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Journal de Physique I

Field Theory of Finite-Size Effects in O(n) symmetric systems

X.S. Chen, V. Dohm, Alexander Esser

Published Feb 1, 1995• 20 citations
Publisher

Abstract

We propose a new perturbation approach to finite-size effects near the critical point within the ϕ4 field theory for an n-component order parameter with periodic boundary conditions. Our approach is applicable both above and below Tc. Calculations of finite-size scaling functions of the specific heat, susceptibility, order parameter, and a cumulant ratio are compared with Monte Carlo data for the three-dimensional XY model. Good overall agreement with the Monte Carlo data is found.

DOI
10.1051/jp1:1995121

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We propose a new perturbation approach to finite-size effects near the critical point within the ϕ4 field theory for an n-component order parameter with periodic boundary conditions. Our approach is applicable both above and below Tc. Calculations of finite-size scaling functions of the specific heat, susceptibility, order parameter, and a cumulant ratio are compared with Monte Carlo data for the three-dimensional XY model. Good overall agreement with the Monte Carlo data is found.
We present a renormalization-group study of the order-parameter distribution function near the critical point of O(n) symmetric three-dimensional (3D) systems in a finite geometry. The distribution function is calculated within the φ4 field theory for a 3D cube with periodic boundary conditions by means of a novel approach that appropriately deals with the Goldstone modes below T c . Results are given for both vanishing and finite external field h. The results describe finite-size effects near the critical point in the h– T-plane including the first-order transition at the coexistence line at h = 0 below T c . Quantitative theoretical predictions of the finite-size scaling function are presented for the Ising (n=1), XY(n=2) and Heisenberg (n=3) models. Good agreement is found with recent Monte Carlo 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 present analytic and numerical studies of the order-parameter distribution function near the critical point of O(n)O(n)O(n) symmetric three-dimensional (3D) systems in a finite geometry. The distribution function is calculated within the ${\ensuremath{\phi}}^{4}$ field theory for a 3D cube with periodic boundary conditions by means of a new approach that appropriately deals with the Goldstone modes below Tc{T}_{c}Tc​. Good agreement is found with new Monte Carlo data for the distribution function of the magnetization of the 3D XY\mathrm{XY}XY (n=2)(n\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2)(n=2) and Heisenberg (n=3)(n\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3)(n=3) models.
We develop a detailed finite-size-scaling theory at a general, asymmetric, temperature-driven, strongly first-order phase transition in a system with periodic boundary conditions. We compute scaling functions for various cumulants of energy in the form U(L,t)=U0{\mathit{U}}_{0}U0​(tLd{\mathit{tL}}^{\mathit{d}}tLd)+${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$U1{\mathit{U}}_{1}U1​(tLd{\mathit{tL}}^{\mathit{d}}tLd) with t=1-Tc{\mathit{T}}_{\mathit{c}}Tc​/T. In particular, we consider the specific heat and Binder's fourth cumulant and show this has a minimum value of 2/3-(e1{\mathit{e}}_{1}e1​/e2{\mathit{e}}_{2}e2​-e2{\mathit{e}}_{2}e2​/e1{\mathit{e}}_{1}e1​)2{)}^{2})2/12+O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$) at a temperature Tc(2){\mathit{T}}_{\mathit{c}}^{(2)}Tc(2)​(L)-Tc{\mathit{T}}_{\mathit{c}}Tc​=O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$). Various other pseudocritical temperatures corresponding to extrema of other cumulants are evaluated. We compare these theoretical predictions with extensive Monte Carlo simulations of the nominally strong first-order transitions in the eight- and ten-state Potts models in two dimensions for system sizes L\ensuremath{\le}50. The ten-state simulations agree with theory in all details in contrast to the eight-state data, and we give estimates for the bulk specific heats at Tc{\mathit{T}}_{\mathit{c}}Tc​ using all exactly known analytic results. A criterion is developed to estimate numerically whether or not system sizes used in a simulation of a first-order transition are in the finite-size-scaling regime.
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.