Spectral properties of two-body random matrix ensembles for boson systems with spin
arXiv:1010.6054
Abstract
For number of bosons, carrying spin ($\cs=\spin$) degree of freedom, in 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 () 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 with SU(2) generating spin. A method for constructing the ensembles in fixed-() spaces has been developed. Numerical calculations show that for BEGOE(2)-$\cs$, the fixed- 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 . Covariances in energy centroids and spectral variances are analyzed. Propagation formula is derived for the variance propagator for the fixed- 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 . This is further corroborated by analyzing the structure of the ground states in the presence of the exchange interaction 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.
44 pages, 10 figures, 1 table, enlarged version with additional results