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Finite-Size Tests of Hyperscaling
PHYSICAL REVIEW B
VOLUME 31, NUMBER 3
1 FEBRUARY 1985
Finite-size tests of hyperscaling K. Binder Institut für Physik, Universität Mainz, D-6500 Mainz, Postfach 3980, Germany and Institut für Festkörperforschung der Kernforschungsanlage Jülich, D5170 Jülich, Germany
M. Nauenberg Natural Sciences, University of California, Santa Cruz, California 95064 V. Privman Baker Laboratory, Cornell University, Ithaca, New York 14853 A. P. Young Mathematics Department, Imperial College, London, England (Received 10 April 1984) The possible form of hyperscaling violations in finite-size scaling theory is discussed. The impli cations for recent tests in Monte Carlo simulations of the d = 3 Ising model are examined, and new results for the d = 5 Ising model are presented.
Recently two extensive Monte Carlo calculations1'2 have been carried out which test the validity of hyperscal ing in the three-dimensional Ising model. In the calcula tion by Freedman and Baker,1 they "observe a systematic downward trend by more than twice the statistical error for a quantity which should be constant if hyperscaling is satisfied," while Barber, Pearson, Toussaint, and Richard son2 conclude that their results require "the anomalous di mension to be small, and are consistent with hyperscal ing." The purpose of this paper is to resolve this apparent conflict by showing that these two collaborations evaluat ed different anomalous exponents, and that the exponent evaluated by Barber et al., which they erroneously identi fied with Fisher's anomalous dimension,3 actually van ishes on theoretical grounds. New Monte Carlo results are also presented for the five-dimensional Ising model which confirm the theoretical predictions for hyperscaling violation in this case. To start, we investigate the consequence of the existence of a "dangerous irrelevant variable" on the finite-size scal ing form of the free energy, and on the hyperscaling rela tions for critical exponents. We then show that consisten cy conditions require that the anomalous exponent intro duced by Barber et al.2 must be zero. Finally, we discuss the finite-size scaling form of the renormalized coupling constant for the Ising model,4 and the results of a Monte Carlo simulation for the dimensionality d = 5. According to the renormalization-group derivations5'6 of finite-size scaling, the singular part of the free energy fL and the correlation length7 %L have the form fL=L-df(tLyT,hLyH,uLyu) £L=L£(tLyT,hLyH,uLyu),
,
(1) (2)
magnetic field, and u is an irrelevant variable, while yr > 0» yH > 0> a n d yu < 0 are renormalization-group ex ponents. We use equality signs to indicate here asymptot ic scaling relations. If the free-energy scaling function f(x,y,z) is singular in the limit z—»0, then u is called a dangerous irrelevant variable.3'6 For simplicity we as sume here that the correlation scaling function £(x,y,z) is regular in this limit, but later we consider the possibility that it is singular. We assume that for small z, f(x,y,z)=zPlf(xzP\yzPi).
The choice of this particular pattern, multiplicative singu lar powers of z, is motivated by the known3 mechanism for the bulk scaling at d > dc = 4 . Equation (3) implies fL=L-d*F(tLy\hLy"),
144
CPSC 2 - paper
1.10
(4)
where scale factors have been absorbed in t and A, and d* =d -p{yUy yr^yr+Piyu, and y*H =yH +ρ&ν are the effective exponents. Note that the anomalous exponent 2 introduced by Barber et al. corresponds to d — d*, while Fisher's exponent3 ω* is given by af=d-d*AyTv).
(5)
These two exponents would be the same if yjv= 1, which was implicitly assumed by Barber et al.2 but has not been proven. We will show instead that d*=d for cubic or similarly shaped systems,6 as usually employed in Monte Carlo simulations.1'2 The existence of the limit L —► oo of fL corresponding to the bulk free energy / „ implies that for x—»± oo and y | x | ~ Δ fixed,
where t is the reduced temperature t = (T — Tc )/Tc ( Tc is the transition temperature of the infinite system), h is the 31
(3)
F(x,y)=\x\d*/y*Y±(y\x\-*), 1498
(6)
© 1985 The American Physical Society
FINITE-SIZE TESTS OF HYPERSCALING
31
where Δ=>>#/y*, and Y± is the scaling function for i < 0 . Likewise the existence of the bulk correlation length ξχ in this limit leads to the asymptotic form \-vx±(y
ξ{χ,γ,0)=\χ
\χ Γ Δ ' ) ,
(18)
asr-+77 .
