Thermal out-of-time-order correlators, KMS relations, and spectral functions

TL;DR

The paper reframes thermal n-point correlation functions in terms of out-of-time-order (OTO) observables by exploiting the KMS condition, showing that all Wightman functions can be organized into cyclic KMS orbits. It introduces a causal basis built from fully nested commutators, derives generalized fluctuation-dissipation relations that express any nested correlator in terms of this basis, and constructs a master generating framework Ψ_σ that encodes KMS relations in frequency space. A systematic OTO classification via δ- and q-necklaces is developed, enabling an explicit decomposition of the (n−1)! independent thermal correlators and clarifying how KMS relations interrelate correlators with different OTO numbers. The harmonic oscillator example and Euclidean-continuation analyses illustrate the concrete realization of these ideas and connect to chaos diagnostics such as the tremolo/chaos correlator. Overall, the work provides a unifying, causality-oriented view of thermal spectral functions and paves the way for efficient spectral representations and applications to thermalization and quantum chaos.

Abstract

We describe general features of thermal correlation functions in quantum systems, with specific focus on the fluctuation-dissipation type relations implied by the KMS condition. These end up relating correlation functions with different time ordering and thus should naturally be viewed in the larger context of out-of-time-ordered (OTO) observables. In particular, eschewing the standard formulation of KMS relations where thermal periodicity is combined with time-reversal to stay within the purview of Schwinger-Keldysh functional integrals, we show that there is a natural way to phrase them directly in terms of OTO correlators. We use these observations to construct a natural causal basis for thermal n-point functions in terms of fully nested commutators. We provide several general results which can be inferred from cyclic orbits of permutations, and exemplify the abstract results using a quantum oscillator as an explicit example.

Figures (3)

• Figure 1: A pictorial collection of the various classes of correlators discussed in this section. The green colored boxes denote classes that are useful for generic states. The KMS condition in thermal states introduces a further reduction to the objects collected in blue colored boxes. Generalized fluctuation-dissipation theorems (FDTs) describe these redundancies. For completeness, note also Appendix \ref{['app:thspec']}, which discusses the relation with standard thermal retarded-advanced Green's functions.
• Figure 2: The k-OTO contour computing the out-of-time-ordered correlation functions encoded in the generating functional.
• Figure 3: The KMS orbit of a six-point Wightman function. The starting permutation is a proper 1-OTO which we have redundantly represented as a $2$-OTO for ease of visualization of the sliding. The subsequent steps represent sliding of successive operators counter-clockwise through the density matrix to complete the KMS orbit. The last picture of the sequence is the same correlator as the first picture (but all time arguments have been shifted by $-i\beta$). The $q$-list for this example is $\{1,2,2,2,2,1\}$ (which gives the proper-OTO numbers if the figure is read from the end to the beginning in reversed order). The $\delta$-list is $\{0,0,0,0,0,1\}$.