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Cambridge University Press eBooks

Analytic Continuation from the Imaginary to the Real-Time Axis

Giancarlo Calvanese Strinati

Published Feb 6, 2026• 0 citations
Publisher

Abstract

This chapter reconsiders the original derivation of the Kadanoff and Baym equations, which relies on a procedure of analytic continuation from imaginary to real time in terms of an “extended” Matsubara approach. The procedure of analytic continuation proves useful for formal developments, like those to be considered in Chapters 28 and 29. The case when the system Hamiltonian does not depend on time is first treated, and the procedure is then extended under appropriate assumptions to the case when the Hamiltonian depends on time.

DOI
10.1017/9781009411509.029

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Cambridge University Press eBooks•Feb 6, 2026•0 citations
This chapter reconsiders the original derivation of the Kadanoff and Baym equations, which relies on a procedure of analytic continuation from imaginary to real time in terms of an “extended” Matsubara approach. The procedure of analytic continuation proves useful for formal developments, like those to be considered in Chapters 28 and 29. The case when the system Hamiltonian does not depend on time is first treated, and the procedure is then extended under appropriate assumptions to the case when the Hamiltonian depends on time.
This chapter considers the closed-time-path Green’s functions approach when specified to equilibrium situations and shows that it offers an alternative to the more standard Matsubara plus analytic continuation procedure for obtaining physical quantities directly in real frequency. In this case, the number of independent components of the single-particle Green’s function (as well as of the related self-energy) reduces considerably, thereby making it easier to solve the Kadanoff–Baym equations. The fluctuation–dissipation theorem and the single-particle spectral function (with its related sum rule) are also considered.
In Chapter 4, we introduced the Kadanoff–Baym equations of motion for the imaginary-time nonequilibrium Green's functions for a Bose gas, as given by (4.59) and (4.60). In this chapter, we will use the generalization of these equations of motion to find the equivalent equations of motion for the real-time Green's functions. These can be written in a natural way in the form of a kinetic equation. Using a simple Hartree–Fock approximation, we show how the coupled equations for the condensate and thermal cloud given in Chapter 3 emerge naturally from the Kadanoff–Baym (KB) formalism. This chapter is based on Imamović-Tomasović and Griffin (2001) and Imamović-Tomasović (2001), building on the pioneering work of Kane and Kadanoff (1965). In this chapter and Chapter 7 we review the KB formalism. However, we also encourage the reader to read the original account given by Kadanoff and Baym (1962). The goals and accomplishments of their seminal book are beautifully captured by the following quote from p. 138: Our rather elaborate Green's function arguments provide a means of describing transport phenomena in a self-contained way, starting from a dynamical approximation, i.e. an approximation for G 2 ( U ) in terms of G 1 ( U ). These calculations require no extra assumptions. The theory provides at the same time a description of the transport processes that occur and a determination of the quantities which appear in the transport equations. A closely related way of treating the nonequilibrium dynamics of a Bose-condensed gas is based on the two-particle irreducible (2PI) effective action together with the Schwinger–Keldysh closed-time path formalism. Berges (2004) gives a detailed review of this approach, which allows one to derive the nonequilibrium action on the basis of controllable approximations.
It is well established that causal linear response functions can be found by computing the much simpler imaginary time-ordered Matsubara functions and performing an analytic continuation. This principle is the basis for much of our understanding of linear response for interacting and disordered systems, via diagrammatic perturbation theory. Similar imaginary-time approaches have recently been introduced for computing nonlinear response functions as well, for example in [Annalen der Physik 536, 2300504 (2024); Physical Review X 11, 041006 (2021)], where the authors analytically continue the Matsubara functions to obtain the Keldysh response functions. In this work, we provide a proof of this connection to all orders in perturbation theory using an equation of motion based approach. We show by induction that causal nonlinear response functions at every order can be obtained from an analytic continuation of an appropriate time-ordered Matsubara function. We demonstrate this connection explicitly for second order response functions in the Lehmann representation. As a byproduct of our approach, we derive an explicit expression for the Lehmann representation of nnn-th order response functions by solving the equations of motion. We also use our result to find an analytic spectral density representation for both causal response functions and Matsubara functions. As an example, we apply our method to derive the non-linear A3A^3A3 term in the SU(2)SU(2)SU(2) spin Hall response of an insulator with spin rotation symmetry. Finally, we show how our results lead to a family of generalized sum rules, focusing explicitly on the asymptotic expression for nnn-th harmonic generation rate. Our work opens the door to using imaginary time approaches to study nonlinear response functions in general condensed matter systems.
Similar to Chapter 20 of Part I, this chapter considers the treatment of a few topics, which are relevant to the general purposes of the book but whose inclusion in previous chapters would have diverted the discussion of the main topics of interest therein. Specifically, it considers the Dyson equations for the contour single-particle Green’s function in the Nambu representation, the relative strength of different frequency terms in the derivation of the time-dependent Gross–Pitaevskii equation, the detailed calculation of an integral occurring in the derivation of the time-dependent Ginzburg–Landau equation, the irrelevance of the reference time t ₀ for the convolutions entering the Kadanoff–Baym equations at equilibrium, and the average energy of the system expressed in terms of the lesser Green’s function even in nonequilibrium situations.
This chapter gives a brief survey about the “time-stepping procedure” and the “predictor-corrector scheme” for solving the Kadanoff–Baym equations with two ( t and t ′) time variables. In this respect, the solutions of the Kadanoff–Baym equations for the greater and lesser Green’s functions are combined with each other in the positive quadrant of the t − t ′ plane, together with the solution of the lesser Green’s function along the time diagonal where t = t ′. The “generalized Kadanoff–Baym ansatz”, which aims at somewhat simplifying the solution of the Kadanoff–Baym equations themselves, is also introduced and derived in detail.