Diversity of critical behavior within a universality class

Abstract

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.

Triviality problem and high-temperature expansions of higher susceptibilities for the Ising and scalar-field models in four-, five-, and six-dimensional lattices

“…(24) and (25), and so on, and observing that the quantities r 2n (K) vanish as K → K − c in the MF approximation [2], the corresponding critical valuesÎ + 2r+4 of the quantities in Eq. (20) can be simply evaluated, obtainingÎ + 6 = 10,Î + 8 = 280,Î + 10 = 15400,Î + 12 = 1 401 400, and so on. It is also not difficult [71] to compute the MF values of the first few terms of the sequences A + 2r+4 and B + 2r+8 ; for example: + 8 = 14/5, + 10 = 55/28B + 10 = 154, and B + 12 = 143/8.…”

“…The detailed exploration of the nearcritical behavior is hampered by the necessity of considering systems of very large sizes and, in particular, at d = 4, by the difficulty of an accurate characterization of the slowly varying logarithmic deviations from MF behavior. Moreover, for d 4, the finite-size-scaling theory and the confluent corrections to scaling have been debated [20][21][22][23][24][25]. Thus, relatively few of the numerous available MC studies [17,22,[26][27][28][29][30][31][32][33][34][35] are likely to be extensive enough to yield a satisfactory overall description of these systems at criticality, in spite of the remarkable progress in the simulation algorithms with reduced critical slowdown.…”

Pronounced minimum of the thermodynamic Casimir forces ofO(n)symmetric film systems: Analytic theory

“…We also predict a minimum for the (n = 3) Heisenberg universality class. Our theory with Dirichlet BC should also be an appropriate basis for describing Casimir forces in superconducting films [7] and finite-size effects in confined magnetic materials provided that the theory includes the effects of lattice anisotropy [21].…”

“…Equations (7)- (9) do not yet correctly describe the finite-size scaling behavior in terms of the scaling variablex with the correct critical exponent ν. This will be achieved by appropriate renormalizations that we perform within the minimal subtraction scheme at fixed dimension d [21,28].…”

Dynamics of the critical Casimir force for a conserved order parameter after a critical quench

“…The Poisson resummation formula in its general form (see, e.g., Refs. [56,76,99]) states that, for a given function f (n) depending on n ∈ Z D ,…”