Test of renormalization predictions for universal finite-size scaling functions
Erik Luijten*Department of Physics, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands; Max-Planck-Institut fu¨r Polymerforschung, Postfach 3148, D-55021 Mainz, Germany;
and Institut fu¨r Physik, WA 331, Johannes Gutenberg-Universita¨t, D-55099 Mainz, Germany† 共Received 9 July 1998兲
We calculate universal finite-size scaling functions for systems with an n-component order parameter and algebraically decaying interactions. Just as previously found for short-range interactions, this leads to a singu-lar expansion, where is the distance to the upper critical dimension. Subsequently, we check the results by numerical simulations of spin models in the same universality class. Our systems offer the essential advantage that can be varied continuously, allowing an accurate examination of the region where is small. The numerical calculations turn out to be in striking disagreement with the predicted singularity.
关S1063-651X共99兲00612-1兴
PACS number共s兲: 05.70.Jk, 64.60.Ak, 64.60.Fr
In order to analyze numerical results obtained by Monte Carlo or transfer-matrix studies of phase transitions and criti-cal phenomena, finite-size scriti-caling 关1兴 is a very widely used technique. This hypothesis, which was properly formulated for the first time by Fisher 关2兴, allows the extrapolation of properties of finite systems, which do not exhibit a phase transition, to the thermodynamic limit. In 1982, Bre´zin 关3兴 achieved a breakthrough by showing that finite-size scaling laws can actually be derived from renormalization-group 共RG兲 theory, provided that the RG equations are not singular at the fixed point. This implies a breakdown of finite-size scaling for dimensionalities d⭓4, and consequently an ex-pansion of the finite-size scaling functions in powers of ⫽4⫺d is singular at ⫽0. This rather surprising result was confirmed by explicit calculations for the n-vector model in the large-n limit. In addition, it follows from Ref.关3兴 that the finite-size scaling relation for the free energy is a universal function depending only on two nonuniversal metric factors, without an additional nonuniversal prefactor. This result was derived from different arguments by Privman and Fisher关4兴, and subsequently confirmed analytically for the spherical model 关5兴. Pioneering work 关6,7兴 then showed that a field-theoretic calculation of finite-size scaling functions is actu-ally possible. Specificactu-ally, Bre´zin and Zinn-Justin关6兴 devel-oped a systematic expansion for these functions. Unlike the standard expansion in powers of for critical exponents and scaling functions of bulk properties, one finds, for a fully finite geometry, an expansion in powers of
冑
. More re-cently, Esser et al. 关8兴 introduced a promising perturbation approach at fixed d which is applicable below the critical temperature as well. However, here we focus on the expan-sion in and in particular on the singular nature of this expansion.The systems under consideration have an n-component order parameterwith O(n) symmetry and periodic bound-ary conditions. A quantity of central interest is the amplitude
ratio Q⫽limL→⬁
具
L2典
2/具
L4典
, which is directly related to the cumulant introduced by Binder 关9兴. At the critical tempera-ture Tc it takes a universal, although geometry-dependent,value共cf. Ref. 关4兴兲. In Ref. 关6兴, an expansion for Q(Tc) was
obtained in powers of
冑
, up to O(), which is shown in Fig. 1 for n⫽1, along with numerical results for integer d. Given the low order of the expansion and the fact that it can only be checked for integer values of, hardly any conclu-sions can be drawn from a comparison to numerical results, and any confirmation of the singular nature of the expan-sion will have to wait until the RG calculation has been carried to substantially higher order. Thus, we propose a dif-ferent route: that is we replace the short-range共SR兲 forces by long-range attractive interactions decaying as a power law, J(r)⬀r⫺(d⫹), where 0⬍⬍2. It was shown in Refs. 关10,11兴 that these systems have an upper critical dimension dc⫽2, and that, for d⬍dc, the critical exponents can be calculated in terms of an expansion in ⬘
⫽2⫺d, very similar to the original expansion, which is recovered for →2. Since the upper critical dimension is now a continu-ous parameter, we have the opportunity to verify ⬘
-expansion results for arbitrarily small⬘
. So, even if ac-tual physical realizations of this system may be scarce, it constitutes a very valuable mathematical model.Interest-*Electronic address: erik.luijten@uni-mainz.de
†Present address.
