- Email: [email protected]
Shape of cross-over between mean-field and asymptotic critical behavior three-dimensional Ising lattice
4 October 1999
Physics Letters A 261 Ž1999. 89–93 www.elsevier.nlrlocaterphysleta
Shape of cross-over between mean-field and asymptotic critical behavior three-dimensional Ising lattice M.A. Anisimov a
a,b
, E. Luijten
c,d
, V.A. Agayan a , J.V. Sengers
a,b,)
, K. Binder
d
Institute for Physical Science and Technology, UniÕersity of Maryland, College Park, MD 20742, USA b Department of Chemical Engineering, UniÕersity of Maryland, College Park, MD 20742, USA c Max-Planck-Institut fur ¨ Polymerforschung, Postfach 3148, D-55021, Mainz, Germany d Institut fur ¨ Physik, WA 331, Johannes Gutenberg-UniÕersitat, ¨ D-55099 Mainz, Germany Received 19 August 1999; accepted 24 August 1999 Communicated by A. Lagendijk
Abstract Recent numerical studies of the susceptibility of the three-dimensional Ising model with various interaction ranges have been analyzed with a cross-over model based on renormalization-group matching theory. It is shown that the model yields an accurate description of the cross-over function for the susceptibility. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 05.70.Jk; 64.60.Fr Keywords: Crossover critical phenomena; Ising model; Susceptibility
Recently, an accurate numerical study of the cross-over from asymptotic ŽIsing-like. critical behavior to classical Žmean-field. behavior has been performed both for two-dimensional w1,2x and threedimensional w3x Ising systems in zero field on either side of the critical temperature with a variety of interaction ranges. It is the objective of the present work to analyze these numerical results within the framework of a cross-over theory that is based on renormalization-group matching and that has already successfully been applied to the description of cross-over in several experimental systems w4,5x.
)
Corresponding author. E-mail: [email protected]
Qualitatively, the cross-over is ruled by the parameter trG where t s ŽT y Tc .rTc is the reduced temperature distance to the critical temperature Tc and G the Ginzburg number w6x. The Ginzburg number depends on the normalized interaction range R as G s G 0 Ry2 d rŽ4yd . ,
Ž 1.
where d is the dimensionality of space and G 0 a constant. Hence, for d s 3 the cross-over occurs as a function of tR 6 . Asymptotic critical behavior takes place for tR 6 < 1 and classical behavior is expected for tR 6 4 1. In real fluids the cross-over is never completed in the critical domain Žwhere t < 1., since the range of interaction is of the same order of magnitude as the distance between molecules Ž R , 1.
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 5 9 1 - 5
90
M.A. AnisimoÕ et al.r Physics Letters A 261 (1999) 89–93
w4x. A new Monte-Carlo algorithm, developed by Luijten and Blote ¨ w7x, offers the advantage that the ratio trG can be tuned over more than eight orders of magnitude allowing one to cover the full cross-over region in three-dimensional spin models w3x. A sensitive description of cross-over behavior is obtained from an analysis of the effective critical exponent of the susceptibility Žthe third derivative of the free energy., defined as
geff"' y dln xˆr dln < t < ,
Ž 2.
where the scaled susceptibility xˆ s k B Tc Ž R .Ž E mr E h.T , k B the Boltzmann constant, m the order parameter, h the ordering field, and where the ‘q’ sign applies for T ) Tc , and the ‘y’ sign for T - Tc . As is seen from Figs. 1 and 2, the variation of geff" reproduces the Ising asymptotic critical behavior Žgeff", 1.24. at tR 6 < 1 as well as the mean-field asymptote Žgeff"s g MF s 1. at tR 6 4 1. Apparently, all data would seem to collapse onto a universal function of the reduced variable tR 6 as predicted by a field-theoretical treatment w8,9x and by the ´-expansion w10x. However, as was noted in Ref. w3x, a more careful look at the data reveals a remarkable discrepancy between the theoretical calculations w8– 12x and the simulation results. Namely, the shape of the cross-over is sharper than predicted by the theory
y Fig. 2. The effective susceptibility exponent geff below Tc . The w x symbols indicate numerical simulation data 3 . The solid curves represent values calculated from the renormalization-group matching cross-over model.
