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The European Physical Journal B

Diagonalisation of finite-size corner transfer matrices and related spin chains

Trong Tuong Truong, Ingo Peschel

Published Mar 1, 1989• 31 citations
Publisher
DOI
10.1007/bf01313574

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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.