An extended scaling analysis of the S=1/2 Ising ferromagnet on the simple cubic lattice

0.65 MB·English

by  I. A. Campbell

An extended scaling analysis of the S = 1/2 Ising ferromagnet on the simple cubic
lattice
I. A. Campbell1 and P. H. Lundow2
1 Laboratoire Charles Coulomb, Universit´e Montpellier II, 34095 Montpellier, France
2Department of Theoretical Physics, Kungliga Tekniska h¨ogskolan, SE-106 91 Stockholm, Sweden
It is often assumed that for treating numerical (or experimental) data on continuous transitions
theformalanalysisderivedfromtherenormalizationgrouptheorycanonlybeappliedoveranarrow
1
temperature range, the ”critical region”; outside this region correction terms proliferate rendering
1
0 attemptstoapplytheformalism hopeless. Thispessimistic conclusion follows largely from achoice
2 ofscalingvariablesandscalingexpressionswhichistraditionalbutveryinefficientfordatacovering
wide temperature ranges. An alternative ”extended scaling” approach can be made where the
n
choiceofscalingvariablesandscalingexpressionsisrationalizedinthelightofwellestablishedhigh
a
temperature series expansion developments. We present the extended scaling approach in detail,
J
and outline the numerical technique used to study the three-dimensional 3d Ising model. After a
1
discussion of the exact expressions for the historic 1d Ising spin chain model as an illustration, an
3
exhaustive analysis of high quality numerical data on the canonical simple cubic lattice 3d Ising

model is given. It is shown that in both models, with appropriate scaling variables and scaling
]
h expressions (in which leading correction terms are taken into account where necessary), critical
c behaviorextendsfrom Tc up to infinitetemperature.
e
m PACSnumbers: 75.50.Lk,05.50.+q,64.60.Cn,75.40.Cx
-
t
a INTRODUCTION orinauthoritativereviewssuchasthoseofPrivman,Ho-
t
s henberg and Aharony [2] or Pelissetto and Vicari [3] the
.
at Understandingtheuniversalcriticalbehaviorobserved scalingvariableis definedast fromthe outset. However,
m at and near continuous transitions is one of the major because t at infinite temperature, when t is chosen
→∞
- achievements of statistical physics; the subject has been asthescalingvariablethecorrectiontermsinFQ(t)each
d studied in depth for many years. It is generally con- individually diverge as temperature is increased. It in-
n sidered howeverthat the formalism basedon the elegant deed becomes extremely awkwardto use the expressions
o
renormalizationgrouptheory(RGT)canonlybeapplied inEq.(3)outsideanarrow”critical”temperatureregion.
c
[ over a narrow temperature range, the ”critical region”, A ”critical-to-classical crossover” has been invoked (e.g.
while outside this region correction terms proliferate so Refs. [4,5])withtheeffectiveexponentγeff(β)tendingto
3
attempts to extend the analysis become pointless. In the mean field values as the high temperature Gaussian
v
4 fact this pessimistic conclusion follows largely because fixed point is approached. The crossover appears as a
4 the traditional choices of scaling variables and scaling consequenceofthe definition ofthe exponentinterms of
2 expressionsarepoorlyadaptedtothestudyofwidetem- the thermodynamic susceptibility and the scaling vari-
6 perature ranges. able t. There is no such crossover when the extended
.
0 The expressionsfor criticaldivergenciesofobservables scaling analysis described below is used.
1
Q(T) near a critical temperature T and in the thermo- Although this is rarely stated explicitly, there is noth-
c
0
dynamic (infinite size) limit are conventionally written ingsacredaboutthescalingvariablet;alternativescaling
1
: variables τ can be legitimately chosen and indeed have
v Q(T)=C t−q(1+F (t)) (1)
Q Q been widely used in practice, see e.g. Refs. [6–11].
i
X
Temperature dependent prefactors can also be intro-
with the scaling variable t defined as
r duced in the scaling expressions on condition that the
a
t=(T T )/T (2) prefactor does not have a critical temperature depen-
c c
−
dence at T .
c
and where F (t) represents an infinite set of confluent
Q An ”extended scaling” approach [12–16] has been in-
and analytic correction terms [1]
troduced which consists in a simple systematic rule for
selecting scaling variables and prefactors, inspired by
F (t)=a tθ+b t+ (3)
Q Q Q
··· the well established high temperature series expansion
The exponents q, the confluent correction exponent θ, (HTSE) method. This approach is a rationalization
and many critical parameters such as amplitude ratios which leads automatically to well behaved high temper-
and finite size scaling functions, are universal, i.e. they ature limits as well as giving the correct critical limit
areidenticalforallmembersofauniversalityclassofsys- behavior.
tems. WhentheRGTformalismisoutlinedintextbooks Here we give a general discussion of this approach.
