Kubo-Martin-Schwinger relation for an interacting mobile impurity

TL;DR

This paper investigates the KMS relation in the integrable Yang-Gaudin model with an interacting mobile impurity, deriving a refined relation for Green's functions at finite temperature through analytical and numerical methods.

Contribution

It introduces a new formulation of the KMS relation for Green's functions in an integrable impurity model, accounting for the Hilbert space's separability and subspace contributions.

Findings

Green's functions obey a refined KMS relation derived as a Riemann-Hilbert problem.

Analytical demonstration of the relation's validity at finite temperature.

Numerical verification supporting the theoretical results.

Abstract

In this work we study the Kubo-Martin-Schwinger (KMS) relation in the Yang-Gaudin model of an interacting mobile impurity. We use the integrability of the model to compute the dynamic injection and ejection Green's functions at finite temperatures. We show that due to separability of the Hilbert space with an impurity, the ejection Green's in a canonical ensemble cannot be reduced to a single expectation value as per microcanonical picture. Instead, it involves a thermal average over contributions from different subspaces of the Hilbert space which, due to the integrability, are resolved using the so-called spin rapidity. It is then natural to consider the injection and ejection Green's functions within each subspace. By means of reformulating the original KMS condition as a Riemann-Hilbert problem, we analytically demonstrate that such Green's functions obey a refined analogous relation, which is finally corroborated by numerical evaluation.