However equation (1.1) does not only make predictions about the position of the suscepti- bility maximum. It also allows _us to make other finite-size scaling predictions for quantities such _as the volume dependence of the susceptibility-maximum, _or the volume dependence of the susceptibility ai the transition point ht, which have not yet ^{been tested for trie mortel}

(1.2). In particular in view of the results of Billoire and cc-workers [18], ^{who found that the} corresponding asymptotic scaling ^form for two dimensional Potts models _was only reached for systems ^{of linear dimensions L} as big _as 5 _to 10 times the correlation length ( of the infinite system, it _seems desirable to check _more finite-size scaling predictions for the mortel considered above. This wiII be clone _m section 4.

As _we wiII _see, the asymptotic FSS-form obtained from equation (1.1) will again only be reached for fairly large volumes. While these volumes _are easily accessible for fl

= o-s and

fl ₌ 0.47 (the corresponding correlation length ( is 2.19238... and 4.35091...), it is not accessible for temperatures near T~ (we ^{dia do} a third simulation for fl ₌ 0.45, corresponding to ( ₌

13.5096...). For these temperatures, however, _a scaling ansatz using the length scale ( to rescale ail lengths in consideration should be reasonable. In fact, _we will _see in section 5 that this ansatz allows _{us to} fit ail data onto a unique _curve which describes the _crossover between first and second order phase transitions.

2. Trie mortel and its relation to trie standard Ising mortel.

In the next three sections, we will analyse the FSS for the mortel (1.2). We recall that the low temperature phases of the model consist of _a phase "minus" corresponding to spin configura-

tions where most spins _{s~ are} equal to -1 with small islands of 0 _or +1, ^and a phase "plus"

corresponding to spin configurations which _{are a} mixture of 0 and +1 with small islands of _-1.

As _we will _see below, it _can be shown that for fl _{> fl~,} where fl~ ^is the critical temperature ^of the Ising mortel, the phase _minus is stable _as long as h is smaller than ht _" -fl~~arsinh(1/2),

while the plus phase _is stable for h _> ht At h

= ht the infinite volume magnetisation jumps from _a value _{m_}

= m_ (fl) to a value _m+

= m+(fl) > m-.

In order to compare the predictions of FSS theory with numerical data, one needs the numerical values of the infinite volume magnetisation _m+ and susceptibility _x+ at the point h = ht + 0. As shown _m [16], ^these constants can be obtained _as follows: Let Z(V,fl,h) be the partition function of the mortel (1.2), and Iet ZI(V,fl,h) be the partition function of the

standard Ising mortel. Then Z and ZI _are related by the formula

Z(V,fl,h)