2
We outline the relationship to the RGT scaling field BelowT ,χ (β,L)tendstotheconnectedreducedsus-
c mod
formalism. As an illustration of the application of the ceptibility in the thermodynamic limit, and
rules, known analytic results on the historically impor-
m (β,L)= χ(β,L) χ (β,L)/√N (7)
tant S = 1/2 Ising ferromagnet chain in dimension one mod
h| |i −
p
(for which the critical temperature is of course T = 0)
c tends to the thermodynamic limit magnetization
are cited. Simple extended scaling expressions for the
m (β,L) at large L.
reduced susceptibility, the second moment correlation h i
(iii) The specific heatwhichis equalto the varianceof
length, and the specific heat are exact over the entire
the energy per spin U(β,L)
temperature range from zero to infinity. An exact sus-
ceptibility finite size scaling function is exhibited. C (β,L)=N U2 U 2 (8)
v
The S = 1/2 nearest neighbor Ising ferromagnet on h i−h i
(cid:0) (cid:1)
the simple cubic lattice is then discussed in detail. This where U = (1/N) ijSi · Sj with the sum over near-
modelisamongtheprincipalcanonicalexamplesofasys- est neighbor bonds.PWe can note that χ(β) and Cv(β)
tem having a continuous phase transition at a non-zero haveconsistentstatisticaldefinitions interms ofthermal
critical temperature. In contrast to the two-dimensional fluctuations. The experimentally observed susceptibility
2d Ising model, in three dimensions no exact values are contains an extraneous factor β.
known for the critical temperature or the critical expo- The thermodynamic limit second moment correlation
nents. Weanalyzehighqualitylargescalenumericaldata length is defined [9, 19] by
whichhavebeen obtainedfor sizes up to L=256,cover-
ξ2(β, )=µ (β, )/2dχ(β, ) (9)
ing wide temperature ranges both above and below the ∞ 2 ∞ ∞
critical temperature [17, 18]. The numerical technique
where the second moment of the correlationfunction is
is outlined. An analysis using the extended scaling ap-
proachprovidescompactcriticalexpressionswithamin- µ (β, )=(1/N) r2 S S (10)
2 ∞ h i,j i· ji
imum ofcorrectionterms, whichareaccurate (if notfor- Xi,j
mally exact) over the entire temperature range from T
c
with r the distance between spins i andj, summing to
to infinity and not only within a narrow critical regime. i,j
infinity.
(The 3d Ising, XY, and Heisenberg ferromagnets have
When the ”thermodynamic limit” condition L
been discussed in Ref. [13]). ≫
ξ(β, )holdsallpropertiesbecomeindependentofLand
∞
so are identical to the thermodynamic limit properties.
DEFINITIONS OF VARIABLES For general L, the Privman-Fisher finite size scaling
ansatz for an observable Q can be written [20, 21]
We study the S = 1/2 nearest neighbor interaction
Q(β,L)/Q(β, )= (11)
ferromagnetic Ising model on the 1d chain and on the ∞
simple cubic lattices of size L3 with periodic boundary FQ(L/ξ(β, )) (1+L−ωGQ(L/ξ(β, )))
∞ ∞
conditions. The Hamiltonian with nearest neighbor in-
The functions F (x), G (x) are universal. F (x) must
Q Q χ
teractions of strength J is tendto1whenx 1,andmustbeproportionaltox2−η
≫
when x 1. We are aware of no generally accepted
= J Si Sj (4) ≪
H − · explicitexpressionsforF (x)validoverthe entirerange
Xij Q
of x.
with the sum over nearest neighbor bonds. As usual we
will use throughout the normalized inverse temperature
β J/kT. EXTENDED SCALING
≡
The observables we have studied are as follows:
(i) The variance of the equilibrium sample moment, In the extended scaling approach[12, 13] a systematic
which is equal to the non-connected reduced susceptibil- choice of scaling variables and scaling expression prefac-
ity torsis made inthe lightofthe HTSE.Basically,anideal
HTSE corresponds to the power series
χ(β,L)=N m2 =(1/N) S Sj (5)
i
h i Xi,j h · i (1 y)−q 1+qy+(q(q+1)/2)y2+ (12)
− ≡ ···
wheremisthemagnetizationperspinm=(1/N) iSi, When a real physical HTSE has the form
N =Ld. P
(ii) The variance of the modulus of the equilibrium Q(x)=CQ 1+a1x+a2x2+ (13)
···
sample moment, or the ”modulus susceptibility” (cid:0) (cid:1)
withageneralstructuresimilartobutnotstrictlyequiv-
χ (β,L)=N m2 m 2 (6) alent to that of Eq.(12) and a prefactor C which can
mod Q
h i−h| |i
(cid:0) (cid:1)
3
be temperature dependent, the asymptotic limit is even- exactclosureconditionsfortheinfinitetemperaturelimit
tually dominated by the closest singularity to the origin τ 1 : C (1+F (1)) = 1 and C /β1/2(1+F (1)) = 1
χ χ ξ c ξ
→
(Darboux’sfirsttheorem[22])leadingtothecriticallimit (or C /(tanhβ )1/2(1+F (1))=1).
ξ c ξ
Q(x) = CQ(1 x)−q. The appropriate critical scaling One can define temperature dependent effective expo-
−
variable is 1 x, and deviations of the series in Eq.(13) nents (introduced by [25]):
−
from the pure Eq.(12) form correspond to confluent and
analytic critical correction terms. γeff(τ)=∂logχ(β)/∂logτ (20)
The extended scaling prescription consists in identify-
see Refs. [9, 26]. For the correlation length,
ing scaling variables and prefactors such that eachseries
is transposed to a form having the same structure as ν (τ)=∂log(ξ(β)/β1/2)/∂logτ (21)
eff
Eq.(13), with the prefactordefined so that the firstterm
of the series is equal to 1. is the extended scaling definition for νeff.