FIG. 1. The amplitude ratio Q(Tc) for systems with short-range
interactions. The dashed line shows the
冑
expansion of Ref. 关6兴. PHYSICAL REVIEW E VOLUME 60, NUMBER 6 DECEMBER 1999PRE 60
ingly, finite-size scaling functions for the spherical model with power-law interactions have been calculated by several authors 关12兴 for d/2⬍⬍d, but the nature of a possible singularity in the limit →d/2 appears not to have been examined. The first part of this paper is therefore devoted to a generalization of the treatment of Ref.关6兴 to systems with algebraically decaying interactions. For notational conve-nience we redefine⫽
⬘
⫽2⫺d. Throughout our analysis we will closely adhere to the approach outlined in Ref.关6兴. Additional details can also be found in Ref.关13兴, Chap. 36. We consider a system with a hypercubic geometry, with linear dimension L and periodic boundary conditions. It is represented by the following Landau-Ginzburg-Wilson Hamiltonian in momentum space,H共k兲/kBT⫽ 1 2
兺
k兺
i 共k ⫹r 0兲i,ki,⫺k⫹ 1 4! 1 Ldu0 ⫻兺
k1,k2,k3兺
i, j i,k1i,k2j,k3j,⫺k1⫺k2⫺k3 , 共1兲 where the factor k (0⬍⬍2) arises from the isotropic long-range interactions. Compared to the SR case, it leads to the general replacement k2→kin all propagators. The indi-ces i and j refer to the components of the field. It has been well established that this system belongs to the same univer-sality class as a discrete spin model with algebraically decay-ing interactions; in particular the-expansion results 关10,11兴 for the critical exponents have been confirmed 共see Refs. 关14,15兴, and references therein兲. Due to the finite geometry all components of the wave vectors are integer multiples of 2/L. The sums run to infinity, which corresponds to a van-ishing lattice spacing a; however, the ratio L/ is finite, whereas L/a and /a are both sent to infinity关6兴. Expecta-tion values are computed from a partiExpecta-tion funcExpecta-tion in which Eq.共1兲 is replaced by an effective Hamiltonian, consisting of the exactly calculated k⫽0 共homogeneous兲 mode contribu-tion and a perturbatively calculated part, which contains the contribution of all nonzero modes. Only the latter contribu-tion is affected by the changeover to long-range interaccontribu-tions, cf. Ref.关14兴. To one-loop order this consists of a shift of the critical temperature and a renormalization of the coupling constant. Higher operators do not contribute at this order. We introduce the dimensionless coupling constant g0⫽⫺u0,and work in the system of units where⫽1. The parameter r0 is split into r0c⫹t, where t⫽r0⫺r0c⬀T⫺Tc, and we
require t⭓0. The RG calculations are carried out in the minimal subtraction scheme 关16,17兴, where it is a crucial ingredient of the calculation that, despite the quantization of all momenta, the UV divergences are taken care of by the bulk renormalization constants. As we do not go beyond one loop, we have ignored the field renormalization constant.
The leading contribution to the shift of Tc is
n⫹2 6 g0 1 Ld
兺
k⬘
1 兩k兩⫹t, 共2兲where the prime indicates that the k⫽0 mode is omitted from the sum. In the Schwinger parametrization this can be rewritten as L⫺d
冕
0 ⬁ ds e⫺st冋
兺
m1⫽⫺⬁ ⬁ •••兺
md⫽⫺⬁ ⬁ e⫺s(m2)/2(2/L)⫺1册
, 共3兲 where m2⫽兺 i⫽1 d m i2 and we have omitted the prefactor 关(n
⫹2)/6兴g0. For d⭓, the UV divergence in Eq. 共2兲 is
re-flected by the divergence of the integral at small s. Thus we isolate this divergence by rewriting Eq. 共3兲 as
L⫺d 共2兲I1共d,,t兲 ⫹ L⫺d 共2兲Sd⫺1 1 ⌫
冉
d 冊
冕
0 ⬁ du u⫺d/e⫺ut(L/2), 共4兲 with I1共d,,t兲⬅冕
0 ⬁ du e⫺ut(L/2)冋
兺
m1 •••兺
md e⫺u(m2)/2⫺1 ⫺Sd⫺1 1 ⌫冉
d 冊
u⫺d/册
, 共5兲which is finite. Sd⫺1⫽2d/2/⌫(d/2) is the surface area of a
d-dimensional unit sphere. The second term in Eq.共4兲 can be continued analytically for Re(d)⭓, and has a simple pole for d⫽2 (⫽0). Upon expansion around this pole we find for Eq. 共4兲,
⫺ 2t 共4兲⌫共兲⫹ t 共4兲⌫共兲
再
2 ln t⫺关ln 4⫹⌿共兲兴冎
⫹ 1 共2L兲I1共2,,t兲⫹O共兲, 共6兲 where⌿() denotes the digamma function. The addition of the2-insertion counterterm leads to the replacement of t by tZ2. To one-loop order, the renormalization constant hasthe same form as for SR interactions 关18兴; the pole is can-celed and the shifted reduced temperature is given by
t˜⫽t⫹n⫹2 6 gˆ0t ln t⫹ 2 12共n⫹2兲⌫共兲gˆ0 1 LI1共2,,t兲 ⫹O共gˆ0 2兲, 共7兲 with gˆ0⫽2关(4)⌫()兴⫺1g0兵1⫹12关ln 4⫹⌿()兴⫹O( 2)其.