= e("(~,~)~~)~~ZI(V,fl,_~1(fl,h)) (2.1a)

~~~~

/1jp, h) ₌ 1/j2p)Inje~~ _{+ e~~~)} j2.lb)

As _a consequence, we obtain the froc energy f(fl, h) of the mortel (1.2) in terms of the free energy fI(fl,_~1) ^of _an Ising mortel with magnetic ^field _~1

flfl, h)

= fI là,~llfl,h)) ₊ h p(p, h) j2.ic)

In _a similar _{way, it} is possible to relate the correlation length ( and the interface tension _a of the mortel (1.2) at h

= ht to those of the Ising mortel at _zero field. Que finds

na) ₌ i~ià) and aià)

= a~ià) _12-2)

Recalling that phase transitions _are associated with singularities of the free _{energy, we con} read off the phase diagram of the mortel (1.2): it has the _sonne critical temperature _as ^the Ising mortel, while ht is just the point where ~1(fl,h) is zero, 1-e-

Differentiating equation (2.1c) with respect to h and taking the limit h _- ht + 0, we finally

obtain _m+ and x+. In terms of the magnetisation _{mI " mI} là) and the susceptibility _{XI " xI} là)

of the Ising mortel at _{~1 =} +0 _one gets

~~ =

l~Ph~ ~ ^lj~ ~ph~~~~ j~~~~

2 2

x+ = (fl + flmI)e~~~~ + (1+ e~~~~)~xI (2.4b)

2 4

Note that the difference _{x+ x-} does not involve the _{constant XI,}

x+ x- " (flmI)e~~~~ (2.4c)

We have simulated the mortel (1.2) in d

= 2, where mI(fl) and (1 " 1/2aI _are exactly known [19],

mI(fl) _" (1 (sinh 2fl)~~)~/~ (2.5a)

i~iô)

j14jjjiijci~ili~i ^Îir flic

i~.~~~

In order _{to compare} FSS theory to numerical data, _we will also need _x+ and _x-. Unfortunately,

the susceptibility _XIlà) of the Ising mortel is _not exactly known. We therefore used the low- temperature ^series for _xI là) found in [20] together with the method of Padé approximants _[21]

to calculate _XIlà)- As in [22], we embodied the knowledge of the _exact transition point and critical exponent in _our approximation. The result is

M N 7/4

XIlà)

" 4flu~

~

£

£

n=1

Î

n=1

where _u

= e~~~, and _{fin, qn are} given in table I. The factor 1 6u + u~ has _a simple _zero at

fl = fl~, _ensuring that XI diverges _as (fl fl~)~R. It is amusing ^{to note} that _a Padé approximant of the form (2.6) delivers the exact formula (2.5a) for _mI if used _on the strong coupling series for magnetisation.

Table I. Low tem peratl~re Fadé coe1Jicients for tl~e suscep tibility of tl~e Ising model.

n fin qn n fin qn

1 -11.422 -9.9941 4 -0.83867 -41.824

2 42.042 27.744 5 -4.9026

3 -49.064 -5.1210

3. Trie position of trie susceptibility maximum.

In order to obtain the FSS of the order pararneter distribution Pv(s) of _an order parameter

s le-g- _s

=

)

= Ld, Binder and Landau approximated the probability distribution Pv(s) by _{a sum} of two Gaussians, centered around the infinite volume

magnetisation _m- and _m+. Some assumption must be made to fix the relative normalisation of these _two Gaussians. In reference _{[6] it was} assumed that _at the transition point ht each of

the two coexisting phases contributes with trie _same weight. Following this "equal weight"-rule

[6] ^one gets

p~j~)çt

(~

+ ~-vp(s-m+)2/2x+ ePs(h-h~)v (~~~)

N /T /T ^'

where x+ are the infinite volume susceptibilities at the point ht 0 and ht +0, respectively, and N

= N(fl, h) is chosen _m such _{a way} that Pv(s) is normalised. Another possibility suggested by mean-field arguments _is that the two Gaussians should be combined in such _a way that the

peak heights _are equal at the transition point ht Following this "equal height"-rule iii Pv(s)

tutus out to be

p~js) çt e-vp(s-m-)2/2x-

+ e-vp(s-m+)2/2x+ ePs(h-h~)v j~ ~~)

