On the reality of spectra of ${\mathbf{U}}_{\mathbf{q}}(\mathbf{s}{\mathbf{l}}_2)$-invariant XXZ Hamiltonians

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 $U_q(s{l}_2)$-invariant XXZ Hamiltonian, for all $q+q^{-1}\in \mathbb{R}$.

30 pages

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• Yang-Baxter Solution of Dimers as a Free-Fermion Six-Vertex Model
• Rational Q-systems at Root of Unity I. Closed Chains
• Yang-Baxter Integrable Dimers on a Strip
• A solvable non-unitary fermionic long-range model with extended symmetry
• Integrability of planar-algebraic models
• Spectrum-preserving deformations of integrable spin chains with open boundaries