To obtain this result from the scaling form, Eq. (14), we must require6
(7)
lim W(x,0)oz \χ \2β
(19)
X-*±oo
where v—\/yT
and Δ'=>>///>>Γ
(8)
and we assumed for simplicity that the scaling function ξ in Eq. (2) is nonsingular as z—►(). Taking suitable deriva tives of the free energy, one finds the following scaling re lations a = 2 - c T / > * , ß=d*/y}-&, and y = 2Δ-*, where α, β, and γ are the thermodynamic exponents for the specific heat, magnetization, and susceptibility, respectively. Consequently y\ and y% can be written as yl=d*/(Y
(9)
+ 2ß) ,
y*H=d*(Y+ß)/(Y
(10)
+ 2ß),
Hyperscaling violations occur if yj and/or Δ differ from the exponents yT( = \/v) and Δ'(=;>#/> Γ ) appearing in the scaling form of the correlation length. However, we will show that even in this case one must have d*—d. We now give three arguments which support d*=d. First, consider the finite-size magnetization mL and sus ceptibility XL: mL = (s)L , d
2
(11) (12)
2
XL=L {(s )L-(s) L),
where s ={\/Ld)^siy s, is the spin at the ith site, and ( ) L denotes the thermal average. According to Eq. (4), mL and XL have the scaling form Mt=
to
i /Ä-'V ( i i/? f / u /i) >
(13)
ß mb =a 11 | and Xb=b
\ t \~Ύ as Γ—Γ"
(15)
so that, by Eq. (12), {s2)L=ml+L-dXb
(i<0,L-
)).
(16)
However, if we first set A = 0 and then take the limit L -¥ oo, then (s ) L = 0 but we expect (see Refs. 4 and 6 for details) lim [{s )L>h=0]
dm=2(yj,+ßyi)-d.
(20)
Substituting Eqs. (9) and (10) in Eq. (20), we obtain the re lation
d*=d ,
(21)
which also implies px = 0 in Eq. (3). Second, we show that this condition also follows from the finite-size scaling form of fL at the ferromagnetic phase boundary due to the existence of a discontinuity fixed point.8 In this case the finite-size magnetization mL below the critical temperature takes the form6 mL = mb tanh(mbhLd)
= ml
(17)
[in fact, Eq. (16) applies in this limit as well4,6]. By Eqs. (12) and (17), the zero-field susceptibility behaves accord ing to
(22) l l d
for L »£b, and | h \ <<(mb£b)~ L ~ , where £b is the bulk correlation length,, and mb is the bulk magnetization, Eq. (15). Hence, XL is given by XL=Ldmb2cosh-2(mbhLd)
.
(23)
For A = 0 , XL reduces to Eq. (18) when T^T~y and therefore implies d* =d, Eq. (21). Finally, we consider the- finite-size scaling properties of the zero-field probability distribution P(s) of the magneti zation below the critical temperature. Binder4 has shown that for large L and s near ±mb, P^is) can be written ap proximately in the form
P , ü ) = T ^ W1/2( e - ( S - ^ , 2 L ^ v
(14)
2 where the oh scaling functions V and W are obtained from F. For the bulk magnetization, mb, as well as the suscepti bility, Xb, special care must be taken in order of the limits L —► oo and h -+0± below the critical temperature. Tak ing first the limit L—»oo, and afterwards the limit A —►O*, one obtains the conventional bulk values,
2
and
2(2ir*a)
on
L—*tx>
XLocLd\t\2ß
1499
-
(24)
Hence the arguments of the exponential functions have the form -ß+a)2(\t\LyT)r+2e 2b which demonstrates the occurrence of the scaling com bination | t \LyT in PL(s), where γτ=ά/{γ+2β). Like wise, by including a magnetic field A, the arguments of the exponential become (s±mb—Xbh)2Ld/2Xb and the term linear in h can be written as
<*l'l
(b \t \Ld/{Y+2ß)r{r+ß)hLy"
.
This shows the dependence of PL(s) on the scaling vari able / / " , where y*H=d{y+ß)/{y + 2ß). These expres sions for j>* and y% are precisely Eqs. (9) and (10), but with d* =d. If we assume that no other scaling variables occur, the scaling form Eq. (4) follows, with d* =d. Next, we discuss some recent Monte Carlo calcula tions1,2 which test possible hyperscaling violation in the
CPSC 2 - paper 1.10
145
d = 3 Ising model. These calculations evaluated the A = 0 finite-size renormalized coupling gL introduced by Binder, 4 (s4)L
gL'-
2 2
(s )
v(4)
-3 =
L
(25)
LdX\
XiL*) =
34/L
(26)
^~L*y*»-d*W(*\tLyh.