N

It is an easy exercise to calculate the FSS for the finite volume magnetisation m(V,h) fsPv(s)ds _following from (3.1a) and (3.1b). For the point h~(V) where the susceptibility=

-1 069

h

-l0691 ~~°.45

-1

-1 0693 ~,~~~""'

,,,""'

-Î

-1 0695

~ ~ ~~~ ~ ~~

lIV a)

-1 O16

~

fl=0.4?

-1 02

-1 ozB

~ ~ ~~ ~ ~~ ~'~~

iIv b)

-o g55

h

fl=0.50

-O 96

,',,

-O 965 ~~~~~~~

-O.97

"""'

~ ~ ~~ ~ ~~ ~ ~~

lIV C)

Fig. l. a) The position hx(V) of the susceptibility maximum for fl

= 0.45. Open points _are Monte Carlo results, solid points are exact results (see appendix). The solid fine _is trie theoretical prediction (3.2a), while trie dashed hne _is trie prediction (3.2b). b) The position hx(V) of trie susceptibility

maximum point hEWIV) for fl

= 0.47. c) The position hx(V) of trie susceptibility maximum point hEWIV) for fl

= 0.50.

x(V,h)

= dm(V, h)/dh has ils maximum, _a lengthy but straight-forward calculation gives the FSS predictions

h~jv) ₌ ht +

(~i~~

j

+ OjV~~) j3.2a)

and

h~(V) _" ~t

2111~~~~

~~ ~_ l

~ O(V~~)

1 ~~~~~

~

~

n~(X+/X-))~2jm+

~-)~ ~~

respectively. An important difference between the mortels is that the shift predicted by the

"equal weight"-rule _{[6] is} proportional to 1/V~ while the shift predicted by trie "equal height"-

rule [7] is proportional to 1IV. (For _symmetric phase transitions both rules _are equivalent and

predict that h~(V)

= ht, 1. _e, no shift.)

We have tested the predictions (3.2) for three different temperatures, namely for fl

= 0.45, fl = 0.47 and fl ₌ o-à- In _our simulation _vie first generated the probability distribution Pvlà)

for L ₌ 4, 5,...,12,14,18, 30 at h

= ht using a multimagnetical version [23] ^{of the heat bath} algonthm. Applying the reweighting technique of Ferrenberg and Swendsen [24], we obtain the probability distribution Pvlà) and hence the magnetisation, m(V, h)

= £ sPv là) and the

susceptibility, x(V, h) ₌ flV(£ s~Pv là) (£ sPv(s))~), for h in the vicinity of ht. In order to

use the vector structure of the CRAY X-MP and Y-MP, _we calculated 64 independent copies

of the system ⁱⁿ parallel, with 15 million sweeps for each of them. Grouping 8 of them together

to save memory, we obtained 8 statistically independent data sets for each L, and therefore 8 statistically independent estimates for h~(V). Statistical _error estimates _are obtained from

these data _m the standard _way.

We have plotted the theoretical _curves (3.2a) and (3.2b) together with _our numerical data in figures 1a-c. We recall that the coefficient in (3.2a) is known exactly, while the coefficients

m (3.2b) have been obtained with the help of _a Padé improved low-temperature expansion.

Within the resolution of figure 1, ^the errors for the numerically determined points h~(V) _are

invisible, except ^for fl

= 0.45. Figure 1 favours the equal weight prescription.

Convinced that the "equal weight"-rule is reasonable _even at higher temperatures we tried the _new method to determme the transition point ht of first-order phase transitions recently proposed by Borgs and Janke [25]. ^{This method} just _uses the fact that both phases should have equal weight at h

= ht To be precise, _one dejines _a finite volume transition point hEwIV)

as the point where

~j

=

L

Here sm,n, s- < sm,n < s+, is the point _s where the distribution Pv(s) has its minimum.

According to reference [25], ^{it is} expected that the systematic _errors (hEw IV) ht _{are expo-} nentially small. Here they _{are so} small that hEwIV _agrees with ht within the statistical _errors

