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Journal of Physics A Mathematical and General

The two-dimensional tricritical Ising model in an external magnetic field

Malte Henkel

Published Oct 7, 1990• 8 citations
Publisher

Abstract

The two-dimensional tricritical Ising model with an order parameter perturbation is studied using two different lattice models. The mass spectrum is found to be the same as in the 2D Ising model with a magnetic field. Results for the scaling functions are in agreement with universality and for the lowest mass a relationship with the scaling function of the Ising model is observed. The numerical results do not support a recent proposal of Dotsenko on the dynamical generation of extra terms in the perturbative calculation of scaling functions.

DOI
10.1088/0305-4470/23/19/020

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The two-dimensional tricritical Ising model with an order parameter perturbation is studied using two different lattice models. The mass spectrum is found to be the same as in the 2D Ising model with a magnetic field. Results for the scaling functions are in agreement with universality and for the lowest mass a relationship with the scaling function of the Ising model is observed. The numerical results do not support a recent proposal of Dotsenko on the dynamical generation of extra terms in the perturbative calculation of scaling functions.
We compute the spectrum and several critical amplitudes of the two dimensional Ising model in a magnetic field with the transfer matrix method. The three lightest masses and their overlaps with the spin and the energy operators are computed on lattices of a width up to L=21. In extracting the continuum results we also take into account the corrections to scaling due to irrelevant operators. In contrast with previous Monte Carlo simulations our final results are in perfect agreement with the predictions of S-matrix and conformal field theory. We also obtain the amplitudes of some of the subleading corrections, for which no S-matrix prediction has yet been obtained.
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.
The two-dimensional Ising model is the simplest model of statistical mechanics exhibiting a second-order phase transition. While in absence of magnetic field it is known to be solvable on the lattice since Onsager's work of the 1940s, exact results for the magnetic case have been missing until the late 1980s, when A Zamolodchikov solved the model in a field at the critical temperature, directly in the scaling limit, within the framework of integrable quantum field theory. In this paper, we review this field theoretical approach to the Ising universality class, with particular attention to the results obtained starting from Zamolodchikov's scattering solution and to their comparison with the numerical estimates on the lattice. The topics discussed include scattering theory, form factors, correlation functions, universal amplitude ratios and perturbations around integrable directions. Although we restrict our discussion to the Ising model, the emphasis is on the general methods of integrable quantum field theory which can be used in the study of all universality classes of critical behaviour in two dimensions.