w11,12x, especially for short ranges of interaction. We will show that this discrepancy is related to the findings of Refs. w4,5x, where it was shown that there is a fundamental problem in describing the cross-over of geff" by a universal function which contains only a single cross-over parameter G A Ry6 . In zero-ordering field above Tc the susceptibility asymptotically close to the critical point behaves as
x s G 0 tyg Ž 1 q G 1 t Ds q G 2 t 2 D s q a1 t q . . . . ,
q Fig. 1. The effective susceptibility exponent geff above Tc . The symbols indicate numerical simulation data w3x. The solid curves represent values calculated from Eq. Ž5.. The dashed-dotted curve corresponds to the limit u™ 0. The dotted curve is a continuation of the cross-over curve for us1.22. For clarity, the error bars have been omitted; they are all of the order of 0.004.
Ž 3.
where g s 1.239 " 0.002 Žsee, e.g., Refs. w13,14x and references therein. and Ds s 0.504 " 0.008 w15x are universal Ising critical exponents, and where G 0 , G 1 , G 2 , and a1 are system-dependent amplitudes. Expansion Ž3. is called the Wegner series w16x. In a universal single-parameter cross-over theory w8–10x, the Ginzburg number is responsible both for the range of validity of the mean-field approximation and for the convergence of the Wegner series Ž3.. However, it is known w17–19x, that the sign of the first Wegner correction amplitude G 1 depends on the difference u y u ) , where u is the scaled coupling constant and u ) s 0.472 is the universal coupling constant at the Ising fixed point w20x. Moreover, Liu and Fisher w18x concluded that the three-dimensional nearest-neighbor Ising model has a negative leading Wegner correction amplitude G 1 , so that geff" asymptotically approaches g , 1.24 from above. Therefore, since the coupling constant itself depends on the
M.A. AnisimoÕ et al.r Physics Letters A 261 (1999) 89–93
interaction range, the shape of geff" cannot be represented by a universal function of the Ginzburg number, since G is not proportional to the difference u y u). In this paper we therefore present an analysis of the numerical data for geff" w3x in terms of a cross-over model based on renormalization-group matching for the free-energy density w17,19,21x. This model contains two cross-over parameters u s uru ) and L Ža dimensionless cut-off wave number., and two rescaled amplitudes c t and cr related to the coefficients of the local density of the classical Landau– Ginzburg free energy D A: Õ0 dŽ D A . k BT
dV
s 12 a 0tw 2 q s 12 c tt M 2 q
1 4! 1 4!
u 0 w 4 q 12 c 0 Ž =w .
2
2
˜ . , u ) u L M 4 q 12 Ž =M Ž 4.
1r2 with t s ŽT y Tc .rT, M s cr w s Ž a 0rc t . w , a0 s ˜ s Õ 01r3 =. cr2 c t , u 0 s u ) u L cr4 , c 0 s cr2 Õ 02r3 , and = The average molecular volume Õ 0 and the prefactor Õ 0rk B T are introduced to make the free-energy density and all the coefficients dimensionless. The inverse cross-over susceptibility xy1 s ˜ E M 2 . t , where D A˜ is the cross-over ŽrenorŽ E 2D Ar malized. free-energy density, in zero field above Tc reads w4x
xy1 s cr2 c tt Y Ž gy1.r D s Ž 1 q y .
Ž 5.
the critical temperature. The cross-over function Y is defined by
L
1 y Ž 1 y u. Y s u 1 q
ys
2 Ds
½ž 2
L
n =
q
Ds 2n y 1
y
Ds
2
k
/
L
k 2 s ct
1q
ž / k
Ž 1 y u. Y 1 y Ž 1 y u. Y y1
5
,
Ž 6.
where n , 0.630 w15,22x is the critical exponent of the asymptotic power law for the correlation length j w4x. Note that xy1 s ŽTcrT . xˆy1 and the relation between geff ' y dln xr dln
ž / k
Y n r Ds
Ž 7.
T Tc
t Y Ž2 ny1.r D s s c t tY Ž2 ny1.r D s .
Ž 8.
