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

On spin systems related to the Temperley–Lieb algebra

P. P. Kulish

Published Sep 9, 2003• 46 citations
Publisher

Abstract

The spectrum of the transfer matrices constructed from the spectral parameter dependent Temperley–Lieb R -matrix is found using functional relations identical to those of the spin 1/2 XXZ -magnet.
DOI
10.1088/0305-4470/36/38/101

Related papers

Spectrum of the Transfer Matrices of the Spin Chains Associated with the $$A^{(2)}_3$$ Lie Algebra

Guang-Liang Li, Junpeng Cao, Kun Hao +4 more

arXiv:2201.00963•Communications in Mathematical Physics•Dec 21, 2022•1 citation

THE TOPOLOGICAL BASIS REALIZATION OF FOUR-QUBIT XXZ SPIN CHAIN

Guijiao Du, Kang Xue, Chunfang Sun +2 more

In this paper, a set of topological basis realization is presented. By acting on the different subspaces, we obtain the new nontrivial six-dimensional (6D) and four-dimensional Temperley–Lieb matrix representations. Then it is shown that XXZ model can be constructed from the Temperley–Lieb algebra (TLA) generators. The eigenstates of XXZ model can be denoted by combinating the set of topological bases. It is interesting that the ground state is closely related to parameter Δ.

International Journal of Quantum Information•Mar 1, 2012•1 citation

A Bethe ansatz solution for the closed Temperley - Lieb quantum spin chains

A. Lima-Santos, R. C. T. Ghiotto

We solve the spectrum of the closed Temperley-Lieb quantum spin chains using the coordinate Bethe ansatz. These models are invariant under the quantum group .

arXiv:solv-int/9809002•Journal of Physics A Mathematical and General•Jan 16, 1998•11 citations

Quantum spin chains of Temperley–Lieb type: periodic boundary conditions, spectral multiplicities and finite temperature

Britta Aufgebauer, Andreas Klümper

We determine the spectra of a class of quantum spin chains of Temperley–Lieb type by utilizing the concept of Temperley–Lieb equivalence with the S = 1/2 XXZ chain as a reference system. We consider open boundary conditions and, in particular, periodic boundary conditions. For both types of boundaries the identification with XXZ spectra is performed within isomorphic representations of the underlying Temperley–Lieb algebra. For open boundaries the spectra of these models differ from the spectrum of the associated XXZ chain only in the multiplicities of the eigenvalues. The periodic case is rather different. Here we show how the spectrum is obtained sector-wise from the spectra of globally twisted XXZ chains. As a spin-off, we obtain a compact formula for the degeneracy of the momentum operator eigenvalues. Our representative theoretical results allow for the study of the thermodynamics by establishing a TL-equivalence at finite temperature and finite field .

arXiv:1003.1932•Journal of Statistical Mechanics Theory and Experiment•May 20, 2010•33 citations

Transfer Matrix Spectrum and Bound States for Lattice Classical Ferromagnetic Spin Systems at High Temperature

Ricardo S. Schor, Michael O’Carroll

Journal of Statistical Physics•Jun 1, 2000•29 citations

A homomorphism between link and XXZ modules over the periodic Temperley–Lieb algebra

Alexi Morin-Duchesne, Yvan Saint-Aubin

We study finite loop models on a lattice wrapped around a cylinder. A section of the cylinder has N sites. We use a family of link modules over the periodic Temperley-Lieb algebra EPTL_N(β, α) introduced by Martin and Saleur, and Graham and Lehrer. These are labeled by the numbers of sites N and of defects d, and extend the standard modules of the original Temperley-Lieb algebra. Beside the defining parameters β=u^2+u^{-2} with u=e^{iλ/2} (weight of contractible loops) and α(weight of non-contractible loops), this family also depends on a twist parameter v that keeps track of how the defects wind around the cylinder. The transfer matrix T_N(λ, ν) depends on the anisotropy νand the spectral parameter λthat fixes the model. (The thermodynamic limit of T_N is believed to describe a conformal field theory of central charge c=1-6λ^2/(π(λ-π)).) The family of periodic XXZ Hamiltonians is extended to depend on this new parameter v and the relationship between this family and the loop models is established. The Gram determinant for the natural bilinear form on these link modules is shown to factorize in terms of an intertwiner i_N^d between these link representations and the eigenspaces of S^z of the XXZ models. This map is shown to be an isomorphism for generic values of u and v and the critical curves in the plane of these parameters for which i_N^d fails to be an isomorphism are given.

arXiv:1203.4996•Journal of Physics A Mathematical and Theoretical•Jun 27, 2013•17 citations
Spectrum of the Transfer Matrices of the Spin Chains Associated with the A3(2)A^{(2)}_3A3(2)​ Lie Algebra
We determine the spectra of a class of quantum spin chains of Temperley–Lieb type by utilizing the concept of Temperley–Lieb equivalence with the S = 1/2 XXZ chain as a reference system. We consider open boundary conditions and, in particular, periodic boundary conditions. For both types of boundaries the identification with XXZ spectra is performed within isomorphic representations of the underlying Temperley–Lieb algebra. For open boundaries the spectra of these models differ from the spectrum of the associated XXZ chain only in the multiplicities of the eigenvalues. The periodic case is rather different. Here we show how the spectrum is obtained sector-wise from the spectra of globally twisted XXZ chains. As a spin-off, we obtain a compact formula for the degeneracy of the momentum operator eigenvalues. Our representative theoretical results allow for the study of the thermodynamics by establishing a TL-equivalence at finite temperature and finite field .
We study finite loop models on a lattice wrapped around a cylinder. A section of the cylinder has N sites. We use a family of link modules over the periodic Temperley-Lieb algebra EPTL_N(β, α) introduced by Martin and Saleur, and Graham and Lehrer. These are labeled by the numbers of sites N and of defects d, and extend the standard modules of the original Temperley-Lieb algebra. Beside the defining parameters β=u^2+u^{-2} with u=e^{iλ/2} (weight of contractible loops) and α(weight of non-contractible loops), this family also depends on a twist parameter v that keeps track of how the defects wind around the cylinder. The transfer matrix T_N(λ, ν) depends on the anisotropy νand the spectral parameter λthat fixes the model. (The thermodynamic limit of T_N is believed to describe a conformal field theory of central charge c=1-6λ^2/(π(λ-π)).) The family of periodic XXZ Hamiltonians is extended to depend on this new parameter v and the relationship between this family and the loop models is established. The Gram determinant for the natural bilinear form on these link modules is shown to factorize in terms of an intertwiner i_N^d between these link representations and the eigenspaces of S^z of the XXZ models. This map is shown to be an isomorphism for generic values of u and v and the critical curves in the plane of these parameters for which i_N^d fails to be an isomorphism are given.