The HTSE spin S = 1/2 expressions for the reduced ForaspinS =1/2Isingferromagnetonalatticewhere
susceptibility and the second moment of the correlation each spin has z neighbors, the high temperature limit
can be written generically in the form [9, 19, 23] of the effective exponents defined by Eqns. 20 and 21
are γ (1) = zβ and ν (1) = γ (1)/2. A comparison
eff c eff eff
χ(β)=1+a1x+a2x2+a3x3+ (14) between these values and the critical exponents γ and
···
ν gives a good indication of the overall influence of the
and
correctionterms. Iftheleadingconfluentcorrectionterm
µ2(β)=b1x+b2x2+b3x3+ (15) inEq.(19)dominatesthenγeff(1)−γ ≈aχθ,andνeff(1)−
··· ν a θ. An analysis along these lines of γ for Ising
ξ eff
≈
where x is a normalized variable which tends to 1 as systems with large z was sketched out in Ref.[26]. The
β β and to zero when β 0. caseofgeneralS isdiscussedinAppendixA.Forallnear
c
→ →
For ferromagnets, (e.g. [9, 19, 23]) possible natural neighborIsingferromagnetsonscorbcclatticescovering
choices for x are x = β/β or x= tanhβ/tanhβ . Scal- the entire range ofspin values S =1/2to S = (which
c c ∞
ing variables for χ(β) are τ = 1 x = 1 β/β or are all in the same 3d universality class), see Ref. [9],
c
− −
τ = 1 x = 1 tanhβ/tanhβc. The former is stan- γeff(1)andνeff(1)differfromthecriticalγ andν byafew
− −
dard when T is non-zero; when T = 0, it is convenient percent at most. For both observables, the total sum of
c c
to use x=tanhβ (as β = , tanhβ =1). the correctionterms is weak overthe entire temperature
c c
∞
Forµ (β)the Eq.(13)formwiththe samexcanbere- range.
2
trievedby extractinga temperature dependent prefactor It should be noted that traditional and widely used
b x so as to write finite size scaling expressions
1
µ (β)=b x(1+(b /b )x+(b /b )x2+ ) (16) Q(β,L)/Lq/ν =F L1/ν(T T )/T (22)
2 1 2 1 3 1 Q c c
··· (cid:16) − (cid:17)
The critical expressions for the reduced susceptibility assumeimplicitlyscalingwiththescalingvariablet. Asa
and the second moment correlation length can then be general rule these expressions should not be used except
written inthelimitoftemperaturesveryclosetoT ;theyrapidly
c
becomes misleading and can suggest incorrect values of
χ(β, )=C τ−γ(1+F (τ)) (17)
∞ χ χ the exponent ν if global fits are made to data covering a
wider range of temperatures. The extended scaling FSS
(c.f. Eq.(1))andfromtherelationEq.(9)betweenµ and
2
expressions [12]
ξ,
ξ(β,∞)=Cξx1/2τ−ν(1+Fξ(τ)) (18) Q(β,L)/(LT1/2)q/ν =FQ(cid:16)(LT1/2)1/ν(T −Tc)/T(cid:17)
(23)
with the temperature scaling variable τ =1 x and the
arevalidatalltemperaturesaboveT towithintheweak
− c
standard definitions for the critical amplitudes C and
χ corrections to scaling.
C . The χ(β, ) expression has been widely used; the
ξ ForspinS =1/2Isingspinsonabipartitelattice(such
∞
ξ(β, ) expressionis specific to the extended scalingap-
as the 1d and 3d sc lattices we will discuss below) there
∞
proach[12,13]. TheF functionscontainalltheconfluent
are only even terms in the HTSE for the specific heat
and analytic correction to scaling terms [1, 24]
[9, 23]
FQ(τ)=aQτθ +bQτ +··· (19) Cv(β)=1+d1x2+d2x4+d3x6+··· (24)
It is important that τ tends to 1 at infinite temperature A natural scaling expression for the specific heat is
(whereas t tends to infinity); the F (τ) thus remainwell
behaved over the entire temperatuQre range. There are C (β)=C +C 1 x2 −α 1+F (1 x2) (25)
v 0 c c
− −
(cid:0) (cid:1) (cid:0) (cid:1)
4
TheconstanttermC ispresentinstandardanalysesand essentially identicalfor β >2.5 but arequite different at
0
plays an important rˆole in 3d ferromagnets because the higher temperatures.)
exponent α is small. The extended scaling expression The internal energy per spin is just
Eq.(25)isnotorthodoxasitusesascalingvariable,τ =
2
1 x2 =τ(2 τ), whichis notthe sameasthe τ =1 x U(β)=tanhβ (32)
− − −
used as scaling variable for χ(β) and ξ(β).
so the specific heat
ANALYTIC RESULTS IN ONE DIMENSION
C (β)=cosh−2β (33)
v
TheoriginalIsingferromagnet[27]consistsofasystem
Though not immediately recognizablethese can all be
ofS =1/2spins with nearestneighborferromagneticin-
re-written in precisely the form of the extended scaling
teractions on a one dimensional chain. Because analytic
Eqns. 17,18,25, with the choice x = tanhβ so τ = (1
results exist for many of the statistical properties of this −
tanhβ);
system, it is often used as a ”textbook” model in intro-
ductions to critical behavior. We will use it to illustrate
χ(β)=2(1 tanhβ)−1(1 (1/2)(1 tanhβ)) (34)
the extended scaling approach (see Ref. [14]). − − −
The model orders only at T = 0 (Ref. [27]); when
T =0thecriticalexponentsdependonthechoiceofthe
c
scaling variable. Baxter [28] states : ”[in one dimension] tanh1/2β
ξ(β)= (35)
it is more sensible to replace t = (T T )/T by t = 1 tanhβ
− c c −
exp( 2β)”; with this scaling variable the exponents are
γ =1−,ν =1,α= 1 [α= νd when T =0 Ref. [29]]. and
c
− −
Expressionsfor ξ(β) and χ(β) in the infinite-size limit
arereadilycalculatedfollowingstandardHTSErules(see Cv(β, )=(1 tanh2β) (36)
∞ −
e.g. Ref. [19]). The reduced susceptibility HTSE can be
written as and so with the same critical exponents γ = 1, ν = 1,
α= 1togetherwithcriticalamplitudesC =2,C =1,
χ ξ
χ(β)=1+2(tanhβ+tanh2β+tanh3β+...) (26) C =−1,C =0. Therearenoanalyticcorrectionstoξ(β)
v 0
or to C (β) and there is only a single simple analytic
and the HTSE for the second-moment of the correlation v
correction to χ(β). There are no confluent corrections.
is
Note againthat these expressionsare validfor the entire
µ (β)=2(tanhβ+22tanh2β+32tanh3β+...) (27) temperature range from T =0 to T = .
