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Proceedings of the Physical Society

A general analytic continuation response formalism for large quantum systems

W. A. B. EVANS

Published Jul 1, 1966• 10 citations
Publisher

Abstract

We develop a general nth-order analytic continuation formalism for evaluating the nth-order response of a many-body system which is subject to an arbitrary time-dependent mechanical perturbation K(t) and which is initially in a state of thermal equilibrium (i.e. its initial density matrix is e-βH/Z). The formalism is a generalization of the linear analytic continuation response formalism as presented by Luttinger and Nozieres and similarly entails the analytic continuation of certain suitably defined correlation functions which may, in principle, be evaluated exactly by the techniques of temperature perturbation theory. In this way the time development of the system is, in principle, exactly prescribed. Finally, as an example of the use of the generalized formalism, we evaluate the Hall effect in a metal with impurity scattering by means of the second-order formalism and the result justifies the usual expression.

DOI
10.1088/0370-1328/88/3/320

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We develop a general nth-order analytic continuation formalism for evaluating the nth-order response of a many-body system which is subject to an arbitrary time-dependent mechanical perturbation K(t) and which is initially in a state of thermal equilibrium (i.e. its initial density matrix is e-βH/Z). The formalism is a generalization of the linear analytic continuation response formalism as presented by Luttinger and Nozieres and similarly entails the analytic continuation of certain suitably defined correlation functions which may, in principle, be evaluated exactly by the techniques of temperature perturbation theory. In this way the time development of the system is, in principle, exactly prescribed. Finally, as an example of the use of the generalized formalism, we evaluate the Hall effect in a metal with impurity scattering by means of the second-order formalism and the result justifies the usual expression.
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.
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.
The quantum geometric tensor and quantum Fisher information have recently been shown to provide a unified geometric description of the linear response of many-body systems. However, a similar geometric description of higher-order perturbative phenomena including nonlinear response in generic quantum systems is lacking. In this work, we develop a general framework for the time-dependent quantum geometry of many-body systems by treating external perturbing fields as coordinates on the space of density matrices. We use the Bures distance between the initial and time-evolved density matrix to define geometric quantities through a perturbative expansion. To lowest order, we derive a time-dependent generalization of the Bures metric related to the spectral density of linear response functions, unifying previous results for the quantum metric in various limits and providing a geometric interpretation of Fermi’s golden rule. At next order in the expansion, we define a time-dependent Bures-Levi-Civita connection for general many-body systems. We show that the connection is the sum of one contribution that is related to a second-order nonlinear response function, and a second contribution that captures the higher geometric structure of first-order perturbation theory. We show that in the quasistatic, zero-temperature limit for noninteracting fermions, this Bures connection reduces to the known expression for band-theoretic Christoffel symbols. Our work provides a systematic framework to explore many-body quantum geometry beyond the quantum metric and highlights how higher-order correlation functions can probe this geometry.
We present a Green's-function formalism generalized to treat arbitrary time-dependent systems initially prepared with an arbitrary density matrix. On the basis of a simultaneous expansion of the usual real-time development operator and the density matrix, we define a Green's function ordered along an extended contour. In this way we are able to treat real-time many-particle Green's functions and initial correlations on the same footing. Both can be decomposed by Wick's theorem. Thus we may utilize familiar diagrammatic analysis for evaluating the Green's function, and finally arrive at a generalized Dyson's equation. The Green's function can be represented by a (3\ifmmode\times\else\texttimes\fi{}3) matrix, which includes the Green's functions used in the Feynman, Matsubara, and Keldysh theories, if the corresponding statistical average is taken. From a matrix representation of Dyson's equation, a basic set of five coupled equations is derived, which explicitly shows the corrections to the Keldysh theory due to an arbitrary initial many-particle density matrix.
The exact response theory based on the Dissipation Function applies to general dynamical systems and has yielded excellent results in various applications. In this article, we propose a method to apply it to quantum mechanics. In many quantum systems, it has not yet been possible to overcome the perturbative approach, and the most developed theory is the linear one. Extensions of the exact response theory developed in the field of nonequilibrium molecular dynamics could prove useful in quantum mechanics, as perturbations of small systems or far-from-equilibrium states cannot always be taken as small perturbations. Here, we introduce a quantum analogue of the classical Dissipation Function. We then derive a quantum expression for the exact calculation of time-dependent expectation values of observables, in a form analogous to that of the classical theory. We restrict our analysis to finite-dimensional Hilbert spaces, for the sake of simplicity, and we apply our method to specific examples, like qubit systems, for which exact results can be obtained by standard techniques. This way, we prove the consistency of our approach with the existing methods, where they apply. Although not required for open systems, we propose a self-adjoint version of our Dissipation Operator, obtaining a second equivalent expression of response, where the contribution of an anti-self-adjoint operator appears. We conclude by using new formalism to solve the Lindblad equations, obtaining exact results for a specific case of qubit decoherence, and suggesting possible future developments of this work.
This is the second in a series of papers which intends to develop a statistical mechanical theory capable of independently treating responses to thermal as well as mechanical disturbances. The essential idea is to develop a method of solving the Liouvill-von Neumann equation for the density matrix of a many-body system in a time-dependent Gibbsian form. The theory is made determinate by defining the relative entropy, which will reflect the character of the entropy production in our model. In this paper, rigorous expressions are obtained for nonlinear responses, which contain all the most general consequences of statistical mechanics regarding the macroscopic variables. In particular, the Onsager reciprocity relations follow from the formula in nonlinear regions. An attempt to treat a stationary state where heat is produced is made in terms of a nonequilibrium density matrix satisfying the Liouville-von Neumann equation.