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Journal of Physics A Mathematical and General

Corner transfer matrix of a critical free fermion system

Hans-Peter Eckle, Trong Tuong Truong

Published May 7, 1992• 3 citations
Publisher

Abstract

In order to diagonalize the generator of the corner transfer matrix of a critical free-fermion 8-vertex system with magnetic field the authors derive an integral representation for the eigenvector components of the generator. At large N they numerically verify the predictions of conformal invariance for critical systems.
DOI
10.1088/0305-4470/25/9/006

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The eigenvalues of the Corner Transfer Matrix Hamiltonian associated to the elliptic RRR matrix of the eight vertex free fermion model are computed in the anisotropic case for magnetic field smaller than the critical value. An argument based on generating functions is given, and the results are checked numerically. The spectrum consists of equally spaced levels.
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 Hamiltonian limit of the corner transfer matrix (CTM) of a generalised free Fermion vertex system of finite size leads to a quantum spin Hamiltonian of the particular form: HN=−∑n=1N−1{n(σnxσn+1x+λσnyσn+1y+h(σnz+σn+1z))} {\cal H}_N=-\sum_{n=1}^{N-1}\left\{ n\left( σ_n^xσ_{n+1}^x +λσ_n^yσ_{n+1}^y +h(σ_n^z+σ_{n+1}^z) \right)\right\} HN​=−∑n=1N−1​{n(σnx​σn+1x​+λσny​σn+1y​+h(σnz​+σn+1z​))} Diagonalisation may be achieved for all pairs of parameters (λ,h)(λ,h)(λ,h) with the use of some new elliptic polynomials which extend the class of special polynomials known so far in the context of CTM.