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Average-fluctuation separation in energy levels in quantum many-particle systems with $k$-body interactions using $q$-Hermite polynomials

N. D. Chavda

Published Nov 23, 2021• 1 citation
OpenAlex

Abstract

Separation between average and fluctuation parts in the state density in many-particle quantum systems with $k$-body interactions, modeled by the $k$-body embedded Gaussian orthogonal random matrices (EGOE($k$)), is demonstrated using the method of normal mode decomposition of the spectra and also verified through power spectrum analysis, for both fermions and bosons. The smoothed state density is represented by the $q$-normal distribution ($f_{qN}$) (with corrections) which is the weight function for $q$-Hermite polynomials. As the rank of interaction $k$ increases, the fluctuations set in with smaller order of corrections in the smooth state density. They are found to be of GOE type, for all $k$ values, for both fermion and boson systems.

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Average-fluctuation separation in energy levels in quantum many-particle systems with kkk-body interactions using qqq-Hermite polynomials
Separation between average and fluctuation parts in the state density in many-particle quantum systems with kkk-body interactions, modeled by the kkk-body embedded Gaussian orthogonal random matrices (EGOE(kkk)), is demonstrated using the method of normal mode decomposition of the spectra and also verified through power spectrum analysis, for both fermions and bosons. The smoothed state density is represented by the qqq-normal distribution (fqNf_{qN}fqN​) (with corrections) which is the weight function for qqq-Hermite polynomials. As the rank of interaction kkk increases, the fluctuations set in with smaller order of corrections in the smooth state density. They are found to be of GOE type, for all kkk values, for both fermion and boson systems.
Abstract In an m particle quantum system, the rank of interactions and the nature of particles (fermions or bosons) can strongly affect the dynamics of the system. To explore this, we study the nonequilibrium dynamics with particles in a one-body mean-field and quenched by an interaction of body-rank k = 2, 3, …, m . Using fermionic embedded Gaussian orthogonal ensembles (EGOEs) and bosonic embedded Gaussian orthogonal ensembles (BEGOEs) of one plus k -body interactions (also the unitary variants EGUE and BEGUE), it is seen that the short time decay of the survival probability of many-particle systems is given by the Fourier transform of the generating function of the q -Hermite polynomials. Deriving the formulas for q for both fermion and boson systems as a function of m , k and the number of single particle states N , we have verified that the Fourier transform of agrees very well with the numerical ensemble calculations for both fermion and boson systems. These results bridge the gap between the known results for k = 2 and k = m .
Eigenstate structure in many-body bosonic systems: Analysis using random matrices and qqq-Hermite polynomials
We analyze the structure of eigenstates in many-body bosonic systems by modeling the Hamiltonian of these complex systems using Bosonic Embedded Gaussian Orthogonal Ensembles (BEGOE) defined by a mean-field plus kkk-body random interactions. The quantities employed are the number of principal components (NPC), the localization length (lHl_HlH​) and the entropy production S(t)S(t)S(t). The numerical results are compared with the analytical formulas obtained using random matrices which are based on bivariate qqq-Hermite polynomials for local density of states Fk(E∣q)F_k(E|q)Fk​(E∣q) and the bivariate qqq-Hermite polynomial form for bivariate eigenvalue density ρbiv:q(E,Ek)ρ_{biv:q}(E,E_k)ρbiv:q​(E,Ek​) that are valid in the strong interaction domain. We also compare transport efficiency in many-body bosonic systems using BEGOE in absence and presence of centrosymmetry. It is seen that the centrosymmetry enhances quantum efficiency.
Abstract We study the mechanism of thermalization in finite many-fermion systems with random k-body interactions in the presence of a mean-field. The system Hamiltonian H, for m fermions in N single particle states with k-body interactions, is modeled by mean field one-body h(1) and a random k-body interaction V(k) with strength λ. Following the recent application of q-Hermite polynomials to these ensembles, a complete analytical description of parameter q, which describes the change in the shape of state density from Gaussian for q = 1 to semi-circle for q = 0 and intermediate for 0 < q < 1 , and variance of the strength function are obtained in terms of model parameters. The latter gives the thermalization marker λ t defining the thermodynamic region. For λ ⩾ λ t , the smooth part of the strength functions is very well represented by conditional q-normal distribution (f CN ), which describes the transition in strength functions from Gaussian to semi-circle as the k-body interaction changes from k = 2 to m in H. In the thermodynamic region, ensemble averaged results for the first four moments of the strength functions and inverse participation ratio are found to be in good agreement with the corresponding smooth forms. For higher body rank of interaction k, the system thermalizes faster.
According to the quantum chaos paradigm, the nature of a system's classical dynamics, whether integrable or chaotic, is universally reflected in the fluctuations of its quantum spectrum. However, since many-body spectra in the mean-field limit are composed of independent single-particle energy levels, their spectral fluctuations always display Poissonian behavior and hence cannot be used to distinguish underlying chaotic from integrable single-particle dynamics. We demonstrate that this distinction can, instead, be revealed from the mean many-body level density (at fixed energy) and its variance after averaging over ensembles representing different types of single-particle dynamics. This is in strong contrast to the energy-averaged mean level density (of a given system) that is assumed not to carry such information and is routinely removed to focus on universal signatures. To support our claim we systematically analyze the role of single-particle level correlations, that enter through Poisson and random matrix statistics (of various symmetry classes) into the ensemble-averaged density of states and its variance, contrasting bosonic and fermionic many-body systems. Our analytical study, together with extensive numerical simulations for systems with N≥5 particles consistently reveal significant differences (up to an order of magnitude for fermions and even larger for bosons) in the mean many-body level densities, depending on the nature of the underlying dynamics. Notably, in the fermionic case Poisson-type single-particle level fluctuations precisely cancel contributions from indistinguishability, such that the average many-body spectral density equals the (Thomas-Fermi) volume term. We further highlight the difference between the mean level density and its variance as functions of the total energy E and the excitation energy Q.
Embedded random matrix ensembles are generic models for describing statistical properties of finite isolated quantum many-particle systems. For the simplest spinless fermion (or boson) systems with say mmm fermions (or bosons) in NNN single particle states and interacting with say kkk-body interactions, we have EGUE(kkk) [embedded GUE of kkk-body interactions) with GUE embedding and the embedding algebra is U(N)U(N)U(N). In this paper, using EGUE(kkk) representation for a Hamiltonian that is kkk-body and an independent EGUE(ttt) representation for a transition operator that is ttt-body and employing the embedding U(N)U(N)U(N) algebra, finite-NNN formulas for moments up to order four are derived, for the first time, for the transition strength densities (transition strengths multiplied by the density of states at the initial and final energies). In the asymptotic limit, these formulas reduce to those derived for the EGOE version and establish that in general bivariate transition strength densities take bivariate Gaussian form for isolated finite quantum systems. Extension of these results for other types of transition operators and EGUE ensembles with further symmetries are discussed.
Embedded ensembles or random matrix ensembles generated by k-body interactions acting in many-particle spaces are now well established to be paradigmatic models for many-body chaos and thermalization in isolated finite quantum (fermion or boson) systems. In this article, briefly discussed are (i) various embedded ensembles with Lie algebraic symmetries for fermion and boson systems and their extensions (for Majorana fermions, with point group symmetries etc.); (ii) results generated by these ensembles for various aspects of chaos, thermalization and statistical relaxation, including the role of q-hermite polynomials in k-body ensembles; and (iii) analyses of numerical and experimental data for level fluctuations for trapped boson systems and results for statistical relaxation and decoherence in these systems with close relations to results from embedded ensembles.