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arXiv:2410.01845 · Journal of Statistical Physics

Green’s Function Perspective on the Nonlinear Density Response of Quantum Many-Body Systems

Jan Vorberger, Tobias Dornheim, Maximilian Böhme, Zhandos A. Moldabekov, Panagiotis B. Tolias

Published May 24, 2025• 6 citations
arXivHTMLPDF

Abstract

We derive equations of motion for higher order density response functions using the theory of thermodynamic Green's functions. We also derive expressions for the higher order generalized dielectric functions and polarization functions. Moreover, we relate higher order response functions and higher order collision integrals within the Martin-Schwinger hierarchy. We expect our results to be highly relevant to the study of a variety of quantum many-body systems such as matter under extreme temperatures, densities, and pressures.

DOI
10.1007/s10955-025-03454-5

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We derive equations of motion for higher order density response functions using the theory of thermodynamic Green's functions. We also derive expressions for the higher order generalized dielectric functions and polarization functions. Moreover, we relate higher order response functions and higher order collision integrals within the Martin-Schwinger hierarchy. We expect our results to be highly relevant to the study of a variety of quantum many-body systems such as matter under extreme temperatures, densities, and pressures.
The present paper is concerned with the retarded many-time thermodynamic Green's function method in classical statistical mechanics. In the formulation of the nonlinear response of a dynamical variable, unaveraged Green's functions are specified and used to discuss the physical meaning of many-time Poisson-bracket nests. Equilibrium ensemble averages of these Green's functions are relevant to the classical Kubo response theory, the latter of which is interpreted from the viewpoints of both the classical ``Heisenberg'' and ``Schrödinger'' pictures. A method for evaluating the many-time thermodynamic Green's functions based on their definition in terms of the resolvent operator is suggested and discussed in relation to the linear dielectric function and the anharmonic oscillator. The many-time Green's function equations of motion are shown to be equivalent to identities based on the resolvent operator. It is shown how microscopic equations of motion can be used to derive the Bogoliubov-Sadovnikov (B-S) hierarchy from the lowest-order many-time Green's function equation of motion.
The quantum dynamics of correlated fermionic or bosonic many-body systems following external excitation can be successfully studied using nonequilibrium Green functions (NEGFs) or reduced density matrix methods. Approximations are introduced via a proper choice of the many-particle self-energy or decoupling of the BBGKY-hierarchy, respectively. These approximations are based on Feynman's diagram approaches or on cluster expansions into single-particle and correlation operators. In a recent paper [E. Schroedter, J.-P. Joost, and M. Bonitz, Condens. Matter Phys. 25, 23401 (2022)] we presented a different approach in which, instead of equations of motion for the many-particle NEGF (or density operators), equations for the correlation functions of fluctuations are analyzed. In particular, we derived the stochastic GWGWGW and polarization approximations that are closely related to the nonequilibrium GWGWGW approximation. Here, we extend this approach to the computation of two-time observables depending on the specific ordering of the underlying operators. In particular, we apply this extension to the calculation of the density correlation function and dynamic structure factor of correlated Hubbard clusters in and out of equilibrium.
Perturbation expansions of the response function S(q,\ensuremath{\omega}) of a many-body system in orders of any chosen component of the Hamiltonian are obtained. The response S(q,\ensuremath{\omega}) describes the inclusive scattering of a projectile interacting weakly with a target, where q and \ensuremath{\omega} are the momentum and energy transfer. Each term of a given order is written as an expectation value of products of paired Green's functions representing each initial- and final-state effects. For several choices of the perturbing operator, close expressions are obtained for the first orders of these expansions. The resulting terms represent partial summations over the Gersch series, whose relation to Green's-function expansions is established. Also, they are finite for singular potentials and are useful in the asymptotic regime of large q. In this regime they can give the dominant contribution to the response and initial- and final-state interactions of the hit particle and other particles in the system may be fully accounted for. Contributions of the n-particle density matrix can be separated for some expansions. In particular, an expression for the complete two-body contribution involving the hit particle is given.
Abstract The method of the equation of motion is used to derive the Martin–Schwinger hierarchy for the nonequilibrium Green’s functions. The formal closure of the hierarchy is reached by using the selfenergy which provides a recipe for how to construct selfenergies from approximations of the two-particle Green’s function. The Langreth–Wilkins rules for a diagrammatic technique are shown to be equivalent to the weakening of initial correlations. The quantum transport equations are derived in the general form of Kadanoff and Baym equations. The information contained in the Green’s function is discussed. In equilibrium this leads to the Matsubara diagrammatic technique.
We consider a finite quantum system under slow driving and weakly coupled to thermal reservoirs at different temperatures. We present a systematic derivation of the quantum master equation for the density matrix and the out-of-time-order correlators. We start from the microscopic Hamiltonian and we formulate the equations ruling the dynamics of these quantities by recourse to the Schwinger-Keldysh nonequilibrium Green's function formalism, performing a perturbative expansion in the coupling between the system and the reservoirs. We focus on the adiabatic dynamics, which corresponds to considering the linear response in the ratio between the relaxation time due to the system-reservoir coupling and the time scale associated to the driving. We calculate the particle and energy fluxes. We illustrate the formalism in the case of a qutrit coupled to bosonic reservoirs and of a pair of interacting quantum dots attached to fermionic reservoirs, also discussing the relevance of coherent effects.
We extend the formalism of the thermodynamic two-time Green's functions to nonextensive quantum statistical mechanics. Working in the optimal Lagrangian multiplier representation, the q -spectral properties and the methods for a direct calculation of the two-time q Green's functions and the related q -spectral density ( q measures the nonextensivity degree) for two generic operators are presented in strict analogy with the extensive (q=1) counterpart. Some emphasis is devoted to the nonextensive version of the less known spectral density method whose effectiveness in exploring equilibrium and transport properties of a wide variety of systems has been well established in conventional classical and quantum many-body physics. To check how both the equations of motion and the spectral density methods work to study the q -induced nonextensivity effects in nontrivial many-body problems, we focus on the equilibrium properties of a second-quantized model for a high-density Bose gas with strong attraction between particles for which exact results exist in extensive conditions. Remarkably, the contributions to several thermodynamic quantities of the q -induced nonextensivity close to the extensive regime are explicitly calculated in the low-temperature regime by overcoming the calculation of the q grand-partition function.