ΘΑ4
Substituting Eqs. (14) and (28) in Eq. (27), we find that for i = 0 , d
gL=L
d
y
*- G(tL T),
(27)
where G is a scaling function. Since d* =d, it follows that gL is a constant (nonzero in general) at T = TC. The finite-size scaling analysis of the Monte Carlo data by Barber et al.2 implies that indeed d —d*=0, with an error of ± 0 . 0 4 due to the uncertainty in their determina tion of the critical temperature. This was interpreted by them as evidence for the validity of hyperscaling, but we have shown here that d*=d even when hyperscaling is violated. For t > 0 and L-+oo,XL and Χ{*] take their bulk values XL oc t ~γ and Xli]«t-γ~2£ί so that
&«<*'·**£')-' forL»;-17'*,
(28)
where we have used y — 2 Δ = — d* /y*. Baker and Freedman1 evaluated gL for values of t > 0 such that the corre lation length ξι =cL, where c is a constant. After some algebra, this yields SL^L'
(29) 3
where ω*, Eq. (5), is Fisher's anomalous dimension, and is positive. For lattices of dimensions L = 3 to 60 they find ω * ~ 0 . 2 . Note, however, that their result depends on the assumption, Eq. (6), that the scaling function ξ in Eq. (2) is not singular as uLyu-+0. We now briefly turn our attention to the possibility that ξ(χ,)?,ζ) is singular in the limit z—»0. In analogy with Eq. (3) we assume that
g(x,y,z)=z'l£(xzq2,yz93)
,
(30)
which implies (31) anc where >>** =yr+92yu * yV—yH+Q&u· the limit x - ± oo and y \ x "Δ** fixed,
Z(xfy)-
>\x\-vZ±(y\x\-^
),
Likewise in (32)
where v=(l-\-qlyu)/y** and Δ**=^*/ν**. Since the finite-size correlation length £L is bounded by L, we re quire q\yu < 0. Even if one adopts a plausible assumption that for t=h=0, the correlation length increases up to the linear dimensions of the lattice, which implies that qx = 0 and v = 1/y**, y$* need not be equal to yT leading to a new possible source for hyperscaling violation.
CPSC 2 - paper
1.10
Finally, we comment on the effect of boundary condi tions on our analysis. The predictions presented here are unchanged in the thermodynamic limit (L —► oo ) if the re duced temperature t in Eq. (4) is replaced by a "shifted" variable tL, where tL=[T-Te(L)]/Te
where XL is given by Eq. (14), and
146
31
BINDER, NAUENBERG, PRIVMAN, AND YOUNG
1500
,
(33)
in which TC(L) is a "pseudo Γ 0 " for the finite system. This could, for example, be defined as the temperature where the probability distribution of the magnetization starts to develop a two-peak structure. One then defines a shift exponent ψ by
[TC(L)-TC]/TC=AL-W*
,
(34)
where A is a constant. For systems with a surface it is ex pected that ψ=ν, because properties of the finite system should differ from their bulk values when £L approaches the system size. In this case the shift in Tc is greater than the range of the finite-size rounding if hyperscaling is violated, because y\ > y. However, we shall argue below that ψ= 1/y* for periodic boundary conditions. A well-known example which violates hyperscaling is the Ising model (more generally the n-component vector model) for c?>4, where critical exponents stick at their mean field values. It is straightforward to show6 that the free energy of a finite system scales as in Eq. (4) with yj=d/2, yfj=d/4 compared with j > r = l / v = 2 , and yH = (d +2)/2. However, these6 renormalization-group arguments do not give information on the size of the shift in Tc, for which an explicit calculation must be per formed. This is possible for the «-component model with periodic boundary conditions in the limit w—>>οο. By a straightforward extension of the work of Brezin9 one can evaluate the scaling function W(x,0) for the susceptibili ty, and see that the scaling form of Eq. (4) is valid, so the coefficient of the L ~ 1 / v term in Eq. (34) must vanish for this model. We observe that this is a general result for periodic boundary conditions provided that tLyT is the only temperature variable which enters thermodynamic scaling, and then ^ = l / y £ follows by a standard argu ment.10 We have tested our claim that d*—d by Monte Carlo simulations on the five-dimensional Ising model on a cu bic lattice with nearest-neighbor interaction, / , for sizes between L = 3 and 7. Figure 1 shows that the data for gL intersect at Γ / / ~ 8 . 7 7 , g L ~ - 1 . 0 0 . A nonzero value of gL is just what is expected from Eq. (27) with d*=d. Furthermore, the temperature agrees precisely with the es timate of Tc from high-temperature series by Fisher and Gaunt.11 In Fig. 2 we show that the data scale well with the predicted exponent y* = y instead of yT = 1 / v = 2 . It is possible to evaluate the scaling function G(x) in Eq. (27) exactly for d > 4 because, in the critical region, the probability distribution for the magnetization per spin 5 has the mean-field form PL(s)ccexp[-Ld(ctLs2
+ us4)]
where c and u are constants. <^ = (wL
(35)
:
By
the
substitution
FINITE-SIZE TESTS OF HYPERSCALING
31
FIG. 1. Monte Carlo results for gL as a function of T/J for sizes L =3 to 7, where 16000 iterations per spin have been per formed for each data point. The arrow marks the value 7 ; / / = 8.77 from Ref. 11.
