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arXiv:0811.3271 · Journal of Physics A Mathematical and Theoretical

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

Vladimir V. Mangazeev, Murray T. Batchelor, Vladimir V. Bazhanov, Michael Yu. Dudalev

Published Dec 19, 2008• 15 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8113/42/4/042005

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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.
The scaling function of the two-dimensional Ising model on the square and triangular lattices is obtained numerically via Baxter's variational corner transfer-matrix approach. The use of Aharony-Fisher nonlinear scaling variables allowed us to perform calculations sufficiently away from the critical point and to confirm all predictions of the scaling and universality hypotheses. Our results are in excellent agreement with quantum field theory calculations of Fonseca and Zamolodchikov as well as with many previously known exact and numerical calculations, including susceptibility results by Barouch, McCoy, Tracy, and Wu.
We study the two-dimensional (2D) Ising model in a complex magnetic field in the vicinity of the Yang-Lee edge singularity. By using Baxter's variational corner transfer matrix method combined with analytic techniques, we numerically calculate the scaling function and obtain an accurate estimate of the location of the Yang-Lee singularity. The existing series expansions for susceptibility of the 2D Ising model on a triangular lattice by Chan, Guttmann, Nickel, and Perk allowed us to substantially enhance the accuracy of our calculations. Our results are in excellent agreement with the Ising field theory calculations by Fonseca, Zamolodchikov, and the recent work by Xu and Zamolodchikov. In particular, we numerically confirm an agreement between the leading singular behavior of the scaling function and the predictions of the M_{2/5} conformal field theory.
In this thesis, we study the critical behaviour of the two-dimensional Ising model on the regular lattices. Using the numerical solution of the model on the square, triangular and honeycomb lattices we compute the universal scaling function, which turns out to be identical on each of the lattices, in addition to being identical to the scaling function of the Ising Field Theory, computed previously by Fonseca and Zamolodchikov. To cope with the lattice contributions we carefully examined series expansions of the lattice free energy derivatives. We included the non-scaling regular part of the free energy as well as non-linear Aharony-Fisher scaling elds, which all have non-universal expansions. Using as many of the previously known exact results as possible, we were able to t the unknown coe cients of the scaling function expansion and obtain some non-universal coe cients. In contrast to the IFT approach of Fonseca and Zamolodchikov, all coe cients were obtained independently from separate datasets, without using dispersion relations. These results show that the Scaling and Universality hypotheses, with the help of the Aharony-Fisher corrections, hold on the lattice to very high precision and so there should be no doubt of their validity. For all numerical computations we used the Corner Transfer Matrix Renormalisation Group (CTMRG) algorithm, introduced by Nishino and Okunishi. The algorithm combines Baxter's variational approach (which gives Corner Transfer Matrix (CTM) equations), and White's Density Matrix Renormalisation Group (DMRG) method to solve the CTM equations e ciently. It was shown that given su cient distance from the critical point, the algorithmic precision is exceptionally good and is unlikely to be exceeded with any other general algorithm using the same amount of numerical computations. While performing tests we also con rmed several critical parameters of the three-state Ising and Blume-Capel models, although no extra precision was gained, compared to previous results from other methods. In addition to the results presented here, we produced an efficient and reusable implementation of the CTMRG algorithm, which after minor modifications could be used for a variety of lattice models, such as the Kashiwara-Miwa and the chiral Potts models.
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.