Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

Corner transfer matrix of generalized free-fermion vertex systems

Hans-Peter Eckle, Trong Tuong Truong

Published Dec 7, 1993• 5 citations
Publisher

Abstract

The Hamiltonian limit of the corner transfer matrix (CTM) of a generalized free-fermion vertex system of finite size leads to a quantum spin Hamiltonian of the particular form H N =- Sigma n=1 N$ - 1 (n( sigma n x sigma n+1 x + lambda sigma n y sigma n+1 y +h( sigma n z + sigma /sub /n +1 z ))). Diagonalization may be achieved for all pairs of parameters ( lambda , 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
10.1088/0305-4470/26/23/027

Related papers

Corner transfer matrix of generalised free Fermion vertex systems

Hans-Peter Eckle, Trong Tuong Truong

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.

arXiv:cond-mat/9308009•Aug 6, 1993•0 citations

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

Trong Tuong Truong, Ingo Peschel

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.

International Journal of Modern Physics B•Apr 1, 1990•15 citations

Corner transfer matrix eigenstates for the six-vertex model

Holger Frahm, H. B. Thacker

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.

Journal of Physics A Mathematical and General•Dec 7, 1991•13 citations

Spectrum of an Elliptic Free Fermionic Corner Transfer Matrix Hamiltonian

Rodolfo Cuerno

The eigenvalues of the Corner Transfer Matrix Hamiltonian associated to the elliptic $R$ 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.

arXiv:hep-th/9308040•Aug 9, 1993•0 citations

Corner transfer matrix of a critical free fermion system

Hans-Peter Eckle, Trong Tuong Truong

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.

Journal of Physics A Mathematical and General•May 7, 1992•3 citations

Corner transfer matrices and quantum affine algebras

Brian Davies

We consider corner transfer matrices of the six-vertex model and of its spin-l/2 generalizations, truncated to a finite lattice, in the anti-ferromagnetic regime. We show that, in the sense of finite size approximation, the low-lying eigenvectors approximate weight vectors of a level-l highest weight U q (sl 2 ) module, whilst the truncated corner transfer matrices approximate the role of a derivation operator. Our method is to first use the theory of the crystal base to prove the result exactly at q=0, and then extend to q not=0. We also consider the relationship between truncated corner transfer matrices, the vertex operator construction of Frenkel and Reshetikhin (1992), and Baxter's original argument for the properties of CTMs (1976).

Journal of Physics A Mathematical and General•Jan 21, 1994•1 citation
The Hamiltonian limit of the corner transfer matrix (CTM) of a generalized free-fermion vertex system of finite size leads to a quantum spin Hamiltonian of the particular form H N =- Sigma n=1 N$ - 1 (n( sigma n x sigma n+1 x + lambda sigma n y sigma n+1 y +h( sigma n z + sigma /sub /n +1 z ))). Diagonalization may be achieved for all pairs of parameters ( lambda , 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 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.
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.
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.
We consider corner transfer matrices of the six-vertex model and of its spin-l/2 generalizations, truncated to a finite lattice, in the anti-ferromagnetic regime. We show that, in the sense of finite size approximation, the low-lying eigenvectors approximate weight vectors of a level-l highest weight U q (sl 2 ) module, whilst the truncated corner transfer matrices approximate the role of a derivation operator. Our method is to first use the theory of the crystal base to prove the result exactly at q=0, and then extend to q not=0. We also consider the relationship between truncated corner transfer matrices, the vertex operator construction of Frenkel and Reshetikhin (1992), and Baxter's original argument for the properties of CTMs (1976).