as soon as L/( is noticeably bigger then _one (see Tab. Il).

Table IIa. Monte Carlo results for fl

= 0.45.

~ ~Xi~) ~i~Xi~)) Xmaxiv)/V Xiht)/V

4 -1.062853(44) 0.291383(07) 0.171484(05) 0.171355(06)

5 -1.066617(38) 0.297833(08) 0.163352(07) 0.163301(06)

6 -1.067975(22) 0.301295(12) 0.157308(08) 0.157282(08)

7 -1.068570(22) 0.303355(08) 0.152602(05) 0.152587(05)

8 -1.068925(25) 0.304703(07) 0.148814(07) 0.148807(06)

9 -1.069085(17) o.305617(12) 0.145699(07) 0.145694(07)

10 -1.069174(15) 0.306261(10) 0.143083(08) 0.143080(09)

11 -1.069264(21) 0.306770(11) 0.140857(06) 0.140856(06)

12 -1.069300(20) 0.307118(08) 0.138951(12) 0.138951(12)

14 -1.069340(17) 0.307640(11) 0.135837(08) 0.135837(08)

18 -1.069338(11) 0.308173(09) 0.131496(11) 0.131495(11)

30 -1.069360(13) 0.308728(30) 0.125416(17) 0.125416(18)

L hu(V) Ui~~ Uv(ht)

4 -1.065260(44) 0.598796(07) 0.598403(12)

5 -1.067640(38) 0.606576(08) 0.606414(08)

6 -1.068483(22) 0.611434(11) 0.611349(12)

7 -1.068851(22) 0.614893(07) 0.614841(10)

8 -1.069093(25) 0.617588(09) 0.617564(10)

9 -1.069191(16) 0.619833(10) 0.619818(11)

10 -1.069244(15) 0.621759(16) 0.621748(18)

11 -1.069313(21) 0.623476(09) 0.623473(10)

12 -1.069335(20) 0.625052(16) 0.625051(16)

14 -1.069359(17) 0.627860(11) 0.627860(10)

18 -1.069345(11) 0.632589(14) 0.632587(13)

30 -1.069361(13) 0.642953(22) 0.642953(36)

L hEwlv) m+lV) m-IV) x+lV) x-Iv)

4 -1.069540(42) 0.28434(02) 0.90380(02) 0.2488(01) 0.1523(01)

5 -1.069546(38) 0.27217(02) 0.89176(02) 0.3214(01) 0.2229(01)

6 -1.069332(24) 0.26350(03) 0.88146(03) 0.3967(02) 0.3093(01)

7 -1.069321(34) 0.25587(21) 0.87381(21) 0.4849(24) 0.3989(23)

8 -1.069398(26) 0.24938(06) 0.86779(08) 0.5832(09) 0.4918(09)

9 -1.069435(27) 0.24364(22) 0.86313(24) 0.6948(42) 0.5828(43)

10 -1.069362(18) 0.24004(18) 0.85797(16) 0.7860(41) 0.7035(38)

11 -1.069407(22) 0.23605(10) 0.85447(09) 0.9016(23) 0.8081(24)

12 -1.069402(26) 0.23292(14) 0.85123(17) 1.0142(48) 0.9216(50)

14 -1.069397(17) 0.22771(15) 0.84606(14) 1.2488(59) 1.1530(61)

18 -1.069361(11) 0.22064(07) 0.83897(10) 1.7316(58) 1.6274(66)

30 -1.069363(13) 0.21183(08) 0.82994(09) 3.036(16) 2.944(16)

co -1.06935961.. 0.2087522.. 0.8267862.. 4.75 4.67

Table IIb. Monte Carlo results for fl ₌ 0.47.

L hxlv) mlhxlv)) xmaxlv)/V xlht)/V

4 -1.018099(59) 0.2917063(77) 0.1879631(51) 0.1878343(56)

5 -1.021451(34) 0.2980674(69) 0.1817612(56) 0.1817091(46)

6 -1.022640(37) 0.3014688(70) 0.1775283(55) 0.1775017(61)

7 -1.023182(23) 0.3035004(77) 0.1744855(53) 0.1744708(53)

8 -1.023500(27) 0.3048250(81) 0.1722426(57) 0.1722357(57)

9 -1.023655(16) 0.3057188(65) 0.1705276(51) 0.1705242(51)

10 -1.023683(14) 0.3063460(64) 0.1692254(71) 0.1692216(67)

11 -1.023747(20) 0.3068320(83) 0.1682018(49) 0.1681995(46)

12 -1.023794(23) 0.3071837(68) 0.1674027(69) 0.1674017(67)

14 -1.023813(14) 0.3076732(72) 0.1662249(70) 0.1662241(71)

18 -1.023858(09) o.3082108(91) 0.1649093(29) 0.1649093(29)

30 -1.023846(06) 0.3087157(58) 0.1636313(95) 0.1636305(96)

L hu(V) Ui~ Uv(ht)

4 -1.020208(59) 0.610042(05) 0.609678(15)

5 -1.022328(34) 0.619780(04) 0.619626(06)

6 -1.023067(37) 0.626326(06) 0.626242(11)

7 -1.023413(23) 0.631209(05) 0.631161(06)

8 -1.023635(27) 0.635124(05) 0.635103(08)

9 -1.023740(16) 0.638356(05) 0.638348(06)

10 -1.023738(14) 0.641133(07) 0.641119(06)

11 -1.023784(20) 0.643530(05) 0.643523(05)

12 -1.023820(23) 0.645639(06) 0.645636(07)

14 -1.023828(14) 0.649127(06) 0.649124(08)

18 -1.023863(09) 0.654133(03) 0.654133(04)

