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

A Numerical Study of the Hierarchical Ising Model: High Temperature Versus Epsilon Expansion

Y. Meurice, G. Ordaz, V.G.J. Rodgers

Published Jan 1, 1994• 4 citations
OpenAlex

Abstract

We study numerically the magnetic susceptibility of the hierarchical model with Ising spins ($\\sigma =\\pm 1$) above the critical temperature and for two values of the epsilon parameter. The integrations are performed exactly, using recursive methods which exploit the symmetries of the model. Lattices with up to $2^18$ sites have been used. Surprisingly, the numerical data can be fitted very well with a simple power law of the form $(1- \\beta /\\beta _c )^{- \\gamma} $for the {\\it whole} temperature range. The numerical values for $\\gamma $ agree within a few percent with the values calculated with a high-temperature expansion but show significant discrepancies with the epsilon-expansion. We would appreciate comments about these results.

Related papers

A numerical study of the hierarchical Ising model: High-temperature versus epsilon expansion

Y. Meurice, G. Ordaz, V. G. J. Rodgers

arXiv:hep-lat/9401016•Journal of Statistical Physics•Nov 1, 1994•8 citations

Evidence for Complex Subleading Exponents from the High-Temperature Expansion of Dyson's Hierarchical Ising Model

Yannick Meurice, G. Ordaz, V.G.J. Rodgers

Using a renormalization group method, we calculate 800 high-temperature coefficients of the magnetic susceptibility of the hierarchical Ising model. The conventional quantities obtained from differences of ratios of coefficients show unexpected smooth oscillations with a period growing logarithmically and can be fitted assuming corrections to the scaling laws with complex exponents.

arXiv:hep-lat/9507016•Physical Review Letters•Dec 18, 1995•26 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

The Oscillatory Behavior of the High-Temperature Expansion of Dyson's Hierarchical Model: A Renormalization Group Analysis

Meurice, Y, Niermann, S M, Ordaz, G

We calculate 800 coefficients of the high-temperature expansion of the magnetic susceptibility of Dyson's hierarchical model with a Landau-Ginzburg measure. Log-periodic corrections to the scaling laws appear as in the case of a Ising measure. The period of oscillation appears to be a universal quantity given in good approximation by the logarithm of the largest eigenvalue of the linearized RG transformation, in agreement with a possibility suggested by K. Wilson and developed by Niemeijer and van Leeuwen. We estimate $\\gamma $ to be 1.300 (with a systematic error of the order of 0.002) in good agreement with the results obtained with other methods such as the $\\epsilon $-expansion. We briefly discuss the relationship between the oscillations and the zeros of the partition function near the critical point in the complex temperature plane.

•Aug 6, 1996•8 citations

Spin- S Ising susceptibility on the triangular lattice

John P. VAN DYKE, William J. Camp

The high-temperature susceptibility of the spin-$S$ Ising model on the triangular net is analyzed as a function of $S$ using tenth-order high-temperature series expansions. In particular the spin dependence of the critical temperature and the amplitude of the dominant singularity are obtained by assuming a universal value $\ensuremath{\gamma}=\frac{7}{4}$ for the critical exponent of the susceptibility. The results, for the critical temperature particularly, provide a valuable guidepost in judging the convergence of recent nonlinear renormalization-group calculations for these models.

Physical review. B, Solid state•Aug 1, 1975•2 citations

Spin Susceptibility of Quantum Magnets from High to Low Temperatures

B. Bernu, C. Lhuillier

We explain how and why all thermodynamic properties of spin systems can be computed in one and two dimensions in the whole range of temperatures overcoming the divergence towards zero temperature of the standard high-temperature series expansions (HTEs). The method relies on an approximation of the entropy versus energy (microcanonical potential function) on the whole range of energies. The success is related to the intrinsic physical constraints on the entropy function and a careful treatment of the boundary behaviors. This method is benchmarked against two one-dimensional solvable models: the Ising model in longitudinal field and the XY model in a transverse field. With ten terms in the HTE, we find a spin susceptibility within a few percent of the exact results over the entire range of temperatures. The method is then applied to two two-dimensional models: the supposedly gapped Heisenberg model and the J(1)-J(2)-J(d) model on the kagome lattice.

arXiv:1407.0949•Physical Review Letters•Feb 4, 2015•38 citations
We study numerically the magnetic susceptibility of the hierarchical model with Ising spins (sigma=pm1\\sigma =\\pm 1sigma=pm1) above the critical temperature and for two values of the epsilon parameter. The integrations are performed exactly, using recursive methods which exploit the symmetries of the model. Lattices with up to 2182^18218 sites have been used. Surprisingly, the numerical data can be fitted very well with a simple power law of the form (1−beta/betac)−gamma(1- \\beta /\\beta _c )^{- \\gamma} (1−beta/betac​)−gammafor the {\it whole} temperature range. The numerical values for gamma\\gamma gamma agree within a few percent with the values calculated with a high-temperature expansion but show significant discrepancies with the epsilon-expansion. We would appreciate comments about these 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 calculate 800 coefficients of the high-temperature expansion of the magnetic susceptibility of Dyson's hierarchical model with a Landau-Ginzburg measure. Log-periodic corrections to the scaling laws appear as in the case of a Ising measure. The period of oscillation appears to be a universal quantity given in good approximation by the logarithm of the largest eigenvalue of the linearized RG transformation, in agreement with a possibility suggested by K. Wilson and developed by Niemeijer and van Leeuwen. We estimate $\\gamma $ to be 1.300 (with a systematic error of the order of 0.002) in good agreement with the results obtained with other methods such as the $\\epsilon $-expansion. We briefly discuss the relationship between the oscillations and the zeros of the partition function near the critical point in the complex temperature plane.
The high-temperature susceptibility of the spin-SSS Ising model on the triangular net is analyzed as a function of SSS using tenth-order high-temperature series expansions. In particular the spin dependence of the critical temperature and the amplitude of the dominant singularity are obtained by assuming a universal value $\ensuremath{\gamma}=\frac{7}{4}$ for the critical exponent of the susceptibility. The results, for the critical temperature particularly, provide a valuable guidepost in judging the convergence of recent nonlinear renormalization-group calculations for these models.
We explain how and why all thermodynamic properties of spin systems can be computed in one and two dimensions in the whole range of temperatures overcoming the divergence towards zero temperature of the standard high-temperature series expansions (HTEs). The method relies on an approximation of the entropy versus energy (microcanonical potential function) on the whole range of energies. The success is related to the intrinsic physical constraints on the entropy function and a careful treatment of the boundary behaviors. This method is benchmarked against two one-dimensional solvable models: the Ising model in longitudinal field and the XY model in a transverse field. With ten terms in the HTE, we find a spin susceptibility within a few percent of the exact results over the entire range of temperatures. The method is then applied to two two-dimensional models: the supposedly gapped Heisenberg model and the J(1)-J(2)-J(d) model on the kagome lattice.