2 ∞
The finite size scaling function can also be considered.
The second-moment correlation length is then given by Withperiodicboundaryconditionsthefinitesizereduced
Eq. (9) with z =2. Using the power series sums susceptibility for a 1d sample of size L is
∞
yn = y (28) 1 tanhLβ
nX=1 (1−y) χ(β,L)=exp(2β)1+−tanhLβ (37)
and
The finite size scaling function is
∞
y(y+1)
n2yn = (29)
χ(β,L)/χ(β, ) (38)
(1 y)3
nX=1 − =tanh(L/2ξ(β, ))(1+L−2G (∞L/2ξ(β, ))
χ
∞ ∞
the exact expressions for reduced susceptibility and cor-
relation length are thus The simple principle expression
χ(β)=exp(2β) (30)
F (L/ξ(β, ))=tanh(L/2ξ(β, )) (39)
χ
∞ ∞
and
is exact.
1
ξ(β) = (exp(4β) 1)1/2 ThehigherordertermGχ(x)inEq.(39)isnumerically
2 −
tiny even for small L. We have not found an analytic
(31)
expression but it can be fitted rather accurately by
(It can be noted that the ”true” correlation length is
G (x)=0.168x2 1+tanh( 0.565x1.18) (40)
ξ = 1/log(tanhβ). The two correlation lengths are χ
true −
− (cid:0) (cid:1)
5
HIGH DIMENSION LIMIT and
u =t(1+c t+c t2+...) (46)
For the Ising ferromagnet in the high dimension hy- t 1 2
percubic lattice limit d , with τ(β) = 1 β/β the
→ ∞ − c where h is the magnetic field. The two series in t are
reduced susceptibility and the correlation length are
analytic.
χ(β, ) τ−1 (41) Ignoringforthemomenttheconfluentcorrectionseries
∞ ≡ F , for phenomenological couplings R
ω
and
R(β,L)=F u L1/ν (47)
ξ(β, ) (β/βc)1/2τ−1/2 (42) R(cid:16) t (cid:17)
∞ ≡
=G L1/νt(1+c t+c t2+ )
R 1 2
exactly over the entire temperature range above Tc; the (cid:16) ··· (cid:17)
exponents γ = 1,ν = 1/2 are of course the mean field
and for χ
exponents. These expressions again follow the extended
scalingformgivenaboveEqns.17,18includingthesquare χ(β,L)=AL2−η 1+b t+b t2+ (48)
1 2
root prefactor in ξ(β, ), with no correction terms. ···
In this high dimensi∞on limit the specific heatabove Tc =Gχ(cid:16)L1/νt(1(cid:0)+a1t+a2t2+···)(cid:1)(cid:17)
is zero (α=0).
In dimensions above the upper critical dimension but Analyses using this formalism are carried out by intro-
not in the extreme high dimension limit the extended ducingaseriesofanalytictermsinpowersoft,adjusting
scaling approach has been used successfully to identify for each particular case the constants an,bn and cn and
the main correction terms in the reduced susceptibility truncating at some power of t.
[30]. Now consider the extended scaling scheme. As a first
Thus analytic expressions for models both in the low step t = (T Tc)/Tc is replaced in the formalism by
−
(1d) and high (d ) dimension limits follow the ex- τ =(T Tc)/T just as for instance in [31]. The variable
→ ∞ −
tended scaling forms. This reinforces the argument that t is replaced by τ(1 τ) everywhere. This leaves the
−
these forms can be considered to be generic and should generic form of the equations unchanged but modifies
beusedatleadingorderalsoforintermediatedimensions, the individual factors in the series for the temperature
where confluent correctionterms and small analytic cor- dependencies of the scaling fields.
rection terms must be allowed for. In the extended scaling approach a second step must
In practice (e.g. Refs. [7, 8, 10, 11, 26]) analyses of then be made due to the (β/β )1/2 prefactor in ξ(β, ).
c
∞
χ(β) have long been carried out using τ as the scaling The extended scaling FSS expressions [12]
variable rather than t. There are analogous advantages
in scaling ξ(β) with Eq.(18), which contains the generic R(β,L)=F τL1/νβ−1/2ν (49)
R
(β/β )1/2 (or (tanhβ/tanhβ )1/2) prefactor. We sug- (cid:16) (cid:17)
c c
gest that this form of scaling expression for ξ(β) could and
profitably become equally standard.
