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For $m$ number of bosons, carrying spin ($\cs=\spin$) degree of freedom, in$\Omega$ number of single particle orbitals, each doubly degenerate, weintroduce and analyze embedded Gaussian orthogonal ensemble of random matricesgenerated by random two-body interactions that are spin ($S$) scalar[BEGOE(2)-$\cs$]. Embedding algebra for the BEGOE(2)-$\cs$ ensemble and alsofor BEGOE(1+2)-$\cs$ that includes the mean-field one-body part is $U(2\Omega)\supset U(\Omega) \otimes SU(2)$ with SU(2) generating spin. A method forconstructing the ensembles in fixed-($m,S$) spaces has been developed.Numerical calculations show that for BEGOE(2)-$\cs$, the fixed-$(m,S)$ densityof states is close to Gaussian and level fluctuations follow GOE in the denselimit. For BEGOE(1+2)-$\cs$, generically there is Poisson to GOE transition inlevel fluctuations as the interaction strength (measured in the units of theaverage spacing of the single particle levels defining the mean-field) isincreased. The interaction strength needed for the onset of the transition isfound to decrease with increasing $S$. Covariances in energy centroids andspectral variances are analyzed. Propagation formula is derived for thevariance propagator for the fixed-$(m,S)$ ensemble averaged spectral variances.Variance propagator clearly shows, by applying the Jacquod and Stoneprescription, that the BEGOE(2)-$\cs$ ensemble generates ground states withspin $S=S_{max}$. This is further corroborated by analyzing the structure ofthe ground states in the presence of the exchange interaction $\hat{S}^2$ inBEGOE(1+2)-$\cs$. Natural spin ordering is also observed with randominteractions. Going beyond these, we also introduce pairing symmetry in thespace defined by BEGOE(2)-$\cs$. Expectation values of the pairing Hamiltonianshow that random interactions exhibit pairing correlations in the ground stateregion.

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