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

TL;DR

This paper establishes an exact spectral equivalence between different boundary conditions in the XXZ chain, explores the algebraic structures involved, and analyzes the implications of indecomposable representations at exceptional points.

Contribution

It introduces a spectral equivalence between boundary conditions in the XXZ chain via one-boundary Temperley-Lieb algebra representations and studies the algebra's centralizer and indecomposable structures.

Findings

Spectral equivalence between boundary conditions in XXZ chain

Identification of indecomposable structures at exceptional points

Construction of a truncated space of 'good' states

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.