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

New method for the Ising model phase diagram

S. M. Girvin

Published May 28, 1978• 8 citations
Publisher

Abstract

An alternative formulation of the cumulant expansion method of Frank and Mitran (see ibid., vol.10, p.2641 (1977) and Solid State Commun., vol.24, p.343 (1977)) is developed for the self-consistent calculation of the two-spin correlation function of the three-dimensional Ising model. The present technique is much simpler and more accurate since the cumulant series is summed exactly and certain other approximations made by Frank and Mitran are avoided. In addition to obtaining critical temperatures for the zero-field ferromagnet which are within 1% of the exact values, the entire phase boundary of the antiferromagnet in an external field is accurately given by the present method. Approximate critical exponents are also obtained. The present method requires considerably less numerical effort than real space renormalisation group methods and gives better results.

DOI
10.1088/0022-3719/11/10/004

Related papers

Comment on Girvin's method for the Ising model

R. Honmura, T. Kaneyoshi

It is proved that Girvin's method (see ibid., vol.11, p.L427 (1978)) for the calculation of the two-spin correlation function of the three-dimensional Ising model is based on the same approximation made by Frank and Mitran (1978).

Journal of Physics C Solid State Physics•Dec 24, 1978•14 citations

Low-Temperature Critical Exponents from High-Temperature Series: The Ising Model

David S. Gaunt, George Allen Baker Jr.

First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.

Physical review. B, Solid state•Feb 1, 1970•113 citations

Thermodynamics of Ising Ferromagnet Studied by the Real-Space Renormalization Group Method

W. Reich, Kamilla Piotrkowska, Ugo Esposito +1 more

The critical exponents of the 3D Ising model were calculated in the approximation of the fourth-order cumulant expansion.Thermodynamic functions in the high-temperature range are obtained.

Acta Physica Polonica A•Aug 1, 1993•0 citations

One-dimensional three-spin cell calculation for two-dimensional Ising exponents

Gregory S. Um

We have calculated the three-spin cell cumulant expansion of Niemeijer and Van Leeuwen for the one-dimensional $\mathrm{XY}$ model in order to obtain the critical exponents of the two-dimensional Ising model. We found that the calculation of this type is not as accurate as the direct cumulant-expansion calculation for the two-dimensional Ising model. Specifically, the results of our present calculation deviate considerably from the exact results in comparison to the results obtained by Hsu, Niemeijer, and Gunton. Moreover, we find that the odd-spin eigenvalue is not physical in this particular renormalization-group transformation.

Physical review. B, Condensed matter•May 1, 1978•5 citations

Position-space studies of the pure and dilute quantum Ising model. III. Finite-temperature extension for higher dimensions

L. S. Campana, Alvaro Caramico D’Auria, Ugo Esposito +1 more

For pt.III see ibid., vol.20, p.1841 (1987). The global real-space renormalisation group approach is applied in two and three dimensions to study the pure and diluted transverse Ising model at finite temperatures. A phase diagram in the temperature-field plane is presented for the pure quantum model on the honeycomb, triangular and square lattices in the first-order cumulant expansion. An improvement of the classical Ising fixed-point data in two and three dimensions is considered by having recourse to larger cells and the second-order cumulant expansion. The zero-temperature percolative behaviour of the bond-diluted system is recovered from the fixed-point data analysis. Some shortcomings and relations to other approaches are also discussed.

Journal of Physics C Solid State Physics•Nov 10, 1987•4 citations

Spin-operator reduced relations in a new representation for the Ising ferromagnet

B. Frank

A stringent test is applied to a set of approximations used, in an earlier work (CCFA) see Frank et al. (ibid., vol.11, p.208 (1978) to calculate T c for the spin- 1 / 2 Ising ferromagnet. In CCFA, all cumulant averages M 2n (2n>or=6) were approximated, for all the cubic lattices, and justified a posteriori. Here, a spin operator reduction relation allows one, for the SC lattice, to approximate only M 6 , and moreover, to understand the approximation used. The extreme closeness at every stage (within 0.01%) of the present T c values to those of CCFA gives one confidence method of approximating the M 2n for the BCC and FCC lattices as well. Here also, as in CCFA, the M 60 value of T c is within 0.5% of the series approximation value of Sykes et al. (1972).

Journal of Physics C Solid State Physics•Sep 28, 1978•8 citations
An alternative formulation of the cumulant expansion method of Frank and Mitran (see ibid., vol.10, p.2641 (1977) and Solid State Commun., vol.24, p.343 (1977)) is developed for the self-consistent calculation of the two-spin correlation function of the three-dimensional Ising model. The present technique is much simpler and more accurate since the cumulant series is summed exactly and certain other approximations made by Frank and Mitran are avoided. In addition to obtaining critical temperatures for the zero-field ferromagnet which are within 1% of the exact values, the entire phase boundary of the antiferromagnet in an external field is accurately given by the present method. Approximate critical exponents are also obtained. The present method requires considerably less numerical effort than real space renormalisation group methods and gives better results.
First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.
We have calculated the three-spin cell cumulant expansion of Niemeijer and Van Leeuwen for the one-dimensional XY\mathrm{XY}XY model in order to obtain the critical exponents of the two-dimensional Ising model. We found that the calculation of this type is not as accurate as the direct cumulant-expansion calculation for the two-dimensional Ising model. Specifically, the results of our present calculation deviate considerably from the exact results in comparison to the results obtained by Hsu, Niemeijer, and Gunton. Moreover, we find that the odd-spin eigenvalue is not physical in this particular renormalization-group transformation.
For pt.III see ibid., vol.20, p.1841 (1987). The global real-space renormalisation group approach is applied in two and three dimensions to study the pure and diluted transverse Ising model at finite temperatures. A phase diagram in the temperature-field plane is presented for the pure quantum model on the honeycomb, triangular and square lattices in the first-order cumulant expansion. An improvement of the classical Ising fixed-point data in two and three dimensions is considered by having recourse to larger cells and the second-order cumulant expansion. The zero-temperature percolative behaviour of the bond-diluted system is recovered from the fixed-point data analysis. Some shortcomings and relations to other approaches are also discussed.
A stringent test is applied to a set of approximations used, in an earlier work (CCFA) see Frank et al. (ibid., vol.11, p.208 (1978) to calculate T c for the spin- 1 / 2 Ising ferromagnet. In CCFA, all cumulant averages M 2n (2n>or=6) were approximated, for all the cubic lattices, and justified a posteriori. Here, a spin operator reduction relation allows one, for the SC lattice, to approximate only M 6 , and moreover, to understand the approximation used. The extreme closeness at every stage (within 0.01%) of the present T c values to those of CCFA gives one confidence method of approximating the M 2n for the BCC and FCC lattices as well. Here also, as in CCFA, the M 60 value of T c is within 0.5% of the series approximation value of Sykes et al. (1972).