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.
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.
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.
The ab initio path integral Monte Carlo (PIMC) approach is one of the most successful methods in quantum many-body theory. A particular strength of this method is its straightforward access to imaginary-time correlation functions (ITCFs). For example, the well-known density–density ITCF F(q, τ) allows one to estimate the linear response of a given system for all wave vectors q from a single simulation of the unperturbed system. Moreover, it constitutes the basis for the reconstruction of the dynamic structure factor S(q, ω)—a key quantity in state-of-the-art scattering experiments. In this work, we present analogous relations between the nonlinear density response in the quadratic and cubic order of the perturbation strength and generalized ITCFs measuring correlations between up to four imaginary-time arguments. As a practical demonstration of our new approach, we carry out simulations of the warm dense electron gas and find excellent agreement with previous PIMC results that had been obtained with substantially larger computational effort. In addition, we give a relation between a cubic ITCF and the triple dynamic structure factor S(q1, ω1; q2, ω2), which evokes the enticing possibility to study dynamic three-body effects on an ab initio level.
The fermion sign problem constitutes one of the most fundamental obstacles in quantum many-body theory. Recently, it has been suggested to circumvent the sign problem by carrying out path integral simulations with a fictitious quantum statistics variable ξ, which allows for a smooth interpolation between the bosonic and fermionic limits [Xiong and Xiong, J. Chem. Phys.157, 094112 (2022)]. This ξ-extrapolation method has subsequently been applied to a variety of systems and has facilitated the analysis of an x-ray scattering measurement taken at the National Ignition Facility with unprecedented accuracy [Dornheim et al., Nat. Commun. 16, 5103 (2025)]. Yet, it comes at the cost of performing an additional 10-20 simulations, which, in combination with the required small error bars, can pose a serious practical limitation. Here, we remove this bottleneck by presenting a new reweighting estimator, which allows the study of the full ξ-dependence from a single path integral Monte Carlo (PIMC) simulation. This is demonstrated for various observables of the uniform electron gas and also warm dense beryllium. We expect our study to be useful for future PIMC simulations of Fermi systems, including ultracold atoms, electrons in quantum dots, and warm dense quantum plasmas.
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.
In general, Quantum Monte Carlo methods suffer from a sign problem when simulating fermionic systems. This causes the efficiency of a simulation to decrease exponentially with the number of particles and inverse temperature. To circumvent this issue, a nodal constraint is often implemented, restricting the Monte Carlo procedure from sampling paths that cause the many-body density matrix to change sign. Unfortunately, this high-dimensional nodal surface is not a priori known unless the system is exactly solvable, resulting in uncontrolled errors. We will discuss two possible routes to extend the applicability of finite-temperatue path integral Monte Carlo. First we extend the regime where signful simulations are possible through a novel permutation sampling scheme. Afterwards, we discuss a method to variationally improve the nodal surface by minimizing a free energy during simulation. Applications of these methods will include both free and interacting electron gases, concluding with discussion concerning extension to inhomogeneous systems.
We present a method to obtain dynamical correlation functions of quantum many-body systems in the real-frequency domain from Monte Carlo data of the corresponding imaginary-time Green's functions. The method is based on a least-squares-fit procedure to solve the integral equation relating imaginary-time and real-frequency correlation functions. To demonstrate its feasibility, we have applied our method to imaginary-time Monte Carlo data for the density correlation function of a simple model of interacting spinless fermions in one dimension. We compare our results to the analytical results, available in the limits of zero and strong interaction. We find that despite the presence of noise in the input data, the real-frequency spectra obtained with our method are qualitatively correct. They reflect accurately the particle-hole and the soliton-antisoliton excitations, present in the noninteracting and in the strongly coupled fermion system, respectively. We also compare our method to other approaches that have been proposed for obtaining dynamical correlation functions via Monte Carlo simulation.