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arXiv · 1502.01859

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

Alexi Morin-Duchesne·Jorgen Rasmussen·Philippe Ruelle·Yvan Saint-Aubin
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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 $q+q^{-1}\in \mathbb R$.

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Alexi Morin-Duchesne, Jorgen Rasmussen, Philippe Ruelle, Yvan Saint-Aubin. 2015-02-06. On the reality of spectra of $\boldsymbol{U_q(sl_2)}$-invariant XXZ Hamiltonians. https://doi.org/10.1088/1742-5468/2016/05/053105

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