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arXiv:1810.03118 · Journal of High Energy Physics

Spectral representation of thermal OTO correlators

Soumyadeep Chaudhuri, Chandramouli Chowdhury, R. Loganayagam

Published Feb 1, 2019• 27 citations
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Abstract

Abstract We study the spectral representation of finite temperature, out of time ordered (OTO) correlators on the multi-time-fold generalised Schwinger-Keldysh contour. We write the contour-ordered correlators as a sum over time-order permutations acting on a fundamental array of Wightman correlators. We decompose this Wightman array in a basis of column vectors, which provide a natural generalisation of the familiar retarded-advanced basis in the finite temperature Schwinger-Keldysh formalism. The coefficients of this decomposition take the form of generalised spectral functions, which are Fourier transforms of nested and double commutators. Our construction extends a variety of classical results on spectral functions in the SK formalism at finite temperature to the OTO case.

DOI
10.1007/jhep02(2019)018

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Abstract We study the spectral representation of finite temperature, out of time ordered (OTO) correlators on the multi-time-fold generalised Schwinger-Keldysh contour. We write the contour-ordered correlators as a sum over time-order permutations acting on a fundamental array of Wightman correlators. We decompose this Wightman array in a basis of column vectors, which provide a natural generalisation of the familiar retarded-advanced basis in the finite temperature Schwinger-Keldysh formalism. The coefficients of this decomposition take the form of generalised spectral functions, which are Fourier transforms of nested and double commutators. Our construction extends a variety of classical results on spectral functions in the SK formalism at finite temperature to the OTO case.
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.
The space of n-point correlation functions, for all possible time-orderings of operators, can be computed by a non-trivial path integral contour, which depends on how many time-ordering violations are present in the correlator. These contours, which have come to be known as timefolds, or out-of-time-order (OTO) contours, are a natural generalization of the Schwinger-Keldysh contour (which computes singly out-of-time-ordered correlation functions). We provide a detailed discussion of such higher OTO functional integrals, explaining their general structure, and the myriad ways in which a particular correlation function may be encoded in such contours. Our discussion may be seen as a natural generalization of the Schwinger-Keldysh formalism to higher OTO correlation functions. We provide explicit illustration for low point correlators (n ≤ 4) to exemplify the general statements.
The space of n-point correlation functions, for all possible time-orderings of operators, can be computed by a non-trivial path integral contour, which depends on how many time-ordering violations are present in the correlator. These contours, which have come to be known as timefolds, or out-of-time-order (OTO) contours, are a natural generalization of the Schwinger-Keldysh contour (which computes singly out-of-time-ordered correlation functions). We provide a detailed discussion of such higher OTO functional integrals, explaining their general structure, and the myriad ways in which a particular correlation function may be encoded in such contours. Our discussion may be seen as a natural generalization of the Schwinger-Keldysh formalism to higher OTO correlation functions. We provide explicit illustration for low point correlators ( n\leq 4 n ≤ 4 ) to exemplify the general statements.
Abstract Multiparticle thermodynamic Green’s functions, defined in terms of grand canonical ensemble averages of time-ordered products of creation and annihilation operators, are interpreted as tracing the amplitude for time-developing correlated interacting particle motions taking place in the background of a thermal ensemble. Under equilibrium conditions, time-translational invariance permits the one-particle thermal Green’s function to be represented in terms of a single frequency, leading to a Lehmann spectral representation whose frequency poles describe the energy spectrum. This Green’s function has finite values for both t>t′ and t<t′ (unlike retarded Green’s functions), and the two parts G1> and G1< (respectively) obey a simple proportionality relation that facilitates the introduction of a spectral weight function: It is also interpreted in terms of a periodicity/antiperiodicity property of a modified Green’s function in imaginary time capable of a Fourier series representation with imaginary (Matsubara) frequencies. The analytic continuation from imaginary time to real time is discussed, as are related commutator/anticommutator functions, also retarded/advanced Green’s functions, and the spectral weight sum rule is derived. Statistical thermodynamic information is shown to be embedded in physical features of the one- and two-particle thermodynamic Green’s functions.
We study the dynamical equation of the time-ordered Green's function at finite temperature. We show that the time-ordered Green's function obeys a conventional Dyson equation only at equilibrium and in the limit of zero temperature. In all other cases, i.e. finite temperature at equilibrium or non-equilibrium, the time-ordered Green's function obeys instead a modified Dyson equation. The derivation of this result is obtained from the general formalism of the non-equilibrium Green's functions on the Keldysh time-loop contour. At equilibrium, our result is fully consistent with the Matsubara temperature Green's function formalism and also justifies rigorously the correction terms introduced in an ad hoc way with Hedin and Lundqvist. Our results show that one should use the appropriate dynamical equation for the time-ordered Green's function when working beyond the equilibrium zero-temperature limit.