The renormalized coupling constant g is calculated in a simi-lar fashion. The leading finite-size contribution is given by ⫺关(n⫹8)/6兴g0
2
L⫺d兺k
⬘
(兩k兩⫹t)⫺2, which has a UV diver-gence for Re(d)⭓2. The 1/ pole is canceled by the coun-terterm, where the renormalization constant to one-loop or-der is given by Zg⫽1⫹关(n⫹8)/6兴gˆ. After some algebra,we find, for the renormalized coupling constant,
g⫽g0
冋
1⫹ n⫹8 6 gˆ0共1⫹ln t兲⫺ n⫹8 12 gˆ0 ⌫共兲 I2共2,,t兲 ⫹O共gˆ0 2兲册
, 共8兲 with I2共d,,t兲⬅冕
0 ⬁ du ue⫺ut(L/2)冋
兺
m1 •••兺
md e⫺u(m2)/2⫺1 ⫺Sd⫺1 1 ⌫冉
d 冊
u⫺d/册
. 共9兲Equations共7兲 and 共8兲 suffice to calculate the finite-size scal-ing functions close to criticality to O(). To this order, the fixed-point value of gˆ0 only differs from the SR case in the
definition of , gˆ0*⫽关6/(n⫹8)兴⫹O(2). We are now
able to calculate t˜Ld/2g⫺1/2at the fixed point, which, as we shall see shortly, is the parameter appearing in the finite-size scaling functions. This also provides a simple consistency check for our calculations, since all factors ln L have to dis-appear upon introduction of the appropriate scaling variable y⫽tL1/. We find 1/⫽⫺关(n⫹2)/(n⫹8)兴⫹O(2),
which indeed agrees with Ref. 关10兴, and the final expression is t˜Ld/2g⫺1/2兩f.p.⫽ 1
冑
g0*冋
y⫺ 1 2y⫹ n⫺4 2共n⫹8兲y ln y ⫹1 4y ⌫共兲 I2共2,, y L⫺1/兲 ⫹2⫺1n⫹8共n⫹2兲⌫共兲I1共2,,y L⫺1/兲 ⫹O共2兲册
. 共10兲We are particularly interested in the amplitude ratio Q at criticality, for n⫽1. The even moments of the magnetiza-tion distribution are calculated from
具
(2)p典
⫽兰⫺⬁⫹⬁d(2)pexp关⫺S()兴/兰⫺⬁
⫹⬁dexp关⫺S()兴, with S()
⫽Ld„1 2t˜
2⫹(1/4!)g4…. We carry out the rescaling
→(Ldg)⫺1/4, and expand in terms of the parameter x
⬅ t˜Ld/2g⫺1/2. Elementary algebra leads to
Q⫽ 4⌫2
冉
3 4冊
⌫2冉
1 4冊
冋
1⫹冉
4 ⌫冉
34冊
⌫冉
1 4冊
⫺1 2 ⌫冉
14冊
⌫冉
3 4冊
冊
冑
6x ⫹冉
13 ⌫2冉
3 4冊
⌫2冉
1 4冊
⫹ 1 16 ⌫2冉
1 4冊
⌫2冉
3 4冊
⫺2冊
6x2⫹O共x3兲册
. 共11兲At criticality, y⫽0 and x takes the value
x0⫽
冑
再
1 2 n⫹2冑
3共n⫹8兲冑
⌫共兲 I1共2,,0兲⫹O共兲冎
. 共12兲 A comparison to Eq. 共3.33兲 in Ref. 关6兴 shows that x0 onlydiffers from the SR case by a redefinition of , a geometric factor, and the integral I1, and thus we see that the singular
structure is preserved in the generalization to long-range forces. For completeness we remark that one may also cal-culate x0by carrying out all manipulations at criticality. This
permits a different parametrization and leads to the same expression for x0, in which I1(2,,0) is replaced by
Iˆ1(2,)/⌫(/2), where Iˆ1(d,)
⬅兰0⬁du u/2⫺1关(兺m⬁⫽⫺⬁e⫺um
2
)d⫺1⫺(/u)d/2兴. As a side note, we conjecture Iˆ1(4,2), which has been evaluated
nu-merically in Refs.关6,13,19兴, to be exactly equal to ⫺8 ln 2. For lower dimensionalities, the upper critical dimension shifts toward smaller values of, and numerical evaluation yields I1(d,d/2,0)⫽⫺2.92, ⫺3.900, and ⫺4.8227 for d
⫽1, 2, and 3, respectively. Substitution into Eqs. 共12兲 and 共11兲 suggests that for each of these values a reasonable con-vergence may be expected for ⬍1.
