Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics C Solid State Physics

Universality and crossover in an Ising-like model

M A Moore, D. M. Saul, Michael Wortis

Published Jan 7, 1974• 17 citations
Publisher

Abstract

The critical behaviour of an anisotropic n-vector model with purely Ising-like coupling is studied on the FCC lattice by means of high-temperature series. The model includes the s= 1 / 2 (n=1) and s= infinity (n=3) Ising models as special cases and exhibits rigorously mean-field behaviour at n= infinity . The crossover region 1/n to 0, t identical to (1-v/v c ) to 0, is examined in detail. The susceptibility exponent gamma remains Ising-like for all finite n. The critical temperature and critical susceptibility amplitude vary as 1/n.

DOI
10.1088/0022-3719/7/1/023

Related papers

Universality and crossover in an extremely anisotropic n-vector model

Amnon Aharony A

The critical behaviour of an anisotropic n-vector model with pure single-component or Ising-like coupling is studied. The non-interacting spin components are integrated over, and the resulting single-component, reduced hamiltonian represents the gaussian model for n= infinity and the Ising-Ginzburg-Landau-Wilson model for finite n. For large n one finds T c (n)-T c ( infinity ) varies as 1/n, while the susceptibility behaves as chi approximately t -1 X(1/nt phi ), with t=T/T c (n)-1 and phi = 1 / 2 (4-d).

Journal of Physics C Solid State Physics•Feb 21, 1974•10 citations

Critical behaviour of the anisotropic Ising model

J. Oitmaa, Ian Graham Enting

The first eleven terms of the high temperature susceptibility series for the anisotropic Ising model on a simple cubic lattice are calculated. The series is analysed to obtain estimates of the critical temperature and critical exponent for various values of the anisotropy parameter eta . Strong evidence is found for an apparent continuous variation of the exponent with eta which is in contradiction with the discontinuous change at eta =0 which has been generally believed to occur. The detailed variation of the critical temperature with eta is also in disagreement with predictions from generalized scaling laws.

Journal of Physics C Solid State Physics•Jan 31, 1972•26 citations

On the Phase Transition and Critical Behaviour in the Two-Dimensional Anisotropic Heisenberg Model

R. Shanker, R. A. Singh

The anisotropic Heisenberg model has been investigated by interpolating anisotropy between the Ising and isotropic Heisenberg limits via general spin high temperature series (HTS) for the zero-field susceptibility of the ferromagnets for the square lattice. This is done by locating the critical temperature ( T c (2) ) and the susceptibility exponent (γ) from the analysis of the series by Pade approximant and ratio methods. The critical temperature is found to be spin dependent. It decreases as anisotropy changes from the Ising limit towards the isotropic limit and remains finite (lower than that of Stanley and Kaplan) for all the spins except for S =1/2 at the isotropic limit. Exponent γ is almost Ising like for the anisotropic system and anomalous jump in it occurs for the isotropic case. In addition, expressions for the magnetization are derived by using spin-wave theory. The limit of complete isotropy has been found not to be a special case for the susceptibility, in contrast to magnetization.

Journal of the Physical Society of Japan•Mar 15, 1976•2 citations

General- R High-Temperature Series for the Susceptibility, Second Moment, and Specific Heat of sc and fcc Ising Models with Lattice Anisotropy

Fredric Harbus, Eugene Stanley

Of particular current interest is the critical behavior of functions on crossing over from one lattice dimensionality to another. To this end, we report high-temperature series for an Ising model with lattice anisotropy-i.e., with different exchange constants for different lattice directions. The Hamiltonian is ${\mathcal{H}}_{l,\mathrm{anis}}=\ensuremath{-}{J}_{\mathrm{xy}}\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{\mathrm{xy}}{s}_{i}{s}_{j}\ensuremath{-}{J}_{z}\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{z}{s}_{i}{s}_{j}\ensuremath{\equiv}\ensuremath{-}{J}_{\mathrm{xy}}(\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{\mathrm{xy}}{s}_{i}{s}_{j}+R\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{z}{s}_{i}{s}_{j})$ where ${s}_{i}=\ifmmode\pm\else\textpm\fi{}1$, the first summation is over all nearest-neighbor pairs in the $\mathrm{xy}$ plane, and the second sum is over pairs coupled in the $z$ direction. The susceptibility, second moment, and specific-heat series are explicitly presented for arbitrary ${J}_{\mathrm{xy}}$ and ${J}_{z}$ for the simple cubic (sc) and face-centered cubic (fcc) lattices to tenth order in inverse temperature. The general-$R$ series are essential if one wishes to study the Riedel-Wegner crossover exponent appropriate to changing lattice dimensionality, since for $R\ensuremath{\equiv}\frac{{J}_{z}}{{J}_{\mathrm{xy}}}=0$, both the sc and fcc lattices reduce to two-dimensional square lattices, while in the limit $R\ensuremath{\rightarrow}\ensuremath{\infty}$, the sc reduces to noninteracting linear chains.

Physical review. B, Solid state•Jan 1, 1973•33 citations

The square-lattice Ising model with first and second neighbour interactions

J. Oitmaa

The critical behaviour of the square-lattice Ising model, with nearest and next-nearest neighbour interactions of either sign, has been investigated by means of high-temperature series. lhe location of the critical lines in the coupling constant plane has been accurately determined. Along the critical line which corresponds to transitions to the layered or superantiferromagnetic state a breakdown of universality is observed and explicit numerical estimates obtained for the exponent of the ordering susceptibility.