We modified the original expression for k 2 , given by Eq. Ž3. in Ref. w4x, by introducing the nonasymptotic factor TrTc in Eq. Ž8. so that k 2 becomes infinite at T ™ ` w23x. Asymptotically close to the critical point Ž Lrk 4 1., the following expression is obtained for the first correction amplitude G 1 in Eq. Ž3.:
G1 s g1
(c
2 Ds t
ž / uL
Ž 1 y u. ,
Ž 9.
where g 1 , 0.62 is a universal constant w21x. In the approximation of an infinite cut-off L ™ `, which physically means neglecting the discrete structure of matter, u s u 0 c t2rŽ u )L a20 . ™ 0 and the two cross-over parameters u and L in the cross-over equations collapse into a single one, u L, which is related to the Ginzburg number G by w21x G s g0
2
2 1r2
and is to be found numerically. The parameter k in Eq. Ž7. is inversely proportional to the fluctuation-induced portion of the correlation length and serves as a measure of the distance to the critical point. In zero field above Tc the expression for k 2 reads:
with u )n
91
Ž u L. ct
2
s g0
u 20 Õ 02 2 Ž u ) . a04 j 06
,
Ž 10 .
where g 0 , 0.028 is a universal constant w21x and j 0 s Õ 01r3 cy1r2 s Ž c 0ra0 .1r2 is the mean-field amt plitude of the power law for the correlation length. Note that the Ginzburg number does not depend explicitly on the cut-off L or on u. This singleparameter cross-over, i.e., the cross-over for u s 0, is universal and is indicated in Fig. 1 by a dasheddotted curve. This simplified description of the cross-over is equivalent to the results of Bagnuls and Bervillier w9x and of Belyakov and Kiselev w10x. In the simulations w3x, each spin interacts equally with its z neighbors lying within a distance R m on a three-dimensional cubic lattice. The effective range
92
M.A. AnisimoÕ et al.r Physics Letters A 261 (1999) 89–93
of interaction R is then defined as R 2 s zy1 Ý j/ i < r i y r j < 2 with < r i y r j < F R m w1x. We have approximated the relation between R and R m by R 2 . s 35 R 2m Ž1 q 23 Ry2 m , as indicated in the insert in Fig. 3. In order to compare the numerical results to the theoretical prediction Eq. Ž5., we need the range dependence of the parameters c t and u. Indeed, the asymptotic R dependence of u follows directly from simple scaling arguments w1x, u s u 0 Ry4 , and c t varies as its square root, c t s c t 0 Ry2 . For a three-dimensional simple cubic lattice, L s p w18,24x, and we obtain for the Ginzburg number G s G 0 Ry6 s 0.28 Ž u 20rc t40 . c t3 s 0.28 Ž u 20rc t 0 . Ry6 . Ž 11 . The non-universal parameters c t 0 and u 0 have to be determined from a least-squares fit to the numerical q data for geff , which yielded c t 0 s 1.72 and u 0 s 1.22 and hence G 0 f 0.24. The solid lines in Fig. 1 indicate the corresponding theoretical curves. It should be noted that these curves are calculated for each value of R m separately; the piecewise continuous character of this description directly reflects the fact that the cross-over cannot be described by a universal single-parameter function. Indeed, Fig. 1 also shows two attempts to describe the data in terms of such a function. The dash-dotted line corresponds to the limit u ™ 0, whereas the dotted curve corre-
Fig. 3. Dependence of the normalized coupling constant u on the normalized interaction range R. Note that u becomes larger than unity for very short interaction ranges. Insert: Effective range of interaction R Žopen circles. plotted as a function of R m . The solid line corresponds to the approximation mentioned in the text and the dashed line represents the asymptotic behavior for large R.
