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

Kubo-Martin-Schwinger relation for an interacting mobile impurity