关于某些Temperley-Lieb自旋链的谱

中文摘要 AI 辅助

我们回顾了关于自旋-$s$链的谱的数学结果，其中最近邻相互作用由两个自旋-$s$自由度的单重态投影给出。在归一化之后，这些相互作用项在环参数$d=2s+1$下生成一个Temperley-Lieb代数。我们还提出了若干新结果。首先，我们证明了该模型各对称扇区中能级的铁磁序，从而推广了Nachtergaele、Spitzer和Starr（2004）关于自旋-$\ frac12$ $XXZ$链的先前结果。这使我们能够确定这些自旋链的有限体积谱隙的一个一致下界。通过将变分估计与Knabe的有限尺寸判据相结合，我们证明了全极化态的GNS哈密顿量的精确谱隙为$1-2/d$。此外，该值作为最大混合基态谱隙的下界。最后，我们证明了开链上的基态投影子满足Jones-Wenzl递推关系，从而恢复了已知结果：简并度为量子整数$[L+1]_q$，其中$q+q^{-1}=d$。我们讨论了基态的高度简并如何使模型在保持有隙的同时打破连续对称性，并使该隙变得不稳定。

英文摘要

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 $q+q^{-1}=d$. 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.

发表机构

• University of California Davis（加州大学戴维斯分校）

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