χ(β,L)=(Lβ1/2)2−ηF τL1/νβ−1/2ν (50)
χ
(cid:16) (cid:17)
RGT FORMALISM AND EXTENDED SCALING
can be translated into the RGT FSS formalism in terms
of explicit built-in leading expressions for the tempera-
InthestandardRGTfinitesizescalingformalism[2,3] ture variationof the scaling fields. The extended scaling
the free energy is written expressions without correction terms are strictly equiv-
alent to leading expressions for the scaling fields u and
t
(β,h,L)= (β,h,L)+ (β,h,L) (43)
sing reg u containing specific infinite analytic series of terms in
F F F h
τn :
wherethesingularpartencodesthecriticalbehaviorand
the regular part is practically L independent. Then u τ(1 τ)−1/2ν (51)
t
∼ −
sing =L−dF uhL(d+2−η)/2,utL1/ν + (44) =τ 1+ 1 τ + (1+2ν)τ2+
F (cid:16) (cid:17) (cid:18) 2ν 8ν2 ···(cid:19)
v L−(d+ω)F u L(d+2−η)/2,u L1/ν +
ω ω h t
(cid:16) (cid:17) ··· and
with the scaling fields uh and ut having temperature de- u h(1 τ)−(2−η)/2 (52)
pendencies h ∼ −
(2 η) (2 η)(4 η)
=h 1+ − τ − − τ2+
uh =hah(1+a1t+a2t2...) (45) (cid:18) 2 − 8 ···(cid:19)
6
In the extended scaling approach these leading expres- all the information needed. Using them one can find the
sions are common to all ferromagnets. The confluent densityofstatesinanenergyintervalaroundthe critical
correctioncontributions willofcoursestillexistwith the regionand that is all that is required for most investiga-
confluent correction terms expressed using τ. Finally, tions of the critical properties of the model.
fine tuning through minor modifications of the analytic For the present analysis a density of states function
scaling field temperature dependence series will usually technique based upon the same method as in [18] was
benecessarytoobtainhigherlevelapproximationstothe used though with considerable numerical improvements
overalltemperature variation of the observables. for all L studied here (adequate improvements to the
NotonlyattemperatureswellaboveT butalreadyat L = 512 data set would unfortunately have been too
c
criticality the extended scaling scheme can aid the data time-consuming). The microcanonical (energy depen-
analysis. For instance, quite generally the critical size dent) data were collected as described in [17]. We use
dependence of the ratio of the derivative of the suscepti- standard Metropolis single spin-flip updates, sweeping
bility to the susceptibility is of the form [31, 32] through the lattice system in a type-writer order. Mea-
surementstake place whenthe expectednumber ofspin-
(∂χ(β,L)/∂β)/χ(β,L)=K1L1/ν 1+cωL−ω+ +K2 flips is at least the number of sites. For high tempera-
···
(cid:0) (cid:1)(53) tures this usually means two sweeps between measure-
AnexplicitleadingordervalueoftheL-independentterm ments and three or four sweeps for the lower tempera-
can be derived from the leading order extended scaling tures we used. Note that in the immediate vicinity of β
c
FSS Eq.(50): the spin-flip probability is very close to 50% for the 3d
simple cubic lattice.
K = (2 η)/2β (54)
2 − − c We report here data on the 3d simple cubic S = 1/2
Ising model with periodic boundary conditions. For
in a ferromagnet. This value will be slightly modified by
L = 256, the largest lattice studied here, we have now
a correction to scaling term.
amassed between 500 and 3500 measurements on an in-
The extended scaling scheme can thus be translated
terval of some 450000 energy levels, where most sam-
unambiguouslyintothestandardRGTFSSformalism. It
plings are near the critical energy U . For L = 128 we
canbeconsideredasprovidinganapriorirationalization c
have between 5000 and 50000 measurements on some
giving explicit leading analyticaltemperature dependen-
150000 energy levels. For L 64 the number of sam-
cies of the scaling fields. At this level the extended scal- ≤
plings are of course vastly bigger.
ing scheme provides compact baseline expressions which
Our measurements at each individual energy level in-
cover the entire temperature region from T to infinity,
c
cludelocalenergystatisticsandmagnetizationmoments.
accurate to within confluent correction terms and resid-
Themicrocanonicaldatawerethenconvertedintocanon-
ual model dependent analytic correction terms.
ical(temperaturedependent)dataaccordingtothetech-
nique in [35]. This gave us energy distributions from
NUMERICAL METHODS APPLIED TO THE whichweobtainenergycumulants(e.g. thespecificheat)
SIMPLE CUBIC MODEL and together with the fixed-energy magnetization mo-
ments we obtain magnetization cumulants (e.g. the sus-
Theequilibriumdistributionsoftheparametersenergy ceptibility).
p(U) for finite size samples from L = 4 up to L = 256 Typicallyaround200differenttemperatureswerecho-
(16,777,216 spins) were estimated using a density of sentocomputethesequantities,withasomewhathigher
statesfunction method. When studying astatisticalme- concentrationnearβc particularlyforthelargerLsothat
chanical model complete information can in principle be one may use standard interpolation techniques on the
obtained through the density of states function. From data to obtain intermediate temperatures.