sponds to u 0 s 1.22 and L s p Ža continuation of the theoretical curve for R s 1.. We see that the actual cross-over lies between these two bounding curves, with u , 0 for large R and u , 1.2 for R s 1. Thus, it is clearly seen that without including the R dependence of u it is impossible to describe data for short interaction ranges R 2m F 5. The dependence of u on R is shown in Fig. 3. The two adjustable parameters c t 0 and u 0 are strongly correlated and if one of them is fixed at a predicted value, the quality of the description remains the same. We hence tried to fit the data while keeping c t 0 fixed at the theoretically predicted value c t 0 s 2 d s 6 w25,26x. In this case a fit of the same quality is obtained with u 0 s 1.22, provided that L , 2p . The value of G 0 f 0.24 then remains unchanged. To describe the data below the critical temperature, a connection between M and t in zero field is ˜ E M .t s 0. The to be found from the condition Ž E D Ar relation between M and t appears to be implicit and x as a function of t cannot be expressed in an explicit form either. Of course, the parameters c t 0 and u 0 should be the same as for T ) Tc and we hence kept them fixed at the above-mentioned values. However, the parameter G 0 appearing in Eq. Ž11. will take a different value. We took this into y account by introducing a factor Gq 0 rG 0 into the q y temperature scale: t ™ t P Ž G 0 rG 0 .. Fig. 2 shows y the results for T - Tc , where the factor Gq 0 rG 0 was included as an adjustable parameter. Our estimate q Gy 0 rG 0 s 2.58 must be compared with the theoretiy q w x cal result Gy 0 rG 0 s 3.125 27 . Interestingly, geff 6 2 clearly shows a minimum around < t < R ; 10 . This corroborates the non-monotonic character of the y cross-over of geff , earlier observed for the two-dimensional Ising lattice w2x, where the effect is much more pronounced. We note that already in Ref. w28x a field-theoretic calculation of the cross-over in the low-temperature regime has been given Žin the limit u ™ 0., but only recently this has been extended to cover the full cross-over region w29x. Actually, also y here a non-monotonicity in geff has been observed. In summary, we remark that although in general the theory contains two cross-over parameters u and L, only one parameter Ž u. changes with the range of interaction. However, this does not mean that the cross-over is a universal function of tR 6 . Indeed, the effective range of interaction R affects the behavior
M.A. AnisimoÕ et al.r Physics Letters A 261 (1999) 89–93
of geff" in a twofold way: through the Ginzburg number, which is proportional to c t3 , and through the first Wegner correction, with an amplitude G 1 that is proportional to Ž1 y u. wEq. Ž9.x. Hence, there is no way to describe the data for short interaction ranges without allowing for u to become larger than unity and correspondingly G 1 to change its sign between R m s 2 and R m s 1 as indicated in Fig. 3. In previous publications we have shown that Eq. Ž5., derived from renormalization-group matching, gives an excellent representation of the experimentally observed cross-over behavior in simple and complex fluids w4,5,30x. From the evidence presented in this paper, we conclude that the same cross-over model also yields a quantitative description of the cross-over critical behavior of a three-dimensional Ising lattice. Acknowledgements We acknowledge valuable discussions with M. E. Fisher and assistance from A. A. Povodyrev. The research at the University of Maryland was supported by DOE Grant No. DE-FG02-95ER-14509. E. Luijten acknowledges the HLRZ Julich for comput¨ ing resources on a Cray-T3E. References w1x E. Luijten, H.W.J. Blote, ¨ K. Binder, Phys. Rev. E 54 Ž1996. 4626. w2x E. Luijten, H.W.J. Blote, ¨ K. Binder, Phys. Rev. Lett. 79 Ž1997. 561. w3x E. Luijten, K. Binder, Phys. Rev. E 58 Ž1998. R4060.
93