In order to verify these predictions, we have carried out extensive Monte Carlo simulations of spin models with n ⫽1 and algebraically decaying interactions, for d⫽1 and 2. Accurate results could be obtained by means of an efficient cluster algorithm 关20兴. We investigated system sizes 10⭐L ⭐150 000 for d⫽1 and 4⭐L⭐400 for d⫽2, for several values of the decay parameter . These were chosen such that 0⬍ⱗ1, where it should be noted that for very small the analysis is hampered by strong corrections to scaling. Simulational details can be found in Ref. 关14兴, where the classical regime 0⬍⭐d/2 was discussed. The numerical results were analyzed using an expression similar to Eq.共13兲 in Ref. 关14兴, QL(T)⫽Q⫹r1tLyt⫹r
2t2L2yt⫹•••⫹s1Lyi
⫹•••. Here ytand yi are the thermal and leading irrelevant
exponents, respectively, ri and si are nonuniversal coeffi-cients, and the ellipses denote higher-order terms. An exten-sive analysis of the data will be presented elsewhere. The resulting estimates for Q(Tc) are shown in Figs. 2 and 3. For
the one-dimensional case 共Fig. 2兲, the region 0.1⭐⭐0.9 has been covered. For →1, the data points approach Q(Tc)⫽1, in agreement with the occurrence of a
Kosterlitz-Thouless transition at ⫽1 关21兴. However, for 0⬍ⱗ0.5, the numerical results are described by a perfectly linear de-pendence on, in strong contrast with the predicted square-root behavior. This discrepancy is reinforced by the two-dimensional results共Fig. 3兲, which are also well described by a linear relation for 0⬍ⱗ1.2. For larger, the error bars increase, signaling a crossover to short-range criticality.
In summary, we have calculated universal finite-size scal-ing functions to second order in
冑
for systems with alge-braically decaying interactions. These calculations are essen-tially a generalization of those for systems with short-range interactions关6兴, and exhibit the same singular dependence on . Subsequently, we have compared our results to accurate simulations for one- and two-dimensional systems belonging to the same universality class as the field-theoreticaltonian. The presence of long-range interactions offers the advantage that is a continuous parameter, so that one can reach the regime where the convergence of the expansion does not have to be doubted. Nevertheless, no agreement is found: the numerical data exhibit a linear rather than a square-root dependence on. Currently, we do not have an explanation for this striking discrepancy. Although higher-order terms might yield some improvement, it is difficult to envisage that this would fully resolve the problems. It would be very remarkable if the apparent linear variation over such a wide range in Figs. 2 and 3, which includes the point from
which our
冑
expansion starts, were accidental. In Ref. 关22兴, it was suggested that Q contains nonuniversal contributions, depending on the cutoff used in the integration of the order parameter probability distribution. However, apart from the validity of this suggestion, it is difficult to envisage how this would lead to the共dis兲appearance of a square-root contribu-tion in the expansion. Furthermore, it is an open question as to what extent the breakdown of the field-theoretic de-scription of finite-size scaling for d⭓4 关19兴 influences the nature of the expansion. We feel that an understanding of these problems is of some significance for an understanding of finite-size scaling of critical phenomena in general.It is a pleasure to thank Professor K. Binder and Professor H. Blo¨te for useful comments.