«"-#£-'■
(36)
where the averages are obtained by integrating φ from — oo to -(- oo with the weight PL{(j))ocexp[-a(x
-ο)φ2-φ4]
(37)
where a=c/uxn and ab=A is the coefficient of the shift relation, Eq. (34) (with ψ-χ=γ*τ=ά/2). The solid curve in Fig. 2 is obtained from Eqs. (36) and (37) with b =0.37, a =0.56, and fits the data very well. To conclude, we propose that the free energy of a finite system with periodic boundary conditions scales as in Eq.
1501
FIG. 2. The data for gL shown in Fig. 1 plotted vs the scal ing variable ( T/Tc — 1 )L5/2, where Tc = 8. 777. The solid line is obtained from the mean-field form of the free energy, as described in the text.
(4) with y* and y^ given by Eqs. (9) and (10) and d* =d. For other boundary conditions, where the system has a surface, it is probably necessary to use both tLyT and tLWv for a complete asymptotic description. When hyperscaling is violated it is not possible to determine the correlation length exponent v from the free energy, and its derivatives, and it is necessary to carry out a separate cal culation, using either renormalization-group techniques or explicitly looking at the spatial dependence of the correla tions. We would like to thank M. E. Fisher and J. Rudnick for helpful comments. M. Nauenberg and V. Privman acknowledge support for this research from the National Science Foundation under Grants Nos. PHY-81-15541 andDMR-81-17011.
!
6 B. A. Freedman and G. A. Baker Jr., J. Phys. A 15, L715 V. Privman and M. E. Fisher, J. Stat. Phys. 33, 385 (1983). 7 (1982). For a finite system of linear dimension L some care must be 2 M. N. Barber, R. B. Pearson, D. Toussaint, and J. L. Richard taken in the definition of the correlation length, particularly son, NSF Report No. NSF-ITP83-144 (unpublished). for temperatures T below the critical temperature Tc. For 3 M. E. Fisher, in Renormalization Group in Critical Phenomena periodic boundary conditions we adopt the relation and Quantum Field Theory, proceedings of a conference, edit 2dil=,2(ri-rj)2({slSj)-cL)/,2«SiSj)-cL)t ed by J. D. Gunton and M. S. Green (Temple University, ij ' i,j Philadelphia, 1974); and in Critical Phenomena, Vol. 186 of where 7, is the position of lattice site /', Lecture Notes in Physics, edited by F. J. W. Hahne (Springer, Berlin, 1983). 4 CL = -£7 2 <*/*/·> , K. Binder, Z. Phys. B 43, 119 (1981). 5 A recent review is given by M. N. Barber, in Phase Transitions and /' is the site with 7,< = Γ/ + γ (1,1, . . . , \)L. Note that and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic, New York, 1983), Vol. 8. for L—»oo, cL-^ml, where mb is the bulk magnetization, so
CPSC 2 - paper 1.10
147
1502
BINDER, NAUENBERG, PRIVMAN, A N D YOUNG
ξί-^ξΐ, the square of the bulk correlation length, defined as the second moment of the connected correlation function 8
148
(siSj)-(si)(Sj).
B. Nienhuis and M. Nauenberg, Phys. Rev. Lett. 35, 477 (1975); T. Niemeiyer and J. M. J. van Leeuwen, in Phase Transitions and Critical Phenomena, edited by C. Domb and
CPSC 2 - paper 1.10
31
M. S. Green (Academic, New York, 1976), Vol. 6. E. Brezin, J. Phys. (Paris) 43, 15 (1982). 10 M. E. Fisher, in Critical Phenomena, Vol. 51 of Proceedings of the "Enrico Fermi" International School of Physics, edited by M. S. Green (Academic, New York, 1971). n M . E. Fisher and D. S. Gaunt, Phys. Rev. 133, A224 (1964).
