One-boundary Temperley-Lieb algebras in the XXZ and loop models

Abstract

We give an exact spectral equivalence between the quantum group invariant XXZ chain with arbitrary left boundary term and the same XXZ chain with purely diagonal boundary terms. This equivalence, and a further one with a link pattern Hamiltonian, can be understood as arising from different representations of the one-boundary Temperley-Lieb algebra. For a system of size L these representations are all of dimension 2^L and, for generic points of the algebra, equivalent. However at exceptional points they can possess different indecomposable structures. We study the centralizer of the one-boundary Temperley-Lieb algebra in the 'non-diagonal' spin-1/2 representation and find its eigenvalues and eigenvectors. In the exceptional cases the centralizer becomes indecomposable. We show how to get a truncated space of 'good' states. The indecomposable part of the centralizer leads to degeneracies in the three mentioned Hamiltonians.

32 pages Latex

References in corpus (1)

• Raise and Peel Models of fluctuating interfaces and combinatorics of Pascal's hexagon

Cited by in corpus (41)

• The half-infinite XXZ chain in Onsager's approach
• Non-diagonal open spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and matrix elements of some quasi-local operators
• Magic in the spectra of the XXZ quantum chain with boundaries at Delta=0 and Delta=-1/2
• The two-boundary Temperley-Lieb algebra
• A physical approach to the classification of indecomposable Virasoro representations from the blob algebra
• Boundary algebras and Kac modules for logarithmic minimal models
• The Temperley-Lieb algebra and its generalizations in the Potts and XXZ models
• Exact solution of the Izergin-Korepin model with general non-diagonal boundary terms
• The open XXZ and associated models at q root of unity
• Combinatorial aspects of boundary loop models
• Boundary operators in the O(n) and RSOS matrix models
• Temperley-Lieb, Birman-Murakami-Wenzl and Askey-Wilson algebras and other centralizers of $U_q({\mathfrak{sl}}_2)$
• Critical dense polymers with Robin boundary conditions, half-integer Kac labels and ${\mathbb{Z}}_4$ fermions
• Structure of the two-boundary XXZ model with non-diagonal boundary terms
• Boundary changing operators in the O(n) matrix model
• Temperley-Lieb, Brauer and Racah algebras and other centralizers of su(2)
• Quantum spin chains from Onsager algebras and reflection $K$-matrices
• $U_q{\mathfrak{sl}}_2$-invariant non-compact boundary conditions for the XXZ spin chain
• Generalized T-Q relations and the open spin-s XXZ chain with nondiagonal boundary terms
• Nested algebraic Bethe ansatz for open GL(N) spin chains with projected K-matrices
• Boundary energy of the general open XXZ chain at roots of unity
• A differential U-module algebra for U=U_q sl(2) at an even root of unity
• Two-point boundary correlation functions of dense loop models
• A representation basis for the quantum integrable spin chain associated with the su(3) algebra
• Pascal's triangle and word bases for blob algebra ideals
• Asymmetric Twin Representation: the Transfer Matrix Symmetry
• Supersymmetry on Jacobstahl lattices
• Original electric-vertex formulation of the symmetric eight-vertex model on the square lattice is fully non-universal
• On the logarithmic bipartite fidelity of the open XXZ spin chain at $\Delta =-1/2$
• Spectral equivalences and symmetry breaking in integrable SU_q(N) spin chains with boundaries
• Finite-size correction and bulk hole-excitations for special case of an open XXZ chain with nondiagonal boundary terms at roots of unity
• Boundary Criticality of Complex Conformal Field Theory: A Case Study in the Non-Hermitian 5-State Potts Model
• A Positive integral property on the ground state of the two-boundary Temperley--Lieb Hamiltonian
• Generic boundary scattering in the open XXZ chain
• The open XXZ chain at $\Delta =-1/2$ and the boundary quantum Knizhnik-Zamolodchikov equations
• Density profiles in the raise and peel model with and without a wall. Physics and combinatorics
• Towers of recollement and bases for diagram algebras: planar diagrams and a little beyond
• NLIE and finite size effects of the spin-1/2 XXZ and sine-Gordon models with two boundaries revisited
• Exotic Springer fibers for orbits corresponding to one-row bipartitions
• Avalanches in the Raise and Peel model in the presence of a wall
• Critical behavior in a non-local interface model