complete knowledge of the density of states one can im- Below Tc the variance of the distribution of m in zero
mediately work with the microcanonical (fixed energy) field,Eq.(5),representsthenon-connectedsusceptibility;
ensemble and of course also compute the partition func- the physicalsusceptibility in the thermodynamic limit is
tion and through it have access to the canonical (fixed the connected susceptibility
temperature) ensemble as well. The main problem here
χ (β,L)=N m2 ( m )2 (55)
conn
is that computing the exactdensity of states for systems h i− h i
(cid:0) (cid:1)
of even modest size is a very hard numerical task. How- For finite L the distribution of m below T is bimodal
c
ever, several sampling schemes have been given for ob- but always symmetrical so in zero applied field m =0,
h i
taining approximate density of states, of which the best which would suggest that supplementary measurements
known are the Wang-Landau [33] and Wang-Swendsen areneededusingsmallappliedfieldsinordertoestimate
[34] methods. In [17] the various methods are described χ . However under the condition L >> ξ (β, ),
conn conn
∞
along with an improved histogram scheme. For work in where ξ (β, ) is the second moment correlation
conn
∞
the microcanonical ensemble the sampling methods give length below T , the two peaks in the distribution of
c
7
m become very well separated and the variance of the
distribution of the absolute value m can be taken as
| | 1.58
essentially equal to the connected susceptibility,
1.56
χ (β,L)=χ (β,L) (56)
conn mod 1.54
The explicit expressionfor ξ is complicated, see [37], 1.52
conn
but the onset of thermodynamic limit conditions can 1.50
3
judged by inspection of the finite size χmod(β,L) data. 1.96 1.48
To estimate the ordering temperature T we have used L
c / 1.46
the size dependence of U (β,L) the kurtosis of the dis-
4
tribution of p(m), frequently expressed in terms of the 1.44
Binder parameter g(β,L). 1.42
Wehaveintroduced[38]analternativeparameterwith
1.40
the same formal properties as g(β,L) which involves
1.38
χmod. The normalized parameter W(β,L) is defined by 10 100
W(β,L)=1 π(χ (β,L)/χ(β,L))/(π 2) (57) L
mod
− −
or
FIG. 1: (Color online) Finite size corrections at the critical
W(β,L)= π m 2/ m2 2 /(π 2) (58) temperature. χ(βc,L)/L2−η against L at βc adopting η =
h| |i h i − − 0.0368. The large black points are measured; the small red
(cid:0) (cid:0) (cid:1) (cid:1)
points are thefit, Eq.(60)
The normalization has been chosen such that, as for the
Binderparameter,W =0inthehightemperatureGaus-
sian limit and W = 1 in the low temperature ferromag-
[38]. This value is consistent with the Monte Carlo es-
neticlimit. AsW(β,L)isalsoaparametercharacteristic
timates β = 0.22165452(8)[10] and β =0.22165463(8)
c c
of the shape of the distribution p(m), it can be consid-
[39], the HTSE estimate β = 0.221655(2) [9], and
c
eredtobeanother”phenomenologicalcoupling”. Itturns
β =0.2216546(3)[18].
c
out that at least for the 3d Ising model the corrections
At criticality, the standard FSS expression [40] for
to scaling for W(β,L) are much weaker than those for
χ(β ,L) is
c
g(β,L), allowing accurate estimates of T and ν from
c
scaling at criticality. The values estimated for the criti- χ(β ,L)=C′L2−η 1+a′L−ω+b′L−ω2 +a′L−2ω+
cal parameters βc and ν are in good agreement with the c χ (cid:16) 1 1 2 ···(cid:17)
(59)
mostaccuratevaluesfromRGT,HTSE,andMonteCarlo
For the 3d Ising ferromagnet, θ =0.504(8)or ω =θ/ν =
methods [38].
0.800(13)[41]so2ω =1.60(3). Thesubleadingirrelevant
exponentisω =1.67(11)[42]sotheω and2ωtermscan
2 2
3D ISING FERROMAGNET SUSCEPTIBILITY betreatedtogetherasasingleeffectivetermb′L−1.65. In
2
AND CORRELATION LENGTH what follows we will assume for convenience θ =0.50.
Fig. 1 shows χ(β ,L)/L2−η against L adopting η =
c
TheIsingferromagnetindimensionthreeisacanonical 0.0368 [10]; the finite size scaling corrections in the
exampleofasystemhavingacontinuousphasetransition present data can be fitted by
at a non-zero critical temperature. In 3d there are no
χ(β ,L)=1.557L1.9632 1 0.218L−0.82 0.256L−1.65
observables which diverge logarithmically in contrast to c
− −
the2dand4dmodels. Thoughtherearenoexactresults (cid:0) (60(cid:1))
The analysis is consistent with that of [10],
for this universality class, rather precise estimates of the
critical exponents (and the critical temperatures) have χ(β ,L)=L2−η 1.559(16) 0.37(5)L−0.8 (61)
beenobtainedandimprovedovertheyearsthankstoex- c −
(cid:0) (cid:1)
tensive analytical, HTSE, and FSS Monte Carlo studies.
Because of the introduction of a next to leading term,
Theessentialaimhasbeentodetermineasaccuratelyas
the fit extends to lower L.
possible the universal critical parameters.
Fig. 2 shows partial data for the ratio
Considerfirstthe finite sizescalingresultsatandvery
close to the critical temperature. The numerical work x(L)=[(∂χ(β,L)/∂β)/χ(β,L)] (62)
βc
[38] provided an estimate β = 0.2216541(2) from inter-
c
sections between curves for phenomenological couplings againstL. Onthisscalethedatacanbewellrepresented
at different sizes L, using data on the Binder cumu- byx(L)= (2 η)/2β +K L−1/ν withβ =0.2216549,
c 1 c
− −
lant g(L) and on the phenomenological coupling W(L) η =0.0368(2),ν =0.6302(1),andK a constant,see the
1
8
100 1.35
1.34
80 1.33
1.32
c
] 60 1.31
)/ 1.30
d 40 1.29
/
d 1.28
(
[ 20 1.27
1.26
1.25
0
1.24
0 20 1.23
0.0 0.2 0.4 0.6 0.8 1.0
1/
L 0.50
FIG. 2: (Color online) The normalized derivative of the sus-
ceptibility [(∂χ(β,L)/∂β))/χ(β,L)] against L1/ν. The ex- FIG. 3: (Color online) An overall plot of the effective expo-
βc
tendedscalingvaluefortheinterceptis−(2−η)/2β tolead- nent γeff(τ,L) fixing βc = 0.221655 and θ = 0.50 for sizes
c
L = 64,32,16 from top to bottom (black, blue, green). The
ing order (red arrow)
thermodynamic limit envelope curveis clearly seen. The red
line corresponds to an HTSE data analysis [9, 43], in full
agreement with the present results over the entire tempera-
extended scaling expression Eq.(53). This form of plot ture range except for a marginal difference near β . The red
c
provides an independent estimate for ν consistent with arrow indicates theconsensus value for γ(βc).
