Random k-Body Ensembles for Chaos and Thermalization in Isolated Systems

Abstract

Abstract: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.

Spin Polarization Oscillations and Coherence Time in the Random Interaction Approach

“…where ⟨⋅ ⋅ ⋅ ⟩ stands for the ensemble average on the Gaussian variables and ⟨⋅ ⋅ ⋅ ⟩ for the thermal average. Concerning the Gaussian ensemble, it is worth mentioning that a wide research on random matrix ensembles underwent a rapid development leading to various modified versions, in particular the embedded ensembles (EE) generated by random -body interactions, = 2 being the most important; see [26,27]. The statistical assumptions that we need in our calculations below are completely characterized by (2) and (12), which, according to [21], are compatible with the postulates of matrix elements given by the two-body random ensemble.…”

“…In the physics of complex systems, it has been frequently found that some processes are insensitive to the details of the interaction, being only a few "gross properties" relevant to describe them. This feature, which is not new to many body problems, has often been used to construct successful and enlightening approaches in terms of ensembles of stochastic interactions [17,[20][21][22][23][24][25][26][27] that make possible satisfying evaluations of ensemble averages for relevant quantities. We will present here a Gaussian stochastic spin-environment interaction approach that strengthens this idea.…”

Structure of wavefunction for interacting bosons in mean-field with random $k$-body interactions

“…A very significant property of EE(k) (and EE(1+k)) is that in general they exhibit Gaussian to semi-circle transition in the eigenvalue density as k increases from 1 to m [21]. This result is now well established from many numerical calculations and analytical proofs obtained via lower order moments [5,9,24,43,45,46]. Very recently, it is shown that generating function for q-Hermite polynomials describes this transition in spectral densities and in strength functions using k-body EGOE and their Unitary variants EGUE, both for fermion and boson systems, as a function of rank of interaction k [36].…”

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

“…In last couple of decades, there is tremorous growth on the study of the statistical properties of isolated finite many-particle quantum systems such as atomic nuclei, atoms, quantum dots and small metallic grains, ultracold atoms, interacting spin models, and quantum black holes with the Sachdev-Ye-Kitaev (SYK) model and so on [7,[14][15][16][17][18][19][20][21][22]. The embedded ensembles of k-body interaction, EGOE(k), operating in many particle spaces [1,7], provide the generic models for finite interacting many-particle systems as inter-particle interactions are known to be predominantly few-body in character.…”