9
VOLUME 31, NUMBER 3
1 FEBRUARY 1985
Finite-size tests of hyperscaling K. Binder Institut für Physik, Universität Mainz, D-6500 Mainz, Postfach 3980, Germany and Institut für Festkörperforschung der Kernforschungsanlage Jülich, D5170 Jülich, Germany
M. Nauenberg Natural Sciences, University of California, Santa Cruz, California 95064 V. Privman Baker Laboratory, Cornell University, Ithaca, New York 14853 A. P. Young Mathematics Department, Imperial College, London, England (Received 10 April 1984) The possible form of hyperscaling violations in finite-size scaling theory is discussed. The impli cations for recent tests in Monte Carlo simulations of the d = 3 Ising model are examined, and new results for the d = 5 Ising model are presented.
Recently two extensive Monte Carlo calculations1'2 have been carried out which test the validity of hyperscal ing in the three-dimensional Ising model. In the calcula tion by Freedman and Baker,1 they "observe a systematic downward trend by more than twice the statistical error for a quantity which should be constant if hyperscaling is satisfied," while Barber, Pearson, Toussaint, and Richard son2 conclude that their results require "the anomalous di mension to be small, and are consistent with hyperscal ing." The purpose of this paper is to resolve this apparent conflict by showing that these two collaborations evaluat ed different anomalous exponents, and that the exponent evaluated by Barber et al., which they erroneously identi fied with Fisher's anomalous dimension,3 actually van ishes on theoretical grounds. New Monte Carlo results are also presented for the five-dimensional Ising model which confirm the theoretical predictions for hyperscaling violation in this case. To start, we investigate the consequence of the existence of a "dangerous irrelevant variable" on the finite-size scal ing form of the free energy, and on the hyperscaling rela tions for critical exponents. We then show that consisten cy conditions require that the anomalous exponent intro duced by Barber et al.2 must be zero. Finally, we discuss the finite-size scaling form of the renormalized coupling constant for the Ising model,4 and the results of a Monte Carlo simulation for the dimensionality d = 5. According to the renormalization-group derivations5'6 of finite-size scaling, the singular part of the free energy fL and the correlation length7 %L have the form fL=L-df(tLyT,hLyH,uLyu) £L=L£(tLyT,hLyH,uLyu),
,
(1) (2)
magnetic field, and u is an irrelevant variable, while yr > 0» yH > 0> a n d yu < 0 are renormalization-group ex ponents. We use equality signs to indicate here asymptot ic scaling relations. If the free-energy scaling function f(x,y,z) is singular in the limit z—»0, then u is called a dangerous irrelevant variable.3'6 For simplicity we as sume here that the correlation scaling function £(x,y,z) is regular in this limit, but later we consider the possibility that it is singular. We assume that for small z, f(x,y,z)=zPlf(xzP\yzPi).
The choice of this particular pattern, multiplicative singu lar powers of z, is motivated by the known3 mechanism for the bulk scaling at d > dc = 4 . Equation (3) implies fL=L-d*F(tLy\hLy"),
144
CPSC 2 - paper
1.10
(4)
where scale factors have been absorbed in t and A, and d* =d -p{yUy yr^yr+Piyu, and y*H =yH +ρ&ν are the effective exponents. Note that the anomalous exponent 2 introduced by Barber et al. corresponds to d — d*, while Fisher's exponent3 ω* is given by af=d-d*AyTv).
(5)
These two exponents would be the same if yjv= 1, which was implicitly assumed by Barber et al.2 but has not been proven. We will show instead that d*=d for cubic or similarly shaped systems,6 as usually employed in Monte Carlo simulations.1'2 The existence of the limit L —► oo of fL corresponding to the bulk free energy / „ implies that for x—»± oo and y | x | ~ Δ fixed,
where t is the reduced temperature t = (T — Tc )/Tc ( Tc is the transition temperature of the infinite system), h is the 31
(3)
F(x,y)=\x\d*/y*Y±(y\x\-*), 1498
(6)
© 1985 The American Physical Society
FINITE-SIZE TESTS OF HYPERSCALING
31
where Δ=>>#/y*, and Y± is the scaling function for i < 0 . Likewise the existence of the bulk correlation length ξχ in this limit leads to the asymptotic form \-vx±(y
ξ{χ,γ,0)=\χ
\χ Γ Δ ' ) ,
(18)
asr-+77 .
