On the Spectrum of Some Temperley-Lieb Spin Chains
Abstract
We review mathematical results about the spectrum of spin-s chains with a nearest neighbor interaction given by the projection onto the singlet state of two spin-s degrees of freedom. Up to normalization, these interaction terms generate a Temperley-Lieb algebra at loop parameter d=2s+1. We also present several new results. First, we prove a ferromagnetic ordering of energy levels across symmetry sectors of the model, thus generalizing a previous result of Nachtergaele, Spitzer and Starr (2004) for the spin-1/2 XXZ chain. This enables us to determine a uniform lower bound on the finite volume spectral gap for these spin chains. By combining a variational estimate with Knabe's finite size criterion, we show that the exact spectral gap of the GNS Hamiltonian of the fully polarized state is 1−2/d. Furthermore, this value serves as a lower bound for the gap of the maximally mixed ground state. Finally, we show that the ground state projectors on open chains satisfy the Jones-Wenzl recursion, thereby recovering a known result that the degeneracies are the quantum integers [L+1]q for q+q−1=d. We discuss how the high degeneracy of the ground states allows the model to break a continuous symmetry while remaining gapped, and renders that gap unstable.
Related
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BibTeX
@misc{ferydouni2026spectrum,
title = {{On the Spectrum of Some Temperley-Lieb Spin Chains}},
author = {Robert Ferydouni and Bruno Nachtergaele},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38074},
eprint = {2609.38074},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38074},
}
RIS
TY - UNPB
TI - On the Spectrum of Some Temperley-Lieb Spin Chains
AU - Robert Ferydouni
AU - Bruno Nachtergaele
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38074
UR - https://arxiv.org/abs/2609.38074
AB - We review mathematical results about the spectrum of spin-s chains with a nearest neighbor interaction given by the projection onto the singlet state of two spin-s degrees of freedom. Up to normalization, these interaction terms generate a Temperley-Lieb algebra at loop parameter d=2s+1. We also present several new results. First, we prove a ferromagnetic ordering of energy levels across symmetry sectors of the model, thus generalizing a previous result of Nachtergaele, Spitzer and Starr (2004) for the spin-1/2 XXZ chain. This enables us to determine a uniform lower bound on the finite volume spectral gap for these spin chains. By combining a variational estimate with Knabe's finite size criterion, we show that the exact spectral gap of the GNS Hamiltonian of the fully polarized state is 1−2/d. Furthermore, this value serves as a lower bound for the gap of the maximally mixed ground state. Finally, we show that the ground state projectors on open chains satisfy the Jones-Wenzl recursion, thereby recovering a known result that the degeneracies are the quantum integers [L+1]_q for q+q^−1=d. We discuss how the high degeneracy of the ground states allows the model to break a continuous symmetry while remaining gapped, and renders that gap unstable.
ER -