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arXiv:cond-mat/9308009

Corner transfer matrix of generalised free Fermion vertex systems

Hans-Peter Eckle, Trong Tuong Truong

Published Aug 6, 1993• 0 citations
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Abstract

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: \[ {\cal H}_N=-\sum_{n=1}^{N-1}\left\{ n\left( σ_n^xσ_{n+1}^x +λσ_n^yσ_{n+1}^y +h(σ_n^z+σ_{n+1}^z) \right)\right\} \] Diagonalisation may be achieved for all pairs of parameters $(λ,h)$ with the use of some new elliptic polynomials which extend the class of special polynomials known so far in the context of CTM.

DOI
10.48550/arxiv.cond-mat/9308009

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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: [ {\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} ] 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 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.
Eigenstates of the corner transfer matrix (CTM) for the six-vertex model are constructed and their relation to the Bethe ansatz eigenstates of the XXZ Hamiltonian is discussed. In the ferromagnetic regime ( Delta >1), eigenstates with any finite number of overturned arrows are constructed. They are related to the Bethe ansatz states by Fourier transformation over spectral (rapidity) variables. This exhibits the role of the CTM as a lattice boost operator. For the antiferromagnetic regime, the relation between CTM eigenstates and those of the XXZ Hamiltonian is complicated by the filling of the sea and associated Bethe ansatz integral equations. For this case, the authors construct CTM eigenstates by first showing that the Hamiltonian ground state is also an eigenstate of the CTM. Other CTM eigenstates are constructed from excited eigenstates of the Hamiltonian by Fourier transforming over the rapidity of each dressed excitation. They discuss the relationship between CTM eigenstates and a Heisenberg algebra of bosonic oscillators.