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Journal of Physics A Mathematical and General

Operator content of the multicritical magnetic hard-square model

Man-Sun Kim, Je-Y Choi, Ki Kwon

Published Jun 7, 1988• 3 citations
Publisher

Abstract

The transfer matrix spectra of the magnetic hard squares along the multicritical line are analysed using the data for the strip width N up to 12. The operator content of the model is exactly the same as that of the Ashkin-Teller model. When N is odd, the spectrum is accounted for by the continuum theory in which the antiperiodic boundary condition is imposed to one of the Ising variables of the Ashkin-Teller model.

DOI
10.1088/0305-4470/21/11/024

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The transfer matrix spectra of the magnetic hard squares along the multicritical line are analysed using the data for the strip width N up to 12. The operator content of the model is exactly the same as that of the Ashkin-Teller model. When N is odd, the spectrum is accounted for by the continuum theory in which the antiperiodic boundary condition is imposed to one of the Ising variables of the Ashkin-Teller model.
Transfer matrix spectra can be used to establish a connection between two dimensional critical lattice models and conformal field theories. Affine–D models are a class of spin models with (L+3) state per site (L=3, 4, ...) generalizing the multicritical line of the magnetic hard square model. There are two continuous parameters characterizing the models; crossing parameter λ with which critical exponents vary continuously and spectral parameter u which controls anisotropy of the interactions. Their transfer matrix spectra combined with finite‐size scaling theory show that the models in the positive u regime belong to the universality class of the Ashkin–Teller model and are related to the c=1 conformal field theory by r={2L2(1−λ/π)}−1/2, where r is the orbifold radius. The models in the negative u regime are found to be related to those in the positive u regime by a certain transformation and also belong to the Ashkin–Teller universality class with the correspondence r={2L2λ/π}−1/2.
We consider solvable lattice models D L +2 (1) (λ) associated with the Dynkin diagrams of the affine Lie algebras D L +2 (1) and parametrized by the crossing parameter λ (0 < λ < 2π/3). The transfer matrix spectra for the strip widths ⩽ 10 are classified according to the irreducible representations of the symmetry groups under several boundary conditions. All L - and λ-dependences of scaling dimensions are embodied in the thermal scaling dimension X T = 2/{ L 2 (1 - λ/π)}. The operator content of the model is the same as that of the Ashkin-Teller model with the same thermal scaling dimension. This generalizes the results of Pasquier (λ = 0) and of the magnetic hard square model ( L = 3).
Based on numerical analysis, the authors conjecture the operator content of the finite-size limit of the spectra of the Ashkin-Teller model with free boundary conditions. The same operator content is obtained from a Hamiltonian with a four-fermion interaction and a U(1) Kac-Moody Sugawara structure. For some special values of the coupling constant the model exhibits N=2 superconformal and Zamolodchikov-Fateev invariance. The operator content in these cases is expressed in terms of irreducible representations of the corresponding algebras.