thevaluesgivenin[9,10,38]. Toobtainanaccuratevalue
for ν it is important to include the non-zero intercept.
and the traditional scaling variable t through
Combining ν and η estimates from FSS at critical-
ity, the present data are almost consistent with the MC
γ =∂logχ (β)/∂logt (64)
and HTSE estimates γ = (2 η)ν = 1.2372(4) [10] th,eff th
−
and γ = 1.2371(1) [9]. Both of these are from meta-
because at high temperatures χ (β) β and t T.
th
analysesonmanysystems inthe same universalityclass, → →
The present data for L = 64, L = 32 and L = 16 are
the latter relying principally on bcc data. A recent very
of very high statistical accuracy. Again assuming β =
c
precise study of the 3d Ising universality class [39] gave
0.221655, γ (τ,L) values in the thermodynamic limit
eff
ν = 0.63002(10) and η = 0.03627(10) so γ = 1.2372(3)
conditions (which arein excellentagreementwith HTSE
together with ω =θ/ν =0.832(6) so θ =0.524(4).
data for γ (τ, ) [9, 43]), can be extrapolated satisfac-
eff
Leaving the pure FSS regime, now consider the over- torily to τ = 0∞assuming γ (τ, ) = γ +a τθ + ,
eff c 1
all temperature and size dependence of χ(β,L). Assum- Fig 3. The fit provides an estima∞te γ =1.239(1), alm·o·s·t
ing β known, the critical exponent γ can be estimated
c c compatible with the HTSE [9] and FSS [10] estimates.
directly and independently from an extrapolation to
ThefluctuationsintheplotforL=64inFig. 4arean
τ = 0 of the derivative γ (τ, ) = ∂logχ(β,L)/∂logτ
eff indicationofhowsensitivetheseplotsaretotheslightest
∞
in the thermodynamic limit conditions i.e. down to
noiseintheoriginaldata. Thetemperatureregioninthe
L-dependent crossover temperatures above which the
far right of Fig. 4 for L = 64 corresponds to a region
χ(β,L)areindependentofL. Thecrossoveroccurswhen
of energy levels measured at least 500,000 times. At the
L 6ξ(β, ), below which the correlation length is
other end the energy levels were measured more than
≈ ∞
no longer negligible compared to the sample size. (As
1,000,000 times. Data for still higher L are not shown
T T below this crossover, χ(β,L) then tends to a
c as the fluctuations become more marked; unfortunately
→
constant for each L).
thesehigherLdatacannotbeusedtorefinetheestimate
There is obviously no ”critical-to-classical crossover” ofγ. Theγ estimatewiththepresentmethodissensitive
as a function of temperature. The crossover would ap- tothevalueassumedforθ. Theγ estimatewouldbecome
pear automaticallyif the effective exponent weredefined incompatible with the consensus value if one assumed
(e.g. Ref. [4]) in terms of the thermodynamic suscepti- significantly higher values for θ, such as 0.54 (estimates
bility of θ are reviewed in [9]).
An advantage of this γ (τ, ) technique is that it is
eff
∞
χth(β)=[∂m(β,h)/∂h]h→0 βχ(β) (63) freefromtheproblemoffinite sizecorrectionstoscaling,
≡
9
1.260 1.08
1.07
1.255
1.06
1.250
2 1.05
1/
-
1.245 1.04
1.03
1.240
1.02
1.235
1.01
1.230 1.00
0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0
0.5
0.5
FIG. 6: (Color online) The normalized correlation length
FIG.4: (Coloronline)AsFig3,blowupofthesmallτ region. ξ(β,∞)τνβ1/2 against τθ inthethermodynamiclimitassum-
ing ν =0.630 and θ =0.50. Raw HTSE data provided by P.
Butera [9, 43].
the scaling expression Eq.(49) and using the data at the
various L but only in the thermodynamic limit, the fit
1.10
χ(β, )τγ =1.106 1 0.080τθ 0.016τ (65)
∞ − −
9 (cid:0) (cid:1)
3
2 givesthe values ofthe criticalamplitude, C =1.106(5),
1. χ
and the coefficient of the leading conformaland analytic
L
1.05 correction terms, aχ = 0.080(3) and bχ = 0.016(3),
− −
readdirectlyofftheplotinFig. 5. Thesevaluesarefully
consistent with but more precise than earlier estimates
from HTSE, C = 1.11(1) and a = 0.10(3) [44], see
χ χ
−
[15]. It can be seen that the extended scaling expression
1.00 with only two leading Wegner correction terms gives a
0.0 0.2 0.4 0.6 0.8 1.0 very accuratefit to the data overthe whole temperature
0.5
range above the critical temperature.