30 -1.023847(06) 0.661167(06) 0.661162(09)

L hEwlv) m+lV) m-IV) x+lV) x-IV)

4 -1.024032(60) 0.303542(24) 0.922636(09) 0.2300(01) 0.1273(01)

5 -1.023969(35) 0.296404(16) 0.915421(15) 0.2807(01) 0.1770(01)

6 -1.023816(49) 0.291949(99) 0.909913(99) 0.3280(11) 0.2320(10)

7 -1.023822(22) 0.288400(97) 0.906357(92) 0.3783(12) 0.2830(12)

8 -1.023887(25) 0.285761(73) 0.903962(76) 0.4281(11) 0.3297(12)

9 -1.023896(17) 0.283968(58) 0.902061(63) 0.4727(12) 0.3761(13)

10 -1.023843(15) 0.282669(71) 0.900775(64) 0.5146(18) 0.4171(17)

11 -1.023858(19) 0.281679(38) 0.899877(42) 0.5533(10) 0.4536(12)

12 -1.023872(23) 0.281054(34) 0.899190(31) 0.5859(11) 0.4874(11)

14 -1.023857(14) 0.280179(28) 0.898372(28) 0.6433(13) 0.5400(15)

18 -1.023873(09) 0.279668(27) 0.897689(30) 0.7059(19) 0.6123(16)

30 -1.023848(06) 0.279645(23) 0.897639(17) 0.7451(10) 0.6528(09)

co -1.02385495.. 0.27967484.. 0.89770883.. 0.7378 0.6433

Table IIC. Monte Carlo results for fl

= 0.50.

L h~iv) mih~iv)) x~~~iv)/v xih~)/v

4 0.957392(41) 0.2920229(56) 0.2112605(43) 0.2111255(57)

5 0.960375(48) 0.2982867(54) 0.2070919(30) 0.2070386(41)

6 0.961410(34) 0.3016339(73) 0.2045653(26) 0.2045385(27)

7 0.961846(20) 0.3036197(50) 0.2029542(30) 0.2029382(34)

8 0.962120(32) 0.3049161(23) 0.2018638(42) 0.2018563(46)

9 0.962230(32) 0.3057789(67) 0.2011162(70) 0.2011113(60)

10 0.962293(23) 0.3064065(53) 0.2005801(45) 0.2005767(48)

11 0.962335(20) 0.3068779(35) 0.2001908(63) 0.2001885(60)

12 0.962391(17) 0.3072287(51) 0.1998887(67) 0.1998882(69)

14 0.962375(06) 0.3076952(53) 0.1994660(22) 0.1994642(24)

18 0.962421(14) 0.3082342(72) 0.1990105(36) 0.1990105(37)

30 0.962427(08) 0.3087290(42) 0.1985327(29) 0.1985326(26)

L hu(V) Ui~~ Uv(ht)

4 0.959187(41) 0.6220964(36) 0.6217458(11)

5 0.961108(48) 0.6328211(25) 0.6326800(12)

6 0.961761(34) 0.6398531(33) 0.6397789(09)

7 0.962034(20) 0.6448716(16) 0.6448240(05)

8 0.962230(32) 0.6486261(28) 0.6486061(09)

9 0.962298(32) 0.6515315(27) 0.6515180(07)

10 0.962337(23) 0.6538180(23) 0.6538081(06)

11 0.962365(20) 0.6556541(31) 0.6556476(04)

12 0.962413(17) 0.6571408(20) 0.6571405(03)

14 0.962387(06) 0.6593655(08) 0.6593586(03)

18 0.962425(14) 0.6620330(08) 0.6620329(07)

30 0.962428(08) 0.6649091(06) 0.6649074(18)

L hEwlv) m+lV) m-IV) x+lV) x-IV)

4 0.962517(40) 0.325905(14) 0.944591(07) 0.20538(04) 0.09385(03)

5 0.962494(48) 0.323234(09) 0.941739(09) 0.23116(02) 0.l1961(03)

6 0.962411(34) 0.321985(05) 0.940006(11) 0.25043(05) 0.14281(06)

7 0.962386(20) 0.321226(27) 0.939220(26) 0.26679(35) 0.15947(34)

8 0.962440(32) 0.320767(09) 0.938869(08) 0.27967(08) 0.17092(05)

9 0.962428(32) 0.320633(17) 0.938654(18) 0.28755(27) 0.17991(31)

10 0.962423(23) 0.320555(16) 0.938597(13) 0.29333(28) 0.18540(22)

11 0.962423(20) 0.320557(10) 0.938592(14) 0.29611(12) 0.18939(14)

12 0.962454(17) 0.320545(10) 0.938622(15) 0.29874(13) 0.19064(17)

14 0.962409(06) 0.320596(07) 0.938641(07) 0.30017(11) 0.19183(09)

18 0.962434(14) 0.320660(09) 0.938720(11) 0.29854(08) 0.19102(10)

30 0.962429(08) 0.320688(08) 0.938717(05) 0.29677(09) 0.18925(11)

cc 0.96242365.. 0.32068921.. 0.93872320 0.29677 0.18920

4. Asymptotic first-order FSS-scaling.

In this section _we test trie FSS predictions following from il.l) by Taylor expanding f~(fl, h)

around h

= ht,

i~jfl, h)

m iifl, h~) m~ih _h~)

jjh

)ih

h~)3 + 14.i)

For trie maximum xmax(V) of trie susceptibility, its position h~(V) and trie magnetisation m(h~(V)) at trie point h~(V), trie FSS predictions following ^from Il.lj _are