w4x M.A. Anisimov, A.A. Povodyrev, V.D. Kulikov, J.V. Sengers, Phys. Rev. Lett. 75 Ž1995. 3146. w5x Y.B. Melnichenko, M.A. Anisimov, A.A. Povodyrev, G.D. Wignall, J.V. Sengers, W.A. Van Hook, Phys. Rev. Lett. 79 Ž1997. 5266. w6x A.Z. Patashinskii, V.L. Pokrovskii, Fluctuation Theory of Phase Transitions ŽPergamon, Oxford, 1979.. w7x E. Luijten, H.W.J. Blote, ¨ Int. J. Mod. Phys. C 6 Ž1995. 359. w8x C. Bagnuls, C. Bervillier, J. Phys. Lett. ŽParis. 45 Ž1984. L-95. w9x C. Bagnuls, C. Bervillier, Phys. Rev. B 32 Ž1985. 7209. w10x M.Y. Belyakov, S.B. Kiselev, Physica A 190 Ž1992. 75. w11x P. Seglar, M.E. Fisher, J. Phys. C 13 Ž1980. 6613. w12x M.E. Fisher, Phys. Rev. Lett. 57 Ž1986. 1911. w13x J.V. Sengers, J.M.H. Levelt Sengers, Ann. Rev. Phys. Chem. 37 Ž1986. 189. w14x A.J. Liu, M.E. Fisher, Physica A 156 Ž1989. 35. w15x R. Guida, J. Zinn-Justin, J. Phys. A 31 Ž1998. 8103. w16x F.J. Wegner, Phys. Rev. B 5 Ž1972. 4529. w17x J.F. Nicoll, P.C. Albright, Phys. Rev. B 31 Ž1985. 4576. w18x A.J. Liu, M.E. Fisher, J. Stat. Phys. 58 Ž1990. 431. w19x S. Tang, J.V. Sengers, Z.Y. Chen, Physica A 179 Ž1991. 344. w20x J.C. Le Guillou, J. Zinn-Justin, J. Phys. ŽParis. 48 Ž1987. 19. w21x M.A. Anisimov, S.B. Kiselev, J.V. Sengers, S. Tang, Physica A 188 Ž1992. 487. w22x H.W.J. Blote, ¨ E. Luijten, J.R. Heringa, J. Phys. A 28 Ž1995. 6289. w23x A. Kostrowicka Wyczalkowska, M.A. Anisimov, J.V. Sengers, Fluid Phase Equilibria 158-160 Ž1999. 523. w24x M.E. Fisher, Rev. Mod. Phys. 46 Ž1974. 597. w25x M.E. Fisher, R.J. Burford, Phys. Rev. 156 Ž1967. 583. w26x E. Luijten, unpublished results. w27x M. Lifschitz, J. Dudowicz, K.F. Freed, J. Chem. Phys. 100 Ž1994. 3957. w28x C. Bagnuls, C. Bervillier, D.I. Meiron, B.G. Nickel, Phys. Rev. B 35 Ž1987. 3585. w29x A. Pelissetto, P. Rossi, E. Vicari, Phys. Rev. E 58 Ž1998. 7146. w30x J. Jacob, A. Kumar, M.A. Anisimov, A.A. Povodyrev, J.V. Sengers, Phys. Rev. E 58 Ž1998. 2188.
Physics Letters A 261 Ž1999. 89–93 www.elsevier.nlrlocaterphysleta
Shape of cross-over between mean-field and asymptotic critical behavior three-dimensional Ising lattice M.A. Anisimov a
a,b
, E. Luijten
c,d
, V.A. Agayan a , J.V. Sengers
a,b,)
, K. Binder
d
Institute for Physical Science and Technology, UniÕersity of Maryland, College Park, MD 20742, USA b Department of Chemical Engineering, UniÕersity of Maryland, College Park, MD 20742, USA c Max-Planck-Institut fur ¨ Polymerforschung, Postfach 3148, D-55021, Mainz, Germany d Institut fur ¨ Physik, WA 331, Johannes Gutenberg-UniÕersitat, ¨ D-55099 Mainz, Germany Received 19 August 1999; accepted 24 August 1999 Communicated by A. Lagendijk
Abstract Recent numerical studies of the susceptibility of the three-dimensional Ising model with various interaction ranges have been analyzed with a cross-over model based on renormalization-group matching theory. It is shown that the model yields an accurate description of the cross-over function for the susceptibility. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 05.70.Jk; 64.60.Fr Keywords: Crossover critical phenomena; Ising model; Susceptibility
Recently, an accurate numerical study of the cross-over from asymptotic ŽIsing-like. critical behavior to classical Žmean-field. behavior has been performed both for two-dimensional w1,2x and threedimensional w3x Ising systems in zero field on either side of the critical temperature with a variety of interaction ranges. It is the objective of the present work to analyze these numerical results within the framework of a cross-over theory that is based on renormalization-group matching and that has already successfully been applied to the description of cross-over in several experimental systems w4,5x.
)
Corresponding author. E-mail: [email protected]
Qualitatively, the cross-over is ruled by the parameter trG where t s ŽT y Tc .rTc is the reduced temperature distance to the critical temperature Tc and G the Ginzburg number w6x. The Ginzburg number depends on the normalized interaction range R as G s G 0 Ry2 d rŽ4yd . ,
Ž 1.
where d is the dimensionality of space and G 0 a constant. Hence, for d s 3 the cross-over occurs as a function of tR 6 . Asymptotic critical behavior takes place for tR 6 < 1 and classical behavior is expected for tR 6 4 1. In real fluids the cross-over is never completed in the critical domain Žwhere t < 1., since the range of interaction is of the same order of magnitude as the distance between molecules Ž R , 1.
