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.
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.
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 .
Average-fluctuation separation in energy levels in quantum many-particle systems with k-body interactions using q-Hermite polynomials
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 (fqN) (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.
Finite interacting Fermi systems with a mean-field and a chaos generating two-body interaction are modeled by one plus two-body embedded Gaussian orthogonal ensemble of random matrices with spin degree of freedom [called EGOE(1+2)-s]. Numerical calculations are used to demonstrate that, as lambda , the strength of the interaction (measured in the units of the average spacing of the single-particle levels defining the mean-field), increases, generically there is Poisson to GOE transition in level fluctuations, Breit-Wigner to Gaussian transition in strength functions (also called local density of states) and also a duality region where information entropy will be the same in both the mean-field and interaction defined basis. Spin dependence of the transition points lambda_{c} , lambdaF, and lambdad , respectively, is described using the propagator for the spectral variances and the formula for the propagator is derived. We further establish that the duality region corresponds to a region of thermalization. For this purpose we compared the single-particle entropy defined by the occupancies of the single-particle orbitals with thermodynamic entropy and information entropy for various lambda values and they are very close to each other at lambda=lambdad.
Embedded random matrix ensembles generated by random interactions (of low body rank and usually two-body) in the presence of a one-body mean field, introduced in nuclear structure physics, are now established to be indispensable in describing statistical properties of a large number of isolated finite quantum many-particle systems. Lie algebra symmetries of the interactions, as identified from nuclear shell model and the interacting boson model, led to the introduction of a variety of embedded ensembles (EEs). These ensembles with a mean field and chaos generating two-body interaction generate in three different stages, delocalization of wave functions in the Fock space of the mean-field basis states. The last stage corresponds to what one may call thermalization and complex nuclei, as seen from many shell model calculations, lie in this region. Besides briefly describing them, their recent applications to nuclear structure are presented and they are (i) nuclear level densities with interactions; (ii) orbit occupancies; (iii) neutrinoless double beta decay nuclear transition matrix elements as transition strengths. In addition, their applications are also presented briefly that go beyond nuclear structure and they are (i) fidelity, decoherence, entanglement and thermalization in isolated finite quantum systems with interactions; (ii) quantum transport in disordered networks connected by many-body interactions with centrosymmetry; (iii) semicircle to Gaussian transition in eigenvalue densities with [Formula: see text]-body random interactions and its relation to the Sachdev–Ye–Kitaev (SYK) model for majorana fermions.
Using the ergodicity principle for the expectation values of several types of observables, we investigate the thermalization process in isolated fermionic systems.These are described by the two-body random ensemble, which is a paradigmatic model to study quantum chaos and especially the dynamical transition from integrability to chaos.By means of exact diagonalizations we analyze the relevance of the eigenstate thermalization hypothesis as well as the influence of other factors, such as the energy and structure of the initial state, or the dimension of the Hilbert space.We also obtain analytical expressions linking the degree of thermalization for a given observable with the so-called number of principal components for transition strengths originating at a given energy, with the dimensions of the whole Hilbert space and microcanonical energy shell, and with the correlations generated by the observable.As the strength of the residual interaction is increased, an order-to-chaos transition takes place, and we show that the onset of Wigner spectral fluctuations, which is the standard signature of chaos, is not sufficient to guarantee thermalization in finite systems.When all the signatures of chaos are fulfilled, including the quasicomplete delocalization of eigenfunctions, the eigenstate thermalization hypothesis is the mechanism responsible for the thermalization of certain types of observables, such as (linear combinations of) occupancies and strength function operators.Our results also suggest that fully chaotic systems will thermalize relative to most observables in the thermodynamic limit.
Using the ergodicity principle for the expectation values of several types of observables, we investigate the thermalization process in isolated fermionic systems. These are described by the two-body random ensemble, which is a paradigmatic model to study quantum chaos and especially the dynamical transition from integrability to chaos. By means of exact diagonalizations we analyze the relevance of the eigenstate thermalization hypothesis as well as the influence of other factors, such as the energy and structure of the initial state, or the dimension of the Hilbert space. We also obtain analytical expressions linking the degree of thermalization for a given observable with the so-called number of principal components for transition strengths originating at a given energy, with the dimensions of the whole Hilbert space and microcanonical energy shell, and with the correlations generated by the observable. As the strength of the residual interaction is increased, an order-to-chaos transition takes place, and we show that the onset of Wigner spectral fluctuations, which is the standard signature of chaos, is not sufficient to guarantee thermalization in finite systems. When all the signatures of chaos are fulfilled, including the quasicomplete delocalization of eigenfunctions, the eigenstate thermalization hypothesis is the mechanism responsible for the thermalization of certain types of observables, such as (linear combinations of) occupancies and strength function operators. Our results also suggest that fully chaotic systems will thermalize relative to most observables in the thermodynamic limit.