Spectral properties of two-body random matrix ensembles for boson systems with spin

For m number of bosons, carrying spin $cs=spin$ degree of freedom, in Ω number of single particle orbitals, each doubly degenerate, we introduce and analyze embedded Gaussian orthogonal ensemble of random matrices generated by random two-body interactions that are spin (S) scalar \[BEGOE(2)-cs\]. Embedding algebra for the BEGOE(2)-cs ensemble and also for BEGOE(1+2)-cs that includes the mean-field one-body part is U(2Ω) supset U(Ω) otimes SU(2) with SU(2) generating spin. A method for constructing the ensembles in fixed-(m,S) spaces has been developed. Numerical calculations show that for BEGOE(2)-cs, the fixed-(m,S) density of states is close to Gaussian and level fluctuations follow GOE in the dense limit. For BEGOE(1+2)-cs, generically there is Poisson to GOE transition in level fluctuations as the interaction strength (measured in the units of the average spacing of the single particle levels defining the mean-field) is increased. The interaction strength needed for the onset of the transition is found to decrease with increasing S. Covariances in energy centroids and spectral variances are analyzed. Propagation formula is derived for the variance propagator for the fixed-(m,S) ensemble averaged spectral variances. Variance propagator clearly shows, by applying the Jacquod and Stone prescription, that the BEGOE(2)-cs ensemble generates ground states with spin S=S_max. This is further corroborated by analyzing the structure of the ground states in the presence of the exchange interaction hat{S}² in BEGOE(1+2)-cs. Natural spin ordering is also observed with random interactions. Going beyond these, we also introduce pairing symmetry in the space defined by BEGOE(2)-cs. Expectation values of the pairing Hamiltonian show that random interactions exhibit pairing correlations in the ground state region.