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 5 9 1 - 5
90
M.A. AnisimoÕ et al.r Physics Letters A 261 (1999) 89–93
w4x. A new Monte-Carlo algorithm, developed by Luijten and Blote ¨ w7x, offers the advantage that the ratio trG can be tuned over more than eight orders of magnitude allowing one to cover the full cross-over region in three-dimensional spin models w3x. A sensitive description of cross-over behavior is obtained from an analysis of the effective critical exponent of the susceptibility Žthe third derivative of the free energy., defined as
geff"' y dln xˆr dln < t < ,
Ž 2.
where the scaled susceptibility xˆ s k B Tc Ž R .Ž E mr E h.T , k B the Boltzmann constant, m the order parameter, h the ordering field, and where the ‘q’ sign applies for T ) Tc , and the ‘y’ sign for T - Tc . As is seen from Figs. 1 and 2, the variation of geff" reproduces the Ising asymptotic critical behavior Žgeff", 1.24. at tR 6 < 1 as well as the mean-field asymptote Žgeff"s g MF s 1. at tR 6 4 1. Apparently, all data would seem to collapse onto a universal function of the reduced variable tR 6 as predicted by a field-theoretical treatment w8,9x and by the ´-expansion w10x. However, as was noted in Ref. w3x, a more careful look at the data reveals a remarkable discrepancy between the theoretical calculations w8– 12x and the simulation results. Namely, the shape of the cross-over is sharper than predicted by the theory
y Fig. 2. The effective susceptibility exponent geff below Tc . The w x symbols indicate numerical simulation data 3 . The solid curves represent values calculated from the renormalization-group matching cross-over model.
w11,12x, especially for short ranges of interaction. We will show that this discrepancy is related to the findings of Refs. w4,5x, where it was shown that there is a fundamental problem in describing the cross-over of geff" by a universal function which contains only a single cross-over parameter G A Ry6 . In zero-ordering field above Tc the susceptibility asymptotically close to the critical point behaves as
x s G 0 tyg Ž 1 q G 1 t Ds q G 2 t 2 D s q a1 t q . . . . ,
q Fig. 1. The effective susceptibility exponent geff above Tc . The symbols indicate numerical simulation data w3x. The solid curves represent values calculated from Eq. Ž5.. The dashed-dotted curve corresponds to the limit u™ 0. The dotted curve is a continuation of the cross-over curve for us1.22. For clarity, the error bars have been omitted; they are all of the order of 0.004.
Ž 3.
where g s 1.239 " 0.002 Žsee, e.g., Refs. w13,14x and references therein. and Ds s 0.504 " 0.008 w15x are universal Ising critical exponents, and where G 0 , G 1 , G 2 , and a1 are system-dependent amplitudes. Expansion Ž3. is called the Wegner series w16x. In a universal single-parameter cross-over theory w8–10x, the Ginzburg number is responsible both for the range of validity of the mean-field approximation and for the convergence of the Wegner series Ž3.. However, it is known w17–19x, that the sign of the first Wegner correction amplitude G 1 depends on the difference u y u ) , where u is the scaled coupling constant and u ) s 0.472 is the universal coupling constant at the Ising fixed point w20x. Moreover, Liu and Fisher w18x concluded that the three-dimensional nearest-neighbor Ising model has a negative leading Wegner correction amplitude G 1 , so that geff" asymptotically approaches g , 1.24 from above. Therefore, since the coupling constant itself depends on the
M.A. AnisimoÕ et al.r Physics Letters A 261 (1999) 89–93
interaction range, the shape of geff" cannot be represented by a universal function of the Ginzburg number, since G is not proportional to the difference u y u). In this paper we therefore present an analysis of the numerical data for geff" w3x in terms of a cross-over model based on renormalization-group matching for the free-energy density w17,19,21x. This model contains two cross-over parameters u s uru ) and L Ža dimensionless cut-off wave number., and two rescaled amplitudes c t and cr related to the coefficients of the local density of the classical Landau– Ginzburg free energy D A: Õ0 dŽ D A . k BT
dV
s 12 a 0tw 2 q s 12 c tt M 2 q
1 4! 1 4!
u 0 w 4 q 12 c 0 Ž =w .
2
2
˜ . , u ) u L M 4 q 12 Ž =M Ž 4.
