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-$\tfrac12$ $XXZ$ 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 $q+q<sup>{-1}=d$. 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.