To obtain this result from the scaling form, Eq. (14), we must require6
(7)
lim W(x,0)oz \χ \2β
(19)
X-*±oo
where v—\/yT
and Δ'=>>///>>Γ
(8)
and we assumed for simplicity that the scaling function ξ in Eq. (2) is nonsingular as z—►(). Taking suitable deriva tives of the free energy, one finds the following scaling re lations a = 2 - c T / > * , ß=d*/y}-&, and y = 2Δ-*, where α, β, and γ are the thermodynamic exponents for the specific heat, magnetization, and susceptibility, respectively. Consequently y\ and y% can be written as yl=d*/(Y
(9)
+ 2ß) ,
y*H=d*(Y+ß)/(Y
(10)
+ 2ß),
Hyperscaling violations occur if yj and/or Δ differ from the exponents yT( = \/v) and Δ'(=;>#/> Γ ) appearing in the scaling form of the correlation length. However, we will show that even in this case one must have d*—d. We now give three arguments which support d*=d. First, consider the finite-size magnetization mL and sus ceptibility XL: mL = (s)L , d
2
(11) (12)
2
XL=L {(s )L-(s) L),
where s ={\/Ld)^siy s, is the spin at the ith site, and ( ) L denotes the thermal average. According to Eq. (4), mL and XL have the scaling form Mt=
to
i /Ä-'V ( i i/? f / u /i) >
(13)
ß mb =a 11 | and Xb=b
\ t \~Ύ as Γ—Γ"
(15)
so that, by Eq. (12), {s2)L=ml+L-dXb
(i<0,L-
)).
(16)
However, if we first set A = 0 and then take the limit L -¥ oo, then (s ) L = 0 but we expect (see Refs. 4 and 6 for details) lim [{s )L>h=0]
dm=2(yj,+ßyi)-d.
(20)
Substituting Eqs. (9) and (10) in Eq. (20), we obtain the re lation
d*=d ,
(21)
which also implies px = 0 in Eq. (3). Second, we show that this condition also follows from the finite-size scaling form of fL at the ferromagnetic phase boundary due to the existence of a discontinuity fixed point.8 In this case the finite-size magnetization mL below the critical temperature takes the form6 mL = mb tanh(mbhLd)
= ml
(17)
[in fact, Eq. (16) applies in this limit as well4,6]. By Eqs. (12) and (17), the zero-field susceptibility behaves accord ing to
(22) l l d
for L »£b, and | h \ <<(mb£b)~ L ~ , where £b is the bulk correlation length,, and mb is the bulk magnetization, Eq. (15). Hence, XL is given by XL=Ldmb2cosh-2(mbhLd)
.
(23)
For A = 0 , XL reduces to Eq. (18) when T^T~y and therefore implies d* =d, Eq. (21). Finally, we consider the- finite-size scaling properties of the zero-field probability distribution P(s) of the magneti zation below the critical temperature. Binder4 has shown that for large L and s near ±mb, P^is) can be written ap proximately in the form
P , ü ) = T ^ W1/2( e - ( S - ^ , 2 L ^ v
(14)
2 where the oh scaling functions V and W are obtained from F. For the bulk magnetization, mb, as well as the suscepti bility, Xb, special care must be taken in order of the limits L —► oo and h -+0± below the critical temperature. Tak ing first the limit L—»oo, and afterwards the limit A —►O*, one obtains the conventional bulk values,
2
and
2(2ir*a)
on
L—*tx>
XLocLd\t\2ß
1499
-
(24)
Hence the arguments of the exponential functions have the form -ß+a)2(\t\LyT)r+2e 2b which demonstrates the occurrence of the scaling com bination | t \LyT in PL(s), where γτ=ά/{γ+2β). Like wise, by including a magnetic field A, the arguments of the exponential become (s±mb—Xbh)2Ld/2Xb and the term linear in h can be written as
<*l'l
(b \t \Ld/{Y+2ß)r{r+ß)hLy"
.
This shows the dependence of PL(s) on the scaling vari able / / " , where y*H=d{y+ß)/{y + 2ß). These expres sions for j>* and y% are precisely Eqs. (9) and (10), but with d* =d. If we assume that no other scaling variables occur, the scaling form Eq. (4) follows, with d* =d. Next, we discuss some recent Monte Carlo calcula tions1,2 which test possible hyperscaling violation in the
CPSC 2 - paper 1.10
145
d = 3 Ising model. These calculations evaluated the A = 0 finite-size renormalized coupling gL introduced by Binder, 4 (s4)L
gL'-
2 2
(s )
v(4)
-3 =
L
(25)
LdX\
XiL*) =
34/L
(26)
^~L*y*»-d*W(*\tLyh.
