Quantum number towers for the Hubbard and Holstein models

TL;DR

This paper reviews Lieb's 1989 and 1995 theorems on quantum number towers in Hubbard and Holstein models, highlighting their significance, implications, and open questions in condensed matter physics.

Contribution

It summarizes and discusses the original proofs of quantum number properties in Hubbard and Holstein models, and explores their impact and remaining open questions.

Findings

Proved ground state properties for Hubbard models using spin-reflection positivity.

Extended results to Holstein and electron-phonon models.

Influenced subsequent research in quantum many-body physics.

Abstract

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 ||\Lambda_A|-|\Lambda_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.