Hubbard和Holstein模型的量子数塔
Quantum number towers for the Hubbard and Holstein models
摘要：
1989年，艾略特·利布发表了《物理评论快报》，证明了关于哈伯德模型的两个定理。本文利用自旋反射正性的概念证明了吸引哈伯德模型的基态总是非简并自旋单态，并证明了二分格点上排斥模型的基态具有自旋||Λ_A|-|A_B||/2，对应于两个子晶格的晶格位置的数量差。此外，这项工作涉及量子数塔-其中具有给定量子数的最小能态，如自旋或赝自旋，根据自旋或赝自旋值进行排序。随后在1995年发表了第二篇论文，将其中一些结果扩展到Holstein模型（以及更一般的电子-声子模型）。这些工作证明了凝聚态物理中关于这些多体模型的量子数的结果，具有重要的影响。在这一章中，我将讨论这些证明的背景，它们的意义，以及与原始工作相关的剩余开放问题。此外，我还将简要讨论这种方法所激发的一些额外工作。
英文摘要：
In 1989, Elliott Lieb published a Physical Review Letter proving two theorems about the Hubbard model. This paper used the concept of spin-reflection positivity to prove that the ground state of the attractive Hubbard model was always a nondegenerate spin singlet and to also prove that the ground state for the repulsive model on a bipartite lattice had spin ||Λ_A|-|Λ_B||/2, corresponding to the difference in number of lattice sites for the two sublattices. In addition, this work relates to quantum number towers -- where the minimal energy state with a given quantum number, such as spin, or pseudospin, is ordered, according to the spin or pseudospin values. It was followed up in 1995 by a second paper that extended some of these results to the Holstein model (and more general electron-phonon models). These works prove results about the quantum numbers of these many-body models in condensed matter physics and have been very influential. In this chapter, I will discuss the context for these proofs, what they mean, and the remaining open questions related to the original work. In addition, I will briefly discuss some of the additional work that this methodology inspired.
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