1r2 with t s ŽT y Tc .rT, M s cr w s Ž a 0rc t . w , a0 s ˜ s Õ 01r3 =. cr2 c t , u 0 s u ) u L cr4 , c 0 s cr2 Õ 02r3 , and = The average molecular volume Õ 0 and the prefactor Õ 0rk B T are introduced to make the free-energy density and all the coefficients dimensionless. The inverse cross-over susceptibility xy1 s ˜ E M 2 . t , where D A˜ is the cross-over ŽrenorŽ E 2D Ar malized. free-energy density, in zero field above Tc reads w4x
xy1 s cr2 c tt Y Ž gy1.r D s Ž 1 q y .
Ž 5.
the critical temperature. The cross-over function Y is defined by
L
1 y Ž 1 y u. Y s u 1 q
ys
2 Ds
½ž 2
L
n =
q
Ds 2n y 1
y
Ds
2
k
/
L
k 2 s ct
1q
ž / k
Ž 1 y u. Y 1 y Ž 1 y u. Y y1
5
,
Ž 6.
where n , 0.630 w15,22x is the critical exponent of the asymptotic power law for the correlation length j w4x. Note that xy1 s ŽTcrT . xˆy1 and the relation between geff ' y dln xr dln
ž / k
Y n r Ds
Ž 7.
T Tc
t Y Ž2 ny1.r D s s c t tY Ž2 ny1.r D s .
Ž 8.
We modified the original expression for k 2 , given by Eq. Ž3. in Ref. w4x, by introducing the nonasymptotic factor TrTc in Eq. Ž8. so that k 2 becomes infinite at T ™ ` w23x. Asymptotically close to the critical point Ž Lrk 4 1., the following expression is obtained for the first correction amplitude G 1 in Eq. Ž3.:
G1 s g1
(c
2 Ds t
ž / uL
Ž 1 y u. ,
Ž 9.
where g 1 , 0.62 is a universal constant w21x. In the approximation of an infinite cut-off L ™ `, which physically means neglecting the discrete structure of matter, u s u 0 c t2rŽ u )L a20 . ™ 0 and the two cross-over parameters u and L in the cross-over equations collapse into a single one, u L, which is related to the Ginzburg number G by w21x G s g0
2
2 1r2
and is to be found numerically. The parameter k in Eq. Ž7. is inversely proportional to the fluctuation-induced portion of the correlation length and serves as a measure of the distance to the critical point. In zero field above Tc the expression for k 2 reads:
with u )n
91
Ž u L. ct
2
s g0
u 20 Õ 02 2 Ž u ) . a04 j 06
,
Ž 10 .
where g 0 , 0.028 is a universal constant w21x and j 0 s Õ 01r3 cy1r2 s Ž c 0ra0 .1r2 is the mean-field amt plitude of the power law for the correlation length. Note that the Ginzburg number does not depend explicitly on the cut-off L or on u. This singleparameter cross-over, i.e., the cross-over for u s 0, is universal and is indicated in Fig. 1 by a dasheddotted curve. This simplified description of the cross-over is equivalent to the results of Bagnuls and Bervillier w9x and of Belyakov and Kiselev w10x. In the simulations w3x, each spin interacts equally with its z neighbors lying within a distance R m on a three-dimensional cubic lattice. The effective range
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of interaction R is then defined as R 2 s zy1 Ý j/ i < r i y r j < 2 with < r i y r j < F R m w1x. We have approximated the relation between R and R m by R 2 . s 35 R 2m Ž1 q 23 Ry2 m , as indicated in the insert in Fig. 3. In order to compare the numerical results to the theoretical prediction Eq. Ž5., we need the range dependence of the parameters c t and u. Indeed, the asymptotic R dependence of u follows directly from simple scaling arguments w1x, u s u 0 Ry4 , and c t varies as its square root, c t s c t 0 Ry2 . For a three-dimensional simple cubic lattice, L s p w18,24x, and we obtain for the Ginzburg number G s G 0 Ry6 s 0.28 Ž u 20rc t40 . c t3 s 0.28 Ž u 20rc t 0 . Ry6 . Ž 11 . The non-universal parameters c t 0 and u 0 have to be determined from a least-squares fit to the numerical q data for geff , which yielded c t 0 s 1.72 and u 0 s 1.22 and hence G 0 f 0.24. The solid lines in Fig. 1 indicate the corresponding theoretical curves. It should be noted that these curves are calculated for each value of R m separately; the piecewise continuous character of this description directly reflects the fact that the cross-over cannot be described by a universal single-parameter function. Indeed, Fig. 1 also shows two attempts to describe the data in terms of such a function. The dash-dotted line corresponds to the limit u ™ 0, whereas the dotted curve corre-
Fig. 3. Dependence of the normalized coupling constant u on the normalized interaction range R. Note that u becomes larger than unity for very short interaction ranges. Insert: Effective range of interaction R Žopen circles. plotted as a function of R m . The solid line corresponds to the approximation mentioned in the text and the dashed line represents the asymptotic behavior for large R.