ΘΑ4
Substituting Eqs. (14) and (28) in Eq. (27), we find that for i = 0 , d
gL=L
d
y
*- G(tL T),
(27)
where G is a scaling function. Since d* =d, it follows that gL is a constant (nonzero in general) at T = TC. The finite-size scaling analysis of the Monte Carlo data by Barber et al.2 implies that indeed d —d*=0, with an error of ± 0 . 0 4 due to the uncertainty in their determina tion of the critical temperature. This was interpreted by them as evidence for the validity of hyperscaling, but we have shown here that d*=d even when hyperscaling is violated. For t > 0 and L-+oo,XL and Χ{*] take their bulk values XL oc t ~γ and Xli]«t-γ~2£ί so that
&«<*'·**£')-' forL»;-17'*,
(28)
where we have used y — 2 Δ = — d* /y*. Baker and Freedman1 evaluated gL for values of t > 0 such that the corre lation length ξι =cL, where c is a constant. After some algebra, this yields SL^L'
(29) 3
where ω*, Eq. (5), is Fisher's anomalous dimension, and is positive. For lattices of dimensions L = 3 to 60 they find ω * ~ 0 . 2 . Note, however, that their result depends on the assumption, Eq. (6), that the scaling function ξ in Eq. (2) is not singular as uLyu-+0. We now briefly turn our attention to the possibility that ξ(χ,)?,ζ) is singular in the limit z—»0. In analogy with Eq. (3) we assume that
g(x,y,z)=z'l£(xzq2,yz93)
,
(30)
which implies (31) anc where >>** =yr+92yu * yV—yH+Q&u· the limit x - ± oo and y \ x "Δ** fixed,
Z(xfy)-
>\x\-vZ±(y\x\-^
),
Likewise in (32)
where v=(l-\-qlyu)/y** and Δ**=^*/ν**. Since the finite-size correlation length £L is bounded by L, we re quire q\yu < 0. Even if one adopts a plausible assumption that for t=h=0, the correlation length increases up to the linear dimensions of the lattice, which implies that qx = 0 and v = 1/y**, y$* need not be equal to yT leading to a new possible source for hyperscaling violation.
CPSC 2 - paper
1.10
Finally, we comment on the effect of boundary condi tions on our analysis. The predictions presented here are unchanged in the thermodynamic limit (L —► oo ) if the re duced temperature t in Eq. (4) is replaced by a "shifted" variable tL, where tL=[T-Te(L)]/Te
where XL is given by Eq. (14), and
146
31
BINDER, NAUENBERG, PRIVMAN, AND YOUNG
1500
,
(33)
in which TC(L) is a "pseudo Γ 0 " for the finite system. This could, for example, be defined as the temperature where the probability distribution of the magnetization starts to develop a two-peak structure. One then defines a shift exponent ψ by
[TC(L)-TC]/TC=AL-W*
,
(34)
where A is a constant. For systems with a surface it is ex pected that ψ=ν, because properties of the finite system should differ from their bulk values when £L approaches the system size. In this case the shift in Tc is greater than the range of the finite-size rounding if hyperscaling is violated, because y\ > y. However, we shall argue below that ψ= 1/y* for periodic boundary conditions. A well-known example which violates hyperscaling is the Ising model (more generally the n-component vector model) for c?>4, where critical exponents stick at their mean field values. It is straightforward to show6 that the free energy of a finite system scales as in Eq. (4) with yj=d/2, yfj=d/4 compared with j > r = l / v = 2 , and yH = (d +2)/2. However, these6 renormalization-group arguments do not give information on the size of the shift in Tc, for which an explicit calculation must be per formed. This is possible for the «-component model with periodic boundary conditions in the limit w—>>οο. By a straightforward extension of the work of Brezin9 one can evaluate the scaling function W(x,0) for the susceptibili ty, and see that the scaling form of Eq. (4) is valid, so the coefficient of the L ~ 1 / v term in Eq. (34) must vanish for this model. We observe that this is a general result for periodic boundary conditions provided that tLyT is the only temperature variable which enters thermodynamic scaling, and then ^ = l / y £ follows by a standard argu ment.10 We have tested our claim that d*—d by Monte Carlo simulations on the five-dimensional Ising model on a cu bic lattice with nearest-neighbor interaction, / , for sizes between L = 3 and 7. Figure 1 shows that the data for gL intersect at Γ / / ~ 8 . 7 7 , g L ~ - 1 . 0 0 . A nonzero value of gL is just what is expected from Eq. (27) with d*=d. Furthermore, the temperature agrees precisely with the es timate of Tc from high-temperature series by Fisher and Gaunt.11 In Fig. 2 we show that the data scale well with the predicted exponent y* = y instead of yT = 1 / v = 2 . It is possible to evaluate the scaling function G(x) in Eq. (27) exactly for d > 4 because, in the critical region, the probability distribution for the magnetization per spin 5 has the mean-field form PL(s)ccexp[-Ld(ctLs2
+ us4)]
where c and u are constants. <^ = (wL
(35)
:
By
the
substitution
FINITE-SIZE TESTS OF HYPERSCALING
31
FIG. 1. Monte Carlo results for gL as a function of T/J for sizes L =3 to 7, where 16000 iterations per spin have been per formed for each data point. The arrow marks the value 7 ; / / = 8.77 from Ref. 11.
