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physica status solidi (b)

On the Higher‐Order Response of a Superconductor

Wolfgang Weller

Published Jan 1, 1969• 2 citations
Publisher

Abstract

Abstract The response of a superconductor to external fields is considered in arbitrary order. The response is calculated for an imaginary time interval and then analytically continued to real times. In this way it is possible to make use of the theorem proved by Kadanoff and Baym on the identity of the correlation functions connected with the physical and the unphysical problem. The result justifies the method of Gorkov and Eliashberg in which the analytic continuation has been performed in the frequency representation.

DOI
10.1002/pssb.19690350204

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Physical review. B, Condensed matter•Jul 1, 1995•2 citations
Abstract The response of a superconductor to external fields is considered in arbitrary order. The response is calculated for an imaginary time interval and then analytically continued to real times. In this way it is possible to make use of the theorem proved by Kadanoff and Baym on the identity of the correlation functions connected with the physical and the unphysical problem. The result justifies the method of Gorkov and Eliashberg in which the analytic continuation has been performed in the frequency representation.
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.
The general expressions for the number density and current are first considered under the action of an external time-dependent perturbing potential of arbitrary strength and cast in terms of the single-particle lesser Green’s function. The expansion of the number density up to linear order in the perturbing potential is then considered, yielding the density–density correlation function of linear-response theory. A connection is also considered with the temperature correlation function of the Matsubara formalism via an analytic continuation in frequency space.
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.
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.
Making use of the inversion method, which is invented to discuss the characteristics of phase transition in a perturbative manner, a fundamental equation for nonequilibrium superconductivity is derived. In the course of the calculation, the generalized contour-ordered Green's-function formalism is used to treat the initial time correlation function and real-time Green's function in a unified manner. The result is a set of self-consistent integral equations and the Gorkov's equation is given a posteriori as one of these components describing the initial time correlation. By virtue of the contour-ordered Green's-function formalism, it is not necessary to employ any assumption to consider the effect of the initial time correlation on the system at subsequent time. Owing to this nature, our result helps in discussing transient behavior of the system induced by a supercritical current that appears in some kinds of superconducting device.