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arXiv:2303.09223 · Europhysics Letters (EPL)

Unravelling the nonlinear ideal density response of many-body systems

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

Published May 1, 2023• 19 citations
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Abstract

Abstract Nonlinear density response theory is revisited focusing on the harmonically perturbed finite temperature uniform electron gas. Within the non-interacting limit, brute force quantum kinetic theory calculations for the quadratic, cubic, quartic and quintic responses reveal a deep connection with the linear response. Careful analysis of the static long wavelength limit led us to conjecture a canonical non-interacting form that expresses arbitrary order nonlinear responses as the weighted sum of the linear responses evaluated at all multiple harmonics. This harmonic expansion is successfully validated against ab initio path integral Monte Carlo simulations.

DOI
10.1209/0295-5075/acd3a6

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Abstract Nonlinear density response theory is revisited focusing on the harmonically perturbed finite temperature uniform electron gas. Within the non-interacting limit, brute force quantum kinetic theory calculations for the quadratic, cubic, quartic and quintic responses reveal a deep connection with the linear response. Careful analysis of the static long wavelength limit led us to conjecture a canonical non-interacting form that expresses arbitrary order nonlinear responses as the weighted sum of the linear responses evaluated at all multiple harmonics. This harmonic expansion is successfully validated against ab initio path integral Monte Carlo simulations.
We present density response estimators for Monte Carlo simulations that are based on a reweighting procedure, where the samples of an unperturbed system are used to estimate the properties of a system perturbed by an external harmonic potential. This allows the linear and nonlinear static density response to be estimated purely from simulations of the unperturbed system. The method is demonstrated for the uniform electron gas under warm dense matter and strongly coupled conditions using ab initio path integral Monte Carlo simulations. The performance of the method with respect to the number of particles and the number of imaginary time slices is investigated. The scheme is generalised to consider multiple external perturbations, acting on different species and with different wavenumbers, giving one access to additional cross-species density response functions and the complete quadratic response function resolved for both wave number arguments through mode coupling. The flexibility of the methodology opens the possibility to investigate numerous new density response properties to further advance our understanding of interacting quantum many-body systems across a broad range of applications.
Ab initio quantum Monte Carlo methods, in principle, allow for the calculation of exact properties of correlated many-electron systems but are, in general, limited to the simulation of a finite number of electrons N under periodic boundary conditions. Therefore, an accurate theory of finite-size effects is indispensable to bridge the gap to realistic applications in the thermodynamic limit. In this work, we revisit the uniform electron gas at finite temperature, as it is relevant to contemporary research, e.g., in the field of warm dense matter. In particular, we present a new scheme to eliminate finite-size effects both in the static structure factor S(q) and in the interaction energy v, which is based on the density response formalism. We demonstrate that this method often allows us to obtain v in the thermodynamic limit within a relative accuracy of ∼0.2% from as few as N = 4 electrons without any empirical choices or knowledge of results for other values of N. Finally, we evaluate the applicability of our method upon increasing the density parameter rs and decreasing the temperature T.
For both, fundamental physics and novel applications, nonlinear effects are crucial.Powerful laser fields require to treat mobile electrons beyond linear response theory.Phenomena familiar from nonlinear optics are manifest in ultracold Rydberg systems, which have a high potential for advancing the understanding of many-body physics via quantum simulations and promote progress in quantum computation and -information.Various devices and plasmonics systems are based on thin, nearly two dimensional (2D) layers, such as quantum wells, graphene-like structures or transition metal dichalcogenides.We here derive an expression for nonlinear response functions in arbitrary order, and evaluate it explicitly in 2D for the quadratic and the cubic density response of non-interacting fermions.As correlations are of enhanced importance in low dimensions, we next incorporate screened interactions.This results in the generation of 2nd and 3rd harmonics of the collective modes (zero sound and plasmons), damped by multiple particle-hole continua.
While the theory of linear response functions in many body theory has developed into a substantial lore over the past decades, rather little is known about the properties and behavior of higher order (quadratic, etc.) response functions of many body systems. In this poster we first review essential elements of quadratic response theory such as the quadratic fluctuation dissipation theorems and sum rules for the quadratic density response function. Next we show how the correlational local field G(k,ω) in the linear response is fundamentally linked to the quadratic response function which opens a way to a non-perturbational analysis of the former. We proceed to present the results of the explicit calculation of the quadratic density response of 2D and 3D electron gas. Both, real and imaginary part of the latter reveal an extremely rich behavior. Of particular interest is the comparison between the two- and the three-dimensional case.
We present density response estimators for Monte Carlo simulations that are based on a reweighting procedure, where the samples of an unperturbed multi-component system are used to estimate the properties of a system perturbed by an external harmonic potential. This allows the species-resolved linear and nonlinear static density response to be estimated purely from simulations of the unperturbed system. The method is demonstrated for the uniform electron gas under warm dense matter and strongly coupled conditions using ab initio path integral Monte Carlo simulations. The performance of the method with respect to the number of particles and the number of imaginary time slices is investigated. The scheme is generalized to consider multiple external perturbations, acting on different species and with different wavenumbers, giving one access to additional cross-species density response functions and the complete quadratic response function resolved for both wavenumber arguments through mode coupling. The flexibility of the methodology opens the possibility to investigate numerous new density response properties to further advance our understanding of interacting quantum many-body systems across a broad range of applications.
A cumulant expansion of the influence functional entering the path integral representation of the reduced density matrix is presented. Truncation of the expansion at the quadratic terms allows mapping of a many-body environment onto an effective harmonic model with the same force autocorrelation function, recovering the conventional linear response limit. It is shown that the latter becomes exact in the thermodynamic limit if the bath consists of independent modes coupled to the system via coefficients that scale appropriately with the size of the bath. Low-order correction terms to the linear response approximation are found to be given by multitime correlation functions of the force exerted by the bath on the system. Within a classical approximation, such correction terms can be obtained from molecular dynamics calculations and incorporated in the path integral, which can be evaluated by means of iterative procedures. Numerical calculations on an intrinsically nonlinear bath of two-level systems confirm the above behaviors and indicate that the deviations from the linear response approximation are more significant at low temperatures where quantum mechanical effects play an important role to the bath dynamics.