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arXiv:cond-mat/0005022 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Universality and φ4 theory of finite-size effects above the upper critical dimension

X.S. Chen, V. Dohm

Published Dec 21, 2000• 14 citations
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Abstract

We analyze finite-size effects in a L(d) geometry above the upper critical dimension d=4 within the O(n) symmetric straight phi(4) theory on the basis of exact results for n-->infinity and one-loop results for n=1. We show that finite-size effects of the straight phi(4) continuum theory with a smooth (rather than sharp) cutoff belong to the same universality class as those of the straight phi(4) lattice theory. Our analysis predicts both universal and nonuniversal features of finite-size effects and resolves long-standing discrepancies in earlier analyses of Monte Carlo (MC) data for the d=5 Ising model. Our estimates of two fundamental length scales xi(0) and l(0) are confirmed by very recent MC data.

DOI
10.1103/physreve.63.016113

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We analyze finite-size effects in a L(d) geometry above the upper critical dimension d=4 within the O(n) symmetric straight phi(4) theory on the basis of exact results for n-->infinity and one-loop results for n=1. We show that finite-size effects of the straight phi(4) continuum theory with a smooth (rather than sharp) cutoff belong to the same universality class as those of the straight phi(4) lattice theory. Our analysis predicts both universal and nonuniversal features of finite-size effects and resolves long-standing discrepancies in earlier analyses of Monte Carlo (MC) data for the d=5 Ising model. Our estimates of two fundamental length scales xi(0) and l(0) are confirmed by very recent MC data.
We analyze finite-size effects in a L-d geometry above the upper critical dimension d=4 within the O(n) symmetric phi (4) theory on the basis of exact results for n-->infinity and one-loop results for n=1. We show that finite-size effects of the phi (4) continuum theory with a smooth (rather than sharp) cutoff belong to the same universality class as those of the phi (4) lattice theory. Our analysis predicts both universal and nonuniversal Features of finite-size effects. and resolves long-standing discrepancies in earlier analyses of Monte Carlo (MC) data for the d = 5 Ising model. Our estimates of two fundamental length scales xi (o) and I-o are confirmed by very recent MC data.
We present a perturbative calculation of finite-size effects near Tc of the φ4 lattice model in a d-dimensional cubic geometry of size L with periodic boundary conditions for d>4. The structural differences between the φ4 lattice theory and the φ4 field theory found previously in the spherical limit are shown to exist also for a finite number of components of the order parameter. The two-variable finite-size scaling functions of the field theory are nonuniversal whereas those of the lattice theory are independent of the nonuniversal model parameters. One-loop results for finite-size scaling functions are derived. Their structure disagrees with the single-variable scaling form of the lowest-mode approximation for any finite ξ/L where ξ is the bulk correlation length. At Tc, the large-L behavior becomes lowest-mode like for the lattice model but not for the field-theoretic model. Characteristic temperatures close to Tc of the lattice model, such as T max (L) of the maximum of the susceptibility χ, are found to scale asymptotically as Tc-T max (L) ~L-d/2, in agreement with previous Monte Carlo (MC) data for the five-dimensional Ising model. We also predict χ max ~Ld/2 asymptotically. On a quantitative level, the asymptotic amplitudes of this large-L behavior close to Tc have not been observed in previous MC simulations at d=5 because of nonnegligible finite-size terms ~L(4-d)/2 caused by the inhomogeneous modes. These terms identify the possible origin of a significant discrepancy between the lowest-mode approximation and previous MC data. MC data of larger systems would be desirable for testing the magnitude of the L(4-d)/2 and L4-d terms predicted by our theory.
We demonstrate that the standard O(n) symmetric φ4 field theory does not correctly describe the leading finite-size effects near the critical point of spin systems with periodic boundary conditions on a d-dimensional lattice with d>4. We show that these finite-size effects require a description in terms of a lattice Hamiltonian. For n →∞ and n=1, explicit results are given for the susceptibility and for the Binder cumulant. They imply that these quantities do not have the universal properties predicted previously and that recent analyses of Monte Carlo results for the five-dimensional Ising model are not conclusive.
We study spatial anisotropy effects on the bulk and finite-size critical behavior of the O(n) symmetric anisotropic phi;{4} lattice model with periodic boundary conditions in a d -dimensional hypercubic geometry above, at, and below Tc. The absence of two-scale factor universality is discussed for the bulk order-parameter correlation function, the bulk scattering intensity, and for several universal bulk amplitude relations. The anisotropy parameters are observable by scattering experiments at Tc. For the confined system, renormalization-group theory within the minimal subtraction scheme at fixed dimension d for 2<d<4 is employed. In contrast to the epsilon=4-d expansion, the fixed- d finite-size approach keeps the exponential form of the order-parameter distribution function unexpanded. For the case of cubic symmetry and for n=1 , our perturbation approach yields excellent agreement with the Monte Carlo (MC) data for the finite-size amplitude of the free energy of the three-dimensional Ising model at Tc by Mon [Phys. Rev. Lett. 54, 2671 (1985)]. The epsilon expansion result is in less good agreement. Below Tc, a minimum of the scaling function of the excess free energy is found. We predict a measurable dependence of this minimum on the anisotropy parameters. The relative anisotropy effect on the free energy is predicted to be significantly larger than that on the Binder cumulant. Our theory agrees quantitatively with the nonmonotonic dependence of the Binder cumulant on the ferromagnetic next-nearest-neighbor (NNN) coupling of the two-dimensional Ising model found by MC simulations of Selke and Shchur [J. Phys. A 38, L739 (2005)]. Our theory also predicts a nonmonotonic dependence for small values of the antiferromagnetic NNN coupling and the existence of a Lifshitz point at a larger value of this coupling. The nonuniversal anisotropy effects in the finite-size scaling regime are predicted to satisfy a kind of restricted universality. The tails of the large- L behavior at T++Tc violate both finite-size scaling and universality even for isotropic systems as they depend on the bare four-point coupling of the phi4 theory, on the cutoff procedure, and on subleading long-range interactions.
. Monte Carlo results for the moments #M k # of the magnetization distribution of the nearestneighbor Ising ferromagnet in a L d geometry, where L (4 # L # 22) is the linear dimension of a hypercubic lattice with periodic boundary conditions in d = 5 dimensions, are analyzed in the critical region and compared to a recent theory of Chen and Dohm (CD) [X.S. Chen and V. Dohm, Int. J. Mod. Phys. C (1998)]. We show that this finite-size scaling theory (formulated in terms of two scaling variables) can account for the longstanding discrepancies between Monte Carlo results and the so-called "lowest-mode" theory, which uses a single scaling variable tL d/2 where t = T/T c - 1 is the temperature distance from the critical temperature, only to a very limited extent. While the CD theory gives a somewhat improved description of corrections to the "lowest-mode" results (to which the CD theory can easily be reduced in the limit t # 0, L ##, tL d/2 fixed) for the fourth-order cum...
The singular part of the finite-size free-energy density f(s) of the O(n) symmetric phi(4) field theory is calculated for confined geometries of linear size L with periodic boundary conditions in the large-n limit and with Dirichlet boundary conditions in one-loop order. We find that both a sharp cutoff and a subleading long-range interaction cause a leading nonuniversal L dependence of f(s) near T(c). This implies a significant restriction for the validity of universal finite-size scaling for model systems and real systems. For film geometry we predict a leading nonuniversal contribution to the critical Casimir force above the superfluid transition of (4)He.