A finite temperature version of the Nagaoka--Thouless theorem in the $\mathrm{SU}(n)$ Hubbard model

TL;DR

This paper extends the Aizenman--Lieb theorem to the $ ext{SU}(n)$ Hubbard model at finite temperatures, demonstrating that magnetization exceeds the paramagnetic value using a novel random-loop representation.

Contribution

It introduces a finite temperature extension of the Aizenman--Lieb theorem for the $ ext{SU}(n)$ Hubbard model, utilizing a new random-loop representation of the partition function.

Findings

Magnetization exceeds the paramagnetic value at finite temperature.

Extension of the Aizenman--Lieb theorem to $ ext{SU}(n)$ Hubbard models.

Development of a random-loop representation for the partition function.

Abstract

The Aizenman--Lieb theorem for the $\mathrm{SU}(2)$ Hubbard model expands upon the Nagaoka--Thouless theorem for the ground state to encompass finite temperatures. It can be succinctly stated that the magnetization $m(\beta, b)$ of the system in the presence of a field $b$ surpasses the pure paramagnetic value $m_0(\beta, b)=\tanh(\beta b)$. In this manuscript, we present an extension of the Aizenman--Lieb theorem to the $\mathrm{SU}(n)$ Hubbard model. Our proof relies on a random-loop representation of the partition function, which becomes accessible when expressing the partition function in terms of path integrals.