Stochastic Aspects of Many-Body Systems: The Embedded Gaussian Ensembles

Hans A. Weidenmüller

Manan Vyas

In finite many-body quantum systems such as nuclei, atoms, mesoscopic systems like quantum dots and small metallic grains, interacting spin systems modeling quantum computing core and BEC, the interparticle interactions are essentially two-body in nature. Therefore, it is more appropriate to represent the complex Hamiltonian of these systems by random ensembles that incorporate the two-body nature of interactions. These ensembles are generically called embedded Gaussian ensembles (EGEs). The aim of the present thesis is to identify and systematically analyze many different physically relevant EGEs with symmetries by considering a variety of quantities and measures that are important for isolated finite interacting quantum systems.

Daniel Neuhauser, Vojtěch Vlček, Eran Rabani +1 more

V. K. B. Kota, N. D. Chavda

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.

Sudip Kumar Haldar, N D Chavda, Manan Vyas +1 more

Abstract We analyze the effect of spin degree of freedom on fidelity decay and entropy production of a many-particle fermionic (bosonic) system in a mean-field, quenched by a random two-body interaction preserving many-particle spin S . The system Hamiltonian is represented by embedded Gaussian orthogonal ensemble (EGOE) of random matrices (for time-reversal and rotationally invariant systems) with one plus two-body interactions preserving S for fermions/bosons. EGOE are paradigmatic models to study the dynamical transition from integrability to chaos in interacting many-body quantum systems. A simple general picture, in which the variances of the eigenvalue density play a central role, is obtained for describing the short-time dynamics of fidelity decay and entropy production. Using some approximations, an EGOE formula for the time ( t sat ) for the onset of saturation of entropy, is also derived. These analytical EGOE results are in good agreement with numerical calculations. Moreover, both fermion and boson systems show significant spin dependence on the relaxation dynamics of the fidelity and entropy.

Sergio A Albeverio, Ph. Combe, M. Sirugue-Collin