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International Journal of Modern Physics B

THE CORNER TRANSFER MATRIX OF SOME FREE-FERMION SYSTEMS AND MEIXNER’S POLYNOMIALS

Trong Tuong Truong, Ingo Peschel

Published Apr 1, 1990• 15 citations
Publisher

Abstract

Corner transfer matrices of some free-fermion vertex systems on a finite lattice, are exactly diagonalised in the Hamiltonian limit with the help of a special class of orthogonal polynomials: the Meixner polynomials. We present the derivation, discuss the asymptotic behavior for a large lattice and compare the results with numerical computation.

DOI
10.1142/s0217979290000413

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Corner transfer matrices of some free-fermion vertex systems on a finite lattice, are exactly diagonalised in the Hamiltonian limit with the help of a special class of orthogonal polynomials: the Meixner polynomials. We present the derivation, discuss the asymptotic behavior for a large lattice and compare the results with numerical computation.
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.
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.