Random matrix ensemble with random two-body interactions in presence of a mean-field for spin one boson systems

TL;DR

This paper introduces a new random matrix ensemble for spin-1 boson systems with two-body interactions and a mean-field, analyzing spectral properties, density of states, and ground state structures.

Contribution

It develops a novel embedded Gaussian orthogonal ensemble for spin-1 bosons, including construction methods, spectral analysis, and pairing symmetry considerations.

Findings

Density of states is approximately Gaussian.

Level fluctuations follow GOE statistics.

Derived propagation formulas for energy centroids and variances.

Abstract

For $m$ number of bosons, carrying spin ($S$=1) degree of freedom, in $\Omega$ number of single particle orbitals, each triply degenerate, we introduce and analyze embedded Gaussian orthogonal ensemble of random matrices generated by random two-body interactions that are spin (S) scalar [BEGOE(2)-$S1$]. The embedding algebra is $U(3) \supset G \supset G1 \otimes SO(3)$ with SO(3) generating spin $S$. A method for constructing the ensembles in fixed-($m$, $S$) space has been developed. Numerical calculations show that the form of the fixed-($m$, $S$) density of states is close to Gaussian and level fluctuations follow GOE. Propagation formulas for the fixed-($m$, $S$) space energy centroids and spectral variances are derived for a general one plus two-body Hamiltonian preserving spin. In addition to these, we also introduce two different pairing symmetry algebras in the space defined by BEGOE(2)-$S1$ and the structure of ground states is studied for each paring symmetry.