Group theory for embedded random matrix ensembles

V K B Kota

doi:10.1088/1742-6596/597/1/012050

Abstract

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 m fermions (or bosons) in N single particle states and interacting with say k-body interactions, we have EGUE(k) [embedded GUE of k-body interactions) with GUE embedding and the embedding algebra is U(N). In this paper, using EGUE(k) representation for a Hamiltonian that is fc-body and an independent EGUE(t) representation for a transition operator that is t-body and employing the embedding U(N) algebra, finite-N 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.

Highlights

Wigner introduced random matrix theory (RMT) in physics in 1955 primarily to understand statistical properties of neutron resonances in heavy nuclei [1, 2]
It is more appropriate to represent an isolated finite interacting quantum system, say with m particles in N single particle states by random matrix models generated by random k-body interactions and propagate the information in the interaction to many particle spaces
We have random matrix ensembles in m-particle spaces - these ensembles are defined by representing the k-particle Hamiltonian (H) by GOE/GUE/GSE and the m particle H matrix is generated by the m-particle Hilbert space geometry

Summary

Introduction

Wigner introduced random matrix theory (RMT) in physics in 1955 primarily to understand statistical properties of neutron resonances in heavy nuclei [1, 2]. This and the various properties of the U (N ) Wigner and Racah coefficients give two formulas for the ensemble average of a product any two m particle matrix elements of H, fmv1 | H(k) | fmv2 fmv3 | H(k) | fmv4

Results

Conclusion