sponds to u 0 s 1.22 and L s p Ža continuation of the theoretical curve for R s 1.. We see that the actual cross-over lies between these two bounding curves, with u , 0 for large R and u , 1.2 for R s 1. Thus, it is clearly seen that without including the R dependence of u it is impossible to describe data for short interaction ranges R 2m F 5. The dependence of u on R is shown in Fig. 3. The two adjustable parameters c t 0 and u 0 are strongly correlated and if one of them is fixed at a predicted value, the quality of the description remains the same. We hence tried to fit the data while keeping c t 0 fixed at the theoretically predicted value c t 0 s 2 d s 6 w25,26x. In this case a fit of the same quality is obtained with u 0 s 1.22, provided that L , 2p . The value of G 0 f 0.24 then remains unchanged. To describe the data below the critical temperature, a connection between M and t in zero field is ˜ E M .t s 0. The to be found from the condition Ž E D Ar relation between M and t appears to be implicit and x as a function of t cannot be expressed in an explicit form either. Of course, the parameters c t 0 and u 0 should be the same as for T ) Tc and we hence kept them fixed at the above-mentioned values. However, the parameter G 0 appearing in Eq. Ž11. will take a different value. We took this into y account by introducing a factor Gq 0 rG 0 into the q y temperature scale: t ™ t P Ž G 0 rG 0 .. Fig. 2 shows y the results for T - Tc , where the factor Gq 0 rG 0 was included as an adjustable parameter. Our estimate q Gy 0 rG 0 s 2.58 must be compared with the theoretiy q w x cal result Gy 0 rG 0 s 3.125 27 . Interestingly, geff 6 2 clearly shows a minimum around < t < R ; 10 . This corroborates the non-monotonic character of the y cross-over of geff , earlier observed for the two-dimensional Ising lattice w2x, where the effect is much more pronounced. We note that already in Ref. w28x a field-theoretic calculation of the cross-over in the low-temperature regime has been given Žin the limit u ™ 0., but only recently this has been extended to cover the full cross-over region w29x. Actually, also y here a non-monotonicity in geff has been observed. In summary, we remark that although in general the theory contains two cross-over parameters u and L, only one parameter Ž u. changes with the range of interaction. However, this does not mean that the cross-over is a universal function of tR 6 . Indeed, the effective range of interaction R affects the behavior
M.A. AnisimoÕ et al.r Physics Letters A 261 (1999) 89–93
of geff" in a twofold way: through the Ginzburg number, which is proportional to c t3 , and through the first Wegner correction, with an amplitude G 1 that is proportional to Ž1 y u. wEq. Ž9.x. Hence, there is no way to describe the data for short interaction ranges without allowing for u to become larger than unity and correspondingly G 1 to change its sign between R m s 2 and R m s 1 as indicated in Fig. 3. In previous publications we have shown that Eq. Ž5., derived from renormalization-group matching, gives an excellent representation of the experimentally observed cross-over behavior in simple and complex fluids w4,5,30x. From the evidence presented in this paper, we conclude that the same cross-over model also yields a quantitative description of the cross-over critical behavior of a three-dimensional Ising lattice. Acknowledgements We acknowledge valuable discussions with M. E. Fisher and assistance from A. A. Povodyrev. The research at the University of Maryland was supported by DOE Grant No. DE-FG02-95ER-14509. E. Luijten acknowledges the HLRZ Julich for comput¨ ing resources on a Cray-T3E. References w1x E. Luijten, H.W.J. Blote, ¨ K. Binder, Phys. Rev. E 54 Ž1996. 4626. w2x E. Luijten, H.W.J. Blote, ¨ K. Binder, Phys. Rev. Lett. 79 Ž1997. 561. w3x E. Luijten, K. Binder, Phys. Rev. E 58 Ž1998. R4060.
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