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Molecular Physics

Spectral density of the Wigner path integral operator correlation function representation. Monte Carlo simulation of the fermion dynamic structure factor

Vladimir S. Filinov, Pavel R. Levashov, Alexander S. Larkin

Published Dec 13, 2024• 2 citations
Publisher

Abstract

An important mathematical function that describes the interparticle correlations and their time evolution is the dynamic structure factor (DSF). According to Van Hove the dynamic structure factor (DSF) is the space and time Fourier transform of the density-density correlation function of the unperturbed system. In the case of classical systems, computations of the DSF typically employ classical molecular dynamics techniques. However, the present study focuses on quantum systems, in particular on the quantum dynamic structure factor. The main idea of this article is to employ the Wigner formulation of quantum mechanics to derive a new path integral description of the quantum DSF. The next important idea is to consider paths in this representation as stationary random processes and apply the Wiener–Khinchin theorem to express the DSF through the expected values of the power spectrum of trajectories in this representation. To calculate the DSF, spin-resolved radial distribution functions (RDFs) and other thermodynamic parameters of a 3D system of soft sphere fermions we developed a Wigner Path Integral Monte Carlo (WPIMC) method. The observed peaks on the RDFs and DSFs were attributed to wave interference resulting from a multiple-scattering and perturbation effects due to particle exchange and interaction. This phenomenon, observed in a soft sphere scattering system, may serve as a precursor to Anderson localisation, a mechanism in which wave interference across multiple scattering channels prevents wave propagation.

DOI
10.1080/00268976.2024.2440477

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An important mathematical function that describes the interparticle correlations and their time evolution is the dynamic structure factor (DSF). According to Van Hove the dynamic structure factor (DSF) is the space and time Fourier transform of the density-density correlation function of the unperturbed system. In the case of classical systems, computations of the DSF typically employ classical molecular dynamics techniques. However, the present study focuses on quantum systems, in particular on the quantum dynamic structure factor. The main idea of this article is to employ the Wigner formulation of quantum mechanics to derive a new path integral description of the quantum DSF. The next important idea is to consider paths in this representation as stationary random processes and apply the Wiener–Khinchin theorem to express the DSF through the expected values of the power spectrum of trajectories in this representation. To calculate the DSF, spin-resolved radial distribution functions (RDFs) and other thermodynamic parameters of a 3D system of soft sphere fermions we developed a Wigner Path Integral Monte Carlo (WPIMC) method. The observed peaks on the RDFs and DSFs were attributed to wave interference resulting from a multiple-scattering and perturbation effects due to particle exchange and interaction. This phenomenon, observed in a soft sphere scattering system, may serve as a precursor to Anderson localisation, a mechanism in which wave interference across multiple scattering channels prevents wave propagation.
The dynamic structure factor (DSF) is a mathematical function that contains information about inter-particle correlations and their time evolution. Mostly the classical molecular dynamics is used to calculate the DSF of the classical systems. On the contrary this article deals with quantum systems and the quantum dynamic structure factor. The Wigner formulation of quantum mechanics was used to derive the path integral representation of the DSF, which is based on the Wiener-Khinchin theorem showing relation of the the power spectrum of a random paths to their correlation function. The $3 {\rm D} $ quantum system of strongly correlated soft-sphere fermions was considered as an interesting physical example. The developed Wigner path integral Monte Carlo (WPIMC) approach has been developed to calculate the spin--resolved DSFs, the radial distribution functions (RDFs) and other thermodynamic functions in a wide range of density and temperatures. The physical meaning of the peaks arising on the RDFs and DSF have been analyzed and explained by the manifestation of the interference effects of the exchange and interparticle interactions and, as well as, the wave interference between multiple-scattering. This phenomenon in the system of the soft-sphere scatterers may to be the precursor effect of the Anderson localization, which finds its origin in the wave interference between multiple-scattering paths.
The Wigner formulation of quantum mechanics is used to derive a path-integral representation of the quantum density of states (DOS) of strongly correlated fermions in the canonical ensemble. A path-integral Monte Carlo approach for the simulation of DOS and other thermodynamic functions is suggested. The derived Wigner function in the phase space resembles the Maxwell-Boltzmann distribution but allows for quantum effects. We consider a three-dimensional quantum system of strongly correlated soft-sphere fermions at different densities and temperatures. The calculated properties include the DOS, momentum distribution functions, spin-resolved radial distribution functions, potentials of mean force, and related energy levels obtained from the Bohr-Sommerfeld condition. We observe sharp peaks on DOS and momentum distribution curves, which are explained by the appearance of fermionic bound states.
Abstract The Wigner formulation of quantum mechanics is used to derive a new path integral representation of quantum density of states (DOSs). A path integral Monte Carlo approach is developed for the numerical investigation of DOSs, internal energy and spin-resolved radial distribution functions for a 2D system of strongly correlated soft-sphere fermions. The peculiarities of the DOSs and internal energy distributions depending on the hardness of the soft-sphere potential and particle density are investigated and explained. In particular, at high enough densities the DOSs rapidly tends to a constant value, as for an ideal system of 2D fermions.
Path-integral Monte Carlo simulations in the Wigner approach to quantum mechanics has been applied to calculate momentum and spin-resolved radial distribution functions of the strongly correlated soft-sphere quantum fermions. The obtained spin-resolved radial distribution functions demonstrate arising triplet clusters of fermions, that is the consequence of the interference of exchange and interparticle interactions. The semiclassical analysis in the framework of the Bohr-Sommerfeld quantization condition, applied to the potential of the mean force corresponding to the same-spin radial distribution functions, allows to detect exchange-correlation bound states in triplet clusters and to estimate corresponding averaged energy levels. The obtained momentum distribution functions demonstrate the narrow sharp separated peaks corresponding to bound states and disturbing the Maxwellian distribution.
A new numerical approach to time correlation functions in the Wigner representation of quantum statistical mechanics has been developed. The time correlation functions have been presented in the form of the integral of the Weil symbol of operators and the Fourier transform of the product of matrix elements of the dynamic propagators. For the last function the integral Wigner-Liouville type equation has been derived. The initial condition for this equation has been obtained in the form of the Fourier transform of the Wiener path integral representation of the matrix elements of the propagators at initial time. The numerical procedure for solving this equation combining both molecular dynamics and Monte Carlo methods has been developed.
For deeper understanding of quantum non-local correlations a new path integral real-time quantum dynamics approach has been developed and applied to the Monte Carlo simulation of the decoherence of pair entangled states in many-body systems in the framework of the ``replica trick''. Calculations have been carried out for the Heisenberg-Wigner path integral representation of the entangled state correlation function (ESCF) in the modified unified electron gas model as example. The Wiener-Khinchin transformation of ESCF has been applied to random paths in path integral allowing for the Monte Carlo simulation of ESCF. Presented results include the spin-resolved radial distribution functions, ESCFs and estimations of the decoherence time of the entangled electron pairs due to thermalization and interparticle interaction.