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Journal of Physics A Mathematical and General

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

Ingo Peschel

Published Feb 7, 1988• 9 citations
Publisher

Abstract

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.
DOI
10.1088/0305-4470/21/3/012

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The finite-size scaling spectra of Ising model corner transfer matrices and their generators are studied at criticality. The generators are diagonalised using fermion algebra. The low-lying eigenvalues, given by the zeros of Meixner polynomials, are equally spaced and collapse like 1/log N for large N as predicted by conformal invariance. The asymptotics are obtained using a generalised Euler-Maclaurin summation formula. The shift in the largest eigenvalue is given analytically as pi c/6 log N with central charge c=1/2. The spectrum generating functions, for both fixed and free boundary conditions, are expressed simply in terms of the c=1/2 Virasoro characters chi Delta (q) with modular parameter q=exp(- pi /log N) and conformal dimensions Delta =0, 1/2, 1/16.
The scaling spectra of finite-size Ising model corner transfer matrices (CTMs) are studied at criticality, using the fermion algebra. The low-lying eigenvalues collapse like 1/ log N for large N as predicted by conformal invariance. The shift in the largest eigenvalue is evaluated analytically using a generalized Euler-Maclaurin summation formula giving πc/6 log N with central charge c=1/2. The spectrum generating functions, for both fixed and free boundary conditions, are expressed simply in terms of the c=1/2 Virasoro characters χ∆(q) with modular parameter q= exp (−π/ log N) and conformal dimensions ∆=0, 1/2, 1/16.
We investigate the hard-square lattice-gas model by means of transfer-matrix calculations and a finite-size-scaling analysis. Using a minimal set of assumptions we find that the spectrum of correction-to-scaling exponents is consistent with that of the exactly solved Ising model, and that the critical exponents and correlation-length amplitudes closely follow the relation predicted by conformal invariance. Assuming that these spectra are exactly identical, and conformal invariance, we determine the critical point, the conformal anomaly, and the temperature and magnetic exponents with numerical margins of 10(-11) or less. These results are in a perfect agreement with the exactly known Ising universal parameters in two dimensions. In order to obtain this degree of precision, we included system sizes as large as feasible, and used extended-precision floating-point arithmetic. The latter resource provided a substantial improvement of the analysis, despite the fact that it restricted the transfer-matrix calculations to finite sizes of at most 34 lattice units.