On the reality of spectra of $\boldsymbol{U_q(sl_2)}$-invariant XXZ Hamiltonians

TL;DR

This paper constructs a new inner product on modules over the Temperley-Lieb algebra, proving the self-adjointness and real spectra of the $U_q(sl_2)$-invariant XXZ Hamiltonian for real parameters, extending known spectral properties.

Contribution

It introduces a new inner product making the XXZ Hamiltonian self-adjoint, ensuring real spectra for all relevant Temperley-Lieb representations, including the $U_q(sl_2)$ case.

Findings

Hamiltonian is self-adjoint with respect to the new inner product

Spectra of the Hamiltonian are real and diagonalisable

Results apply to all Temperley-Lieb representations with real parameters

Abstract

A new inner product is constructed on each standard module over the Temperley-Lieb algebra $\mathsf{TL}_n(\beta)$ for $\beta\in \mathbb R$ and $n \ge 2$. On these modules, the Hamiltonian $h = -\sum_i e_i$ is shown to be self-adjoint with respect to this inner product. This implies that its action on these modules is diagonalisable with real eigenvalues. A representation theoretic argument shows that the reality of spectra of the Hamiltonian extends to all other Temperley-Lieb representations. In particular, this result applies to the celebrated $U_q(sl_2)$-invariant XXZ Hamiltonian, for all $q+q^{-1}\in \mathbb R$.