Ifexactlythesamedatawereexpressedusingt=(T
−
T )/T asthe scalingvariableratherthanτ,becauseτ =
FIG. 5: (Color online) The normalized susceptibility c c
χ(β,L)τγ against τθ assuming γ =1.239 and θ=0.50. Sizes t/(1+t) one would have to write
L = 256,128,64,32,16,8 from top to bottom (black, red,
χ(β, )= (66)
green, blue, olive, orange). The excellent fit (yellow) to the
∞
thermodynamic limit envelope datacorresponds to Eq.(65). 1.106t−1.239(1+1.239t+0.1466t2 0.0373t3+
− ···
0.080t0.5+0.0495t1.5 0.0371t2.5+
− − ···
0.016t+0.016t2 0.016t3+ )
althoughtheWegnerthermalcorrectionstoscalingmust − − ···
betakenintoaccountasabove. Itcanbenotedalsothat Remembering that t diverges at infinite T, each of the
this is a direct measurementof γ rather than anindirect correction terms in the sums is individually diverging at
estimatethroughacombinationofνc and2 ηcestimates high temperatures. Manifestly it is considerably more
−
as is the case for FSS. efficient to scale χ(β, ) with τ rather than with t.
∞
Fig. 5 shows the data for L = 16 to L = 256 in the We have made no correlation length measurements.
form of a normalized plot, χ(β,L)τγ against τ0.50 as- However we have carried out an extended scaling
suming γ = 1.239. Again it can be seen by inspection parametrization of HTSE thermodynamic limit second
atwhichpointforeachLthecurvesleavethe thermody- moment correlation length ξ(β, ) data supplied by P.
∞
namiclimitenvelopecurvewhichisLindependent. With Butera [9, 43].
10
0
10
0.650
-1
10
0.645 2-
1/2 )
)c10-2
/
) 0.640 (
( L/
eff )/( 10-3
0.635 L,
(
-4
10
0.630
-5
10
0.0 0.2 0.4 0.6 0.8 1.0 0.1 1 10 100 1000 10000
0.50 1/21/
(L/ )
FIG. 7: (Color online) The extended scaling effective expo- FIG. 8: (Color online) The leading order extended scaling
nent ν(τ) against τθ in the thermodynamic limit assuming plot χ(L,T)/(L(T/T )1/2)2−η against (LT1/2)1/ντ
c (cid:16) (cid:17)
and θ=0.50. Raw HTSEdata provided by P.Butera [9, 43]
Onthe scale of the plot the scaling is alreadyreasonable
Fig. 6showsaplotofthenormalizedcorrelationlength
for all T above T .
c
ξ(β, )τνβ1/2 against τθ assuming ν = 0.630 and θ =
The conformal correction can then be introduced :
∞
0.50. Thedatacanbefittedwellbytheextendedscaling
Wegner expression with two leading terms only χ(β,L)/χ(β, ) (70)
∞
=F (L/ξ(β, ))(1+a L−ωG (L/ξ(β, )))
ξ(β, )τνβ1/2 =1.074β−1/2 1 0.120τ0.5+0.051τ χ ∞ χ χ ∞
∞ −
(cid:0) (6(cid:1)7) The functionF(x) musthavelimits F(x) 1 atlarge
(note that here the critical amplitude is C /β1/2). The x and F(x) x2−η for small x. An expli→cit compact
ξ c ∼
ansatz which gives these limits automatically is
same equation provides the temperature dependence of
the effective exponent defined by a
F (x)= (1 exp( bx(2−η)/a) (71)
χ
(cid:16) − − (cid:17)
ν (β, )=∂log(ξ(β, )/β1/2)/∂logτ (68)
eff ∞ ∞ where x=L/ξ(β, ). In the critical limit x 1,
∞ ≪
see Fig. 7. The effective exponent varies only by a few F (x)=ba(L/ξ(β, ))2−η. (72)
χ
percent over the whole range from T =T to T = . It ∞
c
is clear that the β1/2 prefactoris an essentialparto∞f the By convention Gχ(0) = 1. Fig. 9 uses the tempera-
temperature dependence of the correlation length. The ture dependence of the thermodynamic limit correlation
compact relationEq.(67) is very useful as it allows finite length, Eq.(67), and the thermodynamic limit suscepti-
size scaling analyses of the entire data set for χ(β,L). bility, Eq.(65), to scale the data for all L and all β using
Eq.(71) for χ(β,L).
The principle scaling function F(x) and the leading
FINITE SIZE SCALING correctionscalingfunction G(x) wereextractedfromthe
data. Withthenumericalconstant2 ηfixedat1.963,an
−
accurateeffectivefunctionalformfortheprincipalscaling
The extrapolation in Fig. 5 concerns only data in the
function is
thermodynamiclimitconditionforeachL. WithEq.(67)
inhandwecanplotallthedataandnotjustthepointsin F (x)=[1 exp( 0.4179x1.963/1.262)]1.262 (73)
χ
− −
the thermodynamic limit condition by appealing to the
Privman-Fisher relation [45], Eq.(12). Onthe scaleofthe figureF(x) withthese fit values(a=
1.262,b = 0.4179) is indistinguishable from the overall
As a first step we ignore corrections to scaling and
curveinFig. 9. BycomparingdataatsmallLwithdata
draw,Fig. 8,theleadingorderextendedscalingFSS[12]
at large L the correction to scaling function can also be
plot for the susceptibility
estimated. A fit gives a 0.22 and
χ
≈−
χ(L,T)/(LT1/2)2−η =Fχ (LT1/2)1/ντ (69) Gχ(x) exp 0.038x2.5 (74)
(cid:16) (cid:17) ≈ −
(cid:0) (cid:1)