O. Gamayun

M. Panfil

F. Taha Sant'Ana

Explicit computation of injection and ejection impurity’s Green’s function reveals a generalisation of the Kubo-Martin-Schwinger relation. 

Shifting time by a factor of temperature lets us equate certain Green’s functions in quantum field theory. This is the famous Kubo-Martin-Schwinger relation (KMS). We consider finite temperature correlation functions for an interacting mobile impurity in the Yang-Gaudin model, and study injection/ejection Green’s functions. These are, in fact, related by a refinement of KMS, which is valid for every value of the so-called spin rapidity.

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.

Mobile impurity

Kubo-Martin-Schwinger relation for an interacting mobile impurity