«"-#£-'■
(36)
where the averages are obtained by integrating φ from — oo to -(- oo with the weight PL{(j))ocexp[-a(x
-ο)φ2-φ4]
(37)
where a=c/uxn and ab=A is the coefficient of the shift relation, Eq. (34) (with ψ-χ=γ*τ=ά/2). The solid curve in Fig. 2 is obtained from Eqs. (36) and (37) with b =0.37, a =0.56, and fits the data very well. To conclude, we propose that the free energy of a finite system with periodic boundary conditions scales as in Eq.
1501
FIG. 2. The data for gL shown in Fig. 1 plotted vs the scal ing variable ( T/Tc — 1 )L5/2, where Tc = 8. 777. The solid line is obtained from the mean-field form of the free energy, as described in the text.
(4) with y* and y^ given by Eqs. (9) and (10) and d* =d. For other boundary conditions, where the system has a surface, it is probably necessary to use both tLyT and tLWv for a complete asymptotic description. When hyperscaling is violated it is not possible to determine the correlation length exponent v from the free energy, and its derivatives, and it is necessary to carry out a separate cal culation, using either renormalization-group techniques or explicitly looking at the spatial dependence of the correla tions. We would like to thank M. E. Fisher and J. Rudnick for helpful comments. M. Nauenberg and V. Privman acknowledge support for this research from the National Science Foundation under Grants Nos. PHY-81-15541 andDMR-81-17011.
!
6 B. A. Freedman and G. A. Baker Jr., J. Phys. A 15, L715 V. Privman and M. E. Fisher, J. Stat. Phys. 33, 385 (1983). 7 (1982). For a finite system of linear dimension L some care must be 2 M. N. Barber, R. B. Pearson, D. Toussaint, and J. L. Richard taken in the definition of the correlation length, particularly son, NSF Report No. NSF-ITP83-144 (unpublished). for temperatures T below the critical temperature Tc. For 3 M. E. Fisher, in Renormalization Group in Critical Phenomena periodic boundary conditions we adopt the relation and Quantum Field Theory, proceedings of a conference, edit 2dil=,2(ri-rj)2({slSj)-cL)/,2«SiSj)-cL)t ed by J. D. Gunton and M. S. Green (Temple University, ij ' i,j Philadelphia, 1974); and in Critical Phenomena, Vol. 186 of where 7, is the position of lattice site /', Lecture Notes in Physics, edited by F. J. W. Hahne (Springer, Berlin, 1983). 4 CL = -£7 2 <*/*/·> , K. Binder, Z. Phys. B 43, 119 (1981). 5 A recent review is given by M. N. Barber, in Phase Transitions and /' is the site with 7,< = Γ/ + γ (1,1, . . . , \)L. Note that and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic, New York, 1983), Vol. 8. for L—»oo, cL-^ml, where mb is the bulk magnetization, so
CPSC 2 - paper 1.10
147
1502
BINDER, NAUENBERG, PRIVMAN, A N D YOUNG
ξί-^ξΐ, the square of the bulk correlation length, defined as the second moment of the connected correlation function 8
148
(siSj)-(si)(Sj).
B. Nienhuis and M. Nauenberg, Phys. Rev. Lett. 35, 477 (1975); T. Niemeiyer and J. M. J. van Leeuwen, in Phase Transitions and Critical Phenomena, edited by C. Domb and
CPSC 2 - paper 1.10
31
M. S. Green (Academic, New York, 1976), Vol. 6. E. Brezin, J. Phys. (Paris) 43, 15 (1982). 10 M. E. Fisher, in Critical Phenomena, Vol. 51 of Proceedings of the "Enrico Fermi" International School of Physics, edited by M. S. Green (Academic, New York, 1971). n M . E. Fisher and D. S. Gaunt, Phys. Rev. 133, A224 (1964).
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