Journal of Physics A Mathematical and General•May 1, 1981•109 citations

Critical behaviour of a generalized Ising model

J. Oitmaa

A high temperature expansion is obtained for the zero field susceptibility of a generalized Ising model, which can be used to describe the critical behaviour of various multicomponent systems. The first seven terms of the series are obtained for the face centred cubic lattice. Estimates of the critical temperature obtained from this series are considerably more precise than those obtained previously from the zero field free energy. Estimates are also obtained for the critical exponents of the susceptibility and the derivatives of the zero field free energy. The results are compared with the predictions of mean field theory.

Journal of Physics C Solid State Physics•Feb 29, 1972•46 citations
The critical behaviour of an anisotropic n-vector model with purely Ising-like coupling is studied on the FCC lattice by means of high-temperature series. The model includes the s= 1 / 2 (n=1) and s= infinity (n=3) Ising models as special cases and exhibits rigorously mean-field behaviour at n= infinity . The crossover region 1/n to 0, t identical to (1-v/v c ) to 0, is examined in detail. The susceptibility exponent gamma remains Ising-like for all finite n. The critical temperature and critical susceptibility amplitude vary as 1/n.
The critical behaviour of an anisotropic n-vector model with pure single-component or Ising-like coupling is studied. The non-interacting spin components are integrated over, and the resulting single-component, reduced hamiltonian represents the gaussian model for n= infinity and the Ising-Ginzburg-Landau-Wilson model for finite n. For large n one finds T c (n)-T c ( infinity ) varies as 1/n, while the susceptibility behaves as chi approximately t -1 X(1/nt phi ), with t=T/T c (n)-1 and phi = 1 / 2 (4-d).
The first eleven terms of the high temperature susceptibility series for the anisotropic Ising model on a simple cubic lattice are calculated. The series is analysed to obtain estimates of the critical temperature and critical exponent for various values of the anisotropy parameter eta . Strong evidence is found for an apparent continuous variation of the exponent with eta which is in contradiction with the discontinuous change at eta =0 which has been generally believed to occur. The detailed variation of the critical temperature with eta is also in disagreement with predictions from generalized scaling laws.
The anisotropic Heisenberg model has been investigated by interpolating anisotropy between the Ising and isotropic Heisenberg limits via general spin high temperature series (HTS) for the zero-field susceptibility of the ferromagnets for the square lattice. This is done by locating the critical temperature ( T c (2) ) and the susceptibility exponent (γ) from the analysis of the series by Pade approximant and ratio methods. The critical temperature is found to be spin dependent. It decreases as anisotropy changes from the Ising limit towards the isotropic limit and remains finite (lower than that of Stanley and Kaplan) for all the spins except for S =1/2 at the isotropic limit. Exponent γ is almost Ising like for the anisotropic system and anomalous jump in it occurs for the isotropic case. In addition, expressions for the magnetization are derived by using spin-wave theory. The limit of complete isotropy has been found not to be a special case for the susceptibility, in contrast to magnetization.
Of particular current interest is the critical behavior of functions on crossing over from one lattice dimensionality to another. To this end, we report high-temperature series for an Ising model with lattice anisotropy-i.e., with different exchange constants for different lattice directions. The Hamiltonian is ${\mathcal{H}}_{l,\mathrm{anis}}=\ensuremath{-}{J}_{\mathrm{xy}}\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{\mathrm{xy}}{s}_{i}{s}_{j}\ensuremath{-}{J}_{z}\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{z}{s}_{i}{s}_{j}\ensuremath{\equiv}\ensuremath{-}{J}_{\mathrm{xy}}(\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{\mathrm{xy}}{s}_{i}{s}_{j}+R\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{z}{s}_{i}{s}_{j})$ where ${s}_{i}=\ifmmode\pm\else\textpm\fi{}1$, the first summation is over all nearest-neighbor pairs in the xy\mathrm{xy}xy plane, and the second sum is over pairs coupled in the zzz direction. The susceptibility, second moment, and specific-heat series are explicitly presented for arbitrary Jxy{J}_{\mathrm{xy}}Jxy​ and Jz{J}_{z}Jz​ for the simple cubic (sc) and face-centered cubic (fcc) lattices to tenth order in inverse temperature. The general-RRR series are essential if one wishes to study the Riedel-Wegner crossover exponent appropriate to changing lattice dimensionality, since for $R\ensuremath{\equiv}\frac{{J}_{z}}{{J}_{\mathrm{xy}}}=0$, both the sc and fcc lattices reduce to two-dimensional square lattices, while in the limit $R\ensuremath{\rightarrow}\ensuremath{\infty}$, the sc reduces to noninteracting linear chains.
The critical behaviour of the square-lattice Ising model, with nearest and next-nearest neighbour interactions of either sign, has been investigated by means of high-temperature series. lhe location of the critical lines in the coupling constant plane has been accurately determined. Along the critical line which corresponds to transitions to the layered or superantiferromagnetic state a breakdown of universality is observed and explicit numerical estimates obtained for the exponent of the ordering susceptibility.
A high temperature expansion is obtained for the zero field susceptibility of a generalized Ising model, which can be used to describe the critical behaviour of various multicomponent systems. The first seven terms of the series are obtained for the face centred cubic lattice. Estimates of the critical temperature obtained from this series are considerably more precise than those obtained previously from the zero field free energy. Estimates are also obtained for the critical exponents of the susceptibility and the derivatives of the zero field free energy. The results are compared with the predictions of mean field theory.