关1兴 M.N. Barber, in Phase Transitions and Critical Phenomena,
edited by C. Domb and J.L. Lebowitz 共Academic, London, 1983兲 Vol. 8; Finite Size Scaling and Numerical Simulation of
Statistical Systems, edited by V. Privman 共World Scientific,
Singapore, 1990兲.
关2兴 M.E. Fisher, in Critical Phenomena, Proceedings of the 51st
Enrico Fermi Summer School, Varenna, Italy, edited by M. S. Green共Academic, New York, 1971兲.
关3兴 E. Bre´zin, J. Phys. 共Paris兲 43, 15 共1982兲.
关4兴 V. Privman and M.E. Fisher, Phys. Rev. B 30, 322 共1984兲. 关5兴 S. Singh and R.K. Pathria, Phys. Rev. B 31, 4483 共1985兲. 关6兴 E. Bre´zin and J. Zinn-Justin, Nucl. Phys. B 257, 867 共1985兲. 关7兴 J. Rudnick, H. Guo, and D. Jasnow, J. Stat. Phys. 41, 353
共1985兲.
关8兴 A. Esser, V. Dohm, M. Hermes, and J.S. Wang, Z. Phys. B:
Condens. Matter 97, 205共1995兲.
关9兴 K. Binder, Z. Phys. B: Condens. Matter 43, 119 共1981兲. 关10兴 M.E. Fisher, S.-k. Ma, and B.G. Nickel, Phys. Rev. Lett. 29,
917共1972兲.
关11兴 M. Suzuki, Y. Yamazaki, and G. Igarashi, Phys. Lett. 42A,
313共1972兲.
关12兴 M.E. Fisher and V. Privman, Commun. Math. Phys. 103, 527 共1986兲; J.G. Brankov and N.S. Tonchev, J. Stat. Phys. 52, 143
共1988兲; S. Singh and R.K. Pathria, Phys. Rev. B 40, 9238 共1989兲.
关13兴 J. Zinn-Justin, Quantum Field Theory and Critical Phenom-ena, 3rd ed.共Clarendon, Oxford, 1996兲.
关14兴 E. Luijten and H.W.J. Blo¨te, Phys. Rev. Lett. 76, 1557,
3662共E兲 共1996兲; Phys. Rev. B 56, 8945 共1997兲.
关15兴 E. Luijten, Interaction Range, Universality and the Upper Critical Dimension共Delft University Press, Delft, 1997兲. 关16兴 E. Bre´zin, J.C. Le Guillou, and J. Zinn-Justin, in Phase
Tran-sitions and Critical Phenomena, edited by C. Domb and M. S.
Green共Academic, London, 1976兲, Vol. 6.
关17兴 D.J. Amit, Field Theory, The Renormalization Group, and Critical Phenomena, 2nd ed. 共World Scientific, Singapore,
1984兲.
关18兴 Y. Yamazaki and M. Suzuki, Prog. Theor. Phys. 57, 1886 共1976兲.
关19兴 X. S. Chen and V. Dohm, Int. J. Mod. Phys. C 9, 1007 共1998兲. 关20兴 E. Luijten and H.W.J. Blo¨te, Int. J. Mod. Phys. C 6, 359
共1995兲.
关21兴 J.M. Kosterlitz, Phys. Rev. Lett. 37, 1577 共1976兲.
关22兴 R. Hilfer, Z. Phys. B: Condens. Matter 96, 63 共1994兲; R. Hilfer
and N.B. Wilding, J. Phys. A 28, L281共1994兲.
FIG. 2. The critical value of the amplitude ratio Q as a function of the decay parameter for d⫽1. The corresponding values of are shown at the upper axis. The data in the regime 0⬍⭐0.5 are taken from Ref.关14兴.
FIG. 3. The analog of Fig. 2 for d⫽2.
