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arXiv:1112.5274 · Physical Review E

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

P. Butera, M. Pernici

Published Feb 6, 2012• 10 citations
arXivHTMLPDF

Abstract

High-temperature expansions are presently the only viable approach to the numerical calculation of the higher susceptibilities for the spin and the scalar-field models on high-dimensional lattices. The critical amplitudes of these quantities enter into a sequence of universal amplitude ratios that determine the critical equation of state. We have obtained a substantial extension, through order 24, of the high-temperature expansions of the free energy (in presence of a magnetic field) for the Ising models with spin s≥1/2 and for the lattice scalar-field theory with quartic self-interaction on the simple-cubic and the body-centered-cubic lattices in four, five, and six spatial dimensions. A numerical analysis of the higher susceptibilities obtained from these expansions yields results consistent with the widely accepted ideas, based on the renormalization group and the constructive approach to Euclidean quantum field theory, concerning the no-interaction ("triviality") property of the continuum (scaling) limit of spin-s Ising and lattice scalar-field models at and above the upper critical dimensionality.

DOI
10.1103/physreve.85.021105

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High-temperature expansions are presently the only viable approach to the numerical calculation of the higher susceptibilities for the spin and the scalar-field models on high-dimensional lattices. The critical amplitudes of these quantities enter into a sequence of universal amplitude ratios that determine the critical equation of state. We have obtained a substantial extension, through order 24, of the high-temperature expansions of the free energy (in presence of a magnetic field) for the Ising models with spin s≥1/2 and for the lattice scalar-field theory with quartic self-interaction on the simple-cubic and the body-centered-cubic lattices in four, five, and six spatial dimensions. A numerical analysis of the higher susceptibilities obtained from these expansions yields results consistent with the widely accepted ideas, based on the renormalization group and the constructive approach to Euclidean quantum field theory, concerning the no-interaction ("triviality") property of the continuum (scaling) limit of spin-s Ising and lattice scalar-field models at and above the upper critical dimensionality.
We have substantially extended the high-temperature and low-magnetic-field (and the related low-temperature and high-magnetic-field) bivariate expansions of the free energy for the conventional three-dimensional Ising model and for a variety of other spin systems generally assumed to belong to the same critical universality class. In particular, we have also derived the analogous expansions for the Ising models with spin s=1,3/2,…s=1,3/2,\dots{}s=1,3/2,… and for the lattice Euclidean scalar-field theory with quartic self-interaction, on the simple-cubic and the body-centered-cubic lattices. Our bivariate high-temperature expansions, which extend through 24th order, enable us to compute, through the same order, all higher derivatives of the free energy with respect to the field, namely, all higher susceptibilities. These data make more accurate checks possible, in critical conditions, both of the scaling and the universality properties with respect to the lattice and the interaction structure and also help to improve an approximate parametric representation of the critical equation of state for the three-dimensional Ising model universality class.
The high-temperature expansion coefficients of the ordinary and the higher susceptibilities of the spin-1/2 nearest-neighbor Ising model are calculated exactly up to the 20th order for the general d-dimensional (hyper)simple-cubical lattices. These series are analyzed to study the dependence of critical parameters on the lattice dimensionality. Using the general d expression of the ordinary susceptibility, we have more than doubled the length of the existing series expansion of the critical temperature in powers of 1/d.
The coordinates of the critical points of spin-S Ising models with coupling constants J and J′ are calculated for 1/2 ≤ S ≤ 13/2. The calculations are performed for several values of S and Δ ≡ J′/J independently by using the phenomenological renormalization-group method or (approximate) self-duality. Numerical results combined with a mean-field analysis show that the critical coupling strength for Δ ∼ 1 (weakly anisotropic lattice) is K c () (Δ) = K c () (1)[1 + a(1 − Δ)], where a = (d − 1)/d is independent of S (d is the space dimension). Both free energy and internal energy are determined at the critical points. An extremum of the critical internal energy is found at Δ* ∈ (0, 1). The parameter Δ* can be used as a criterion that separates quasi-isotropic and quasi-one-dimensional regimes (Δ* < Δ ≤ 1 and Δ < Δ*, respectively). The finite-size scaling amplitudes A s and A e of the inverse spin-spin and energy-energy correlation lengths are estimated. Calculations show that the amplitudes A s and A e are independent of S within the accuracy of the adopted approximations. Moreover, their ratio A e/A s is independent of the anisotropy parameter Δ. These results support the Ising universality hypothesis.
We compute a number of universal amplitude ratios in the three-dimensional Ising universality class. To this end, we perform Monte Carlo simulations of the improved Blume-Capel model on the simple cubic lattice. For example, we obtain ${A}_{+}/{A}_{\ensuremath{-}}=0.536(2)$ and ${C}_{+}/{C}_{\ensuremath{-}}=4.713(7)$, where ${A}_{\ifmmode\pm\else\textpm\fi{}}$ and ${C}_{\ifmmode\pm\else\textpm\fi{}}$ are the amplitudes of the specific heat and the magnetic susceptibility, respectively. The subscripts +++ and $\ensuremath{-}$ indicate the high- and the low-temperature phase, respectively. We compare our results with those obtained from previous Monte Carlo simulations, high- and low-temperature series expansions, field theoretic methods, and experiments.
The critical behavior of the one-fifth-depleted square-lattice Ising model with nearest-neighbor ferromagnetic interaction has been investigated by means of both an exact solution and a high-temperature series expansion study of the zero-field susceptibility. For the exact solution we employ a decoration transformation followed by a mapping to a staggered eight-vertex model. This yields a quartic equation for the critical coupling giving K(c)(≡βJ(c))=0.695. The series expansion for the susceptibility, to O(K(18)), when analyzed via standard Padé approximant methods gives an estimate of K(c), consistent with the exact solution result to at least four significant figures. The series expansion is also analyzed for the leading amplitude and subdominant terms.
We study the ±J transverse-field Ising spin-glass model at zero temperature on d-dimensional hypercubic lattices and in the Sherrington-Kirkpatrick (SK) model, by series expansions around the strong-field limit. In the SK model and in high dimensions our calculated critical properties are in excellent agreement with the exact mean-field results, surprisingly even down to dimension d=6, which is below the upper critical dimension of d=8. In contrast, at lower dimensions we find a rich singular behavior consisting of critical and Griffiths-McCoy singularities. The divergence of the equal-time structure factor allows us to locate the critical coupling where the correlation length diverges, implying the onset of a thermodynamic phase transition. We find that the spin-glass susceptibility as well as various power moments of the local susceptibility become singular in the paramagnetic phase before the critical point. Griffiths-McCoy singularities are very strong in two dimensions but decrease rapidly as the dimension increases. We present evidence that high enough powers of the local susceptibility may become singular at the pure-system critical point.