Magic in the spectra of the XXZ quantum chain with boundaries at Delta=0 and Delta=-1/2

TL;DR

This paper uncovers spectral identities linking the spectra of XXZ quantum chains with different boundary conditions at specific anisotropy parameters, using representation theory and character identities.

Contribution

It reveals new spectral identities between chains with various boundary conditions at special anisotropy values, grounded in Temperley-Lieb algebra representation theory.

Findings

Spectral relations between chains with different boundary conditions at Delta=-1/2 and Delta=0.

Identification of algebraic origins of spectral identities via Temperley-Lieb algebras.

Discovery of additional spectral identities inspired by initial findings.

Abstract

We show that from the spectra of the U_q (sl(2)) symmetric XXZ spin-1/2 finite quantum chain at Delta=-1/2 (q=e^{pi i/3}) one can obtain the spectra of certain XXZ quantum chains with diagonal and non-diagonal boundary conditions. Similar observations are made for Delta=0 (q=e^{pi i/2}). In the finite-size scaling limit the relations among the various spectra are the result of identities satisfied by known character functions. For the finite chains the origin of the remarkable spectral identities can be found in the representation theory of one and two boundaries Temperley-Lieb algebras at exceptional points. Inspired by these observations we have discovered other spectral identities between chains with different boundary conditions.