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

arXiv:cond-mat/9610212 · Annalen der Physik

A unified treatment of Ising model magnetizations

Brian Davies, Ingo Peschel

Published Jan 1, 1997• 10 citations
arXivHTMLPDF

Abstract

Abstract We show how the spontaneous bulk, surface and corner magnetizations in the square lattice Ising model can all be obtained within one approach. The method is based on functional equations which follow from the properties of corner transfer matrices and vertex operators and which can be derived graphically. In all cases, exact analytical expressions for general anisotropy are obtained. Known results, including several for which only numerical computation was previously possible, are verified and new results related to general anisotropy and corner angles are obtained.

DOI
10.1002/andp.19975090304

Related papers

A unified treatment of Ising model magnetizations

Brian Davies, Ingo Peschel, Fachbereich Physik +1 more

Abstract We show how the spontaneous bulk, surface and corner magnetizations in the square lattice Ising model can all be obtained within one approach. The method is based on functional equations which follow from the properties of corner transfer matrices and vertex operators and which can be derived graphically. In all cases, exact analytical expressions for general anisotropy are obtained. Known results, including several for which only numerical computation was previously possible, are verified and new results related to general anisotropy and corner angles are obtained.

•Aug 17, 2016•8 citations

Corner transfer matrices and corner magnetization for the Ising model

Brian Davies, Ingo Peschel

The authors consider the corner magnetization of an Ising square lattice with free edges along a diagonal direction. Using some remarkable properties of the corner transfer matrix, they give a unified treatment for various opening angles. The calculation of the magnetization is reduced to the solution of a matrix equation. In the thermodynamic limit they construct an explicit form of this equation and discuss its numerical and asymptotic solution and the results obtained thereby.

Journal of Physics A Mathematical and General•Mar 21, 1991•23 citations

Variational approach to the scaling function of the 2D Ising model in a magnetic field

Vladimir V. Mangazeev, Murray T. Batchelor, Vladimir V. Bazhanov +1 more

The universal scaling function of the square lattice Ising model in a magnetic field is obtained numerically via Baxter's variational corner transfer matrix approach. The high precision numerical data are in perfect agreement with the remarkable field theory results obtained by Fonseca and Zamolodchikov, as well as with many previously known exact and numerical results for the 2D Ising model. This includes excellent agreement with analytic results for the magnetic susceptibility obtained by Orrick, Nickel, Guttmann and Perk. In general, the high precision of the numerical results underlines the potential and full power of the variational corner transfer matrix approach.

arXiv:0811.3271•Journal of Physics A Mathematical and Theoretical•Dec 19, 2008•15 citations

The bulk, surface and corner free energies of the square lattice Ising model

R. J. Baxter

Abstract We use Kaufman’s spinor method to calculate the bulk, surface and corner free energies f b , f s , f s ′ , f c of the anisotropic square lattice zero-field Ising model for the ordered ferromagnetic case. For f b , f s , f s ′ our results of course agree with the early work of Onsager, McCoy and Wu. We also find agreement with the conjectures made by Vernier and Jacobsen (VJ) for the isotropic case. We note that the corner free energy f c depends only on the elliptic modulus k that enters the working, and not on the argument v , which means that VJ’s conjecture applies for the full anisotropic model. The only aspect of this paper that is new is the actual derivation of f c , but by reporting all four free energies together we can see interesting structures linking them.

arXiv:1606.02029•Journal of Physics A Mathematical and Theoretical•Nov 29, 2016•22 citations

Finite-size scaling of corner transfer matrices for the two-dimensional Ising model

Ingo Peschel

The eigenvalue spectra of generalised corner transfer matrices are calculated for an Ising model on a square lattice. The predictions of conformal invariance at the critical point are thereby verified.

Journal of Physics A Mathematical and General•Feb 7, 1988•9 citations

The bulk, surface and corner free energies of the anisotropic triangular Ising model

R. J. Baxter

We consider the anisotropic Ising model on the triangular lattice with finite boundaries, and use Kaufman’s spinor method to calculate low-temperature series expansions for the partition function to high order. From these, we can obtain 108-term series expansions for the bulk, surface and corner free energies. We extrapolate these to all terms and thereby conjecture the exact results for each. Our results agree with the exactly known bulk-free energy and with Cardy and Peschel’s conformal invariance predictions for the dominant behaviour at criticality. For the isotropic case, they also agree with Vernier and Jacobsen’s conjecture for the 60 ° corners.

Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Feb 1, 2020•3 citations
Abstract We show how the spontaneous bulk, surface and corner magnetizations in the square lattice Ising model can all be obtained within one approach. The method is based on functional equations which follow from the properties of corner transfer matrices and vertex operators and which can be derived graphically. In all cases, exact analytical expressions for general anisotropy are obtained. Known results, including several for which only numerical computation was previously possible, are verified and new results related to general anisotropy and corner angles are obtained.
Abstract We show how the spontaneous bulk, surface and corner magnetizations in the square lattice Ising model can all be obtained within one approach. The method is based on functional equations which follow from the properties of corner transfer matrices and vertex operators and which can be derived graphically. In all cases, exact analytical expressions for general anisotropy are obtained. Known results, including several for which only numerical computation was previously possible, are verified and new results related to general anisotropy and corner angles are obtained.
The universal scaling function of the square lattice Ising model in a magnetic field is obtained numerically via Baxter's variational corner transfer matrix approach. The high precision numerical data are in perfect agreement with the remarkable field theory results obtained by Fonseca and Zamolodchikov, as well as with many previously known exact and numerical results for the 2D Ising model. This includes excellent agreement with analytic results for the magnetic susceptibility obtained by Orrick, Nickel, Guttmann and Perk. In general, the high precision of the numerical results underlines the potential and full power of the variational corner transfer matrix approach.
Abstract We use Kaufman’s spinor method to calculate the bulk, surface and corner free energies f b , f s , f s ′ , f c of the anisotropic square lattice zero-field Ising model for the ordered ferromagnetic case. For f b , f s , f s ′ our results of course agree with the early work of Onsager, McCoy and Wu. We also find agreement with the conjectures made by Vernier and Jacobsen (VJ) for the isotropic case. We note that the corner free energy f c depends only on the elliptic modulus k that enters the working, and not on the argument v , which means that VJ’s conjecture applies for the full anisotropic model. The only aspect of this paper that is new is the actual derivation of f c , but by reporting all four free energies together we can see interesting structures linking them.
We consider the anisotropic Ising model on the triangular lattice with finite boundaries, and use Kaufman’s spinor method to calculate low-temperature series expansions for the partition function to high order. From these, we can obtain 108-term series expansions for the bulk, surface and corner free energies. We extrapolate these to all terms and thereby conjecture the exact results for each. Our results agree with the exactly known bulk-free energy and with Cardy and Peschel’s conformal invariance predictions for the dominant behaviour at criticality. For the isotropic case, they also agree with Vernier and Jacobsen’s conjecture for the 60 ° corners.