~~~~~ ~~ ~

fIÎÎÎÎÎ

)3 /2

/3

~~~~~~~~ ^~~

Î

~

/fl ÎÎ-

Î

~

~(j)

Xmax(V) = fl ^{~+ ~~} V + ~+ ~ ~~

+ O( (4.4)

2

~

2 V

It is also interesting to analyse the FSS of trie susceptibility ^and magnetisation at trie infinite volume transition point ht (for systems where ht is not known exactly, _one ^could _use trie point hEw(V) introduced in (25]). Due to trie exponentially small _error bound in Il.l), the

corrections to the known leading order FSS predictions _are exponentially small

m(ht) ₌ ^~+

)

_(l

+ O(Ve~~/~°)) ^(4.5)

X(ht) ₌

fl

~+ ~~

V ₊ ~+ ~ ~~

(l

2

~

2

Note that the _error bounds in (4.5) and (4.6) _are only bounds. For trie model considered here,

the value of m(ht), _e-g-, is in tact exactly (m

+ m-)/2, due to (2.1) and the fact _{that the} finite volume magnetisation with periodic boundary conditions and _no externat field _{is zero} for the standard Ising model. The corrections for x(ht), _on the other hand, _are expected to be

asymptotically proportional to L"e~~R, _with

a constant a not necessary equal to the worst

case bound (4.6).

In addition to the FSS of the susceptibility and magnetisa,tion, _we also analyse trie FSS of trie cumulant Uv introduced by Binder (Si. It ^is defined _as

~ lm~l~

lim imi)4)

(iM

~~~~

~ 3 lM21) 3 ilm lMll2l~

Here I.ic denotes connected expectation values and M

= £~~~ _s~. By trie mequality (F~) >

(F)~ (applied to F

= (M (Mi_)~) the cumulant Uv cannot exceed the value 2/3. In the high temperature region, where trie asymptotic probability distribution of M is Gaussian, UL _goes

to 0 in trie infinite volume limit for ail h, while in trie low temperature region considered here

~~~~~

~ /ÎÎ

~ÎÎ-Î~ Î

~ ~~

/2 ^Î

'

~~'~~

implying that Uv _goes to 2/3 at trie first order transition point ^h ₌ ht if V _{- cc.} At the second order transition point ^h = ht, fl

" flc the cumulant Uv is expected to go ^to a non-zero

limit strictly smaller than 2/3.