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arXiv:1205.6758 · Physics Letters A

Thermalization in one- plus two-body ensembles for dense interacting boson systems

N. D. Chavda, V. K. B. Kota, V. Potbhare

Published Sep 10, 2012• 19 citations
arXivHTMLPDF
DOI
10.1016/j.physleta.2012.08.054

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Localization to delocalization transitions in eigenfunctions are studied for finite interacting boson systems by employing one‐ plus two‐body embedded Gaussian orthogonal ensemble of random matrices [EGOE(1+2)]. In the first analysis, considered are bosonic EGOE(1+2) for two‐species boson systems with a fictitious ( F ) spin degree of freedom [called BEGOE(1+2)‐ F ]. Numerical calculations are carried out as a function of the two‐body interaction strength (λ). It is shown that, in the region (defined by ) after the onset of Poisson to GOE transition in energy levels, the strength functions exhibit Breit‐Wigner to Gaussian transition for . Further, analyzing information entropy and participation ratio, it is established that there is a region defined by where the system exhibits thermalization. The F ‐spin dependence of the transition markers and follow from the propagator for the spectral variances. These results, well tested near the center of the spectrum and extend to the region within to from the center (σ 2 is the spectral variance), establish universality of the transitions generated by embedded ensembles. In the second analysis, entanglement entropy is studied for spin‐less BEGOE(1+2) ensemble and shown that the results generated are close to the recently reported results for a Bose‐Hubbard model. image
For a two-species boson system, it is possible to introduce a fictitious (F) spin for the bosons such that the two projections of F represent the two species. Then, for m bosons the total fictitious spin F takes values m/2, m/2 ? 1,?, 0 or 1/2. For such a system with m number of bosons in ? number of single-particle levels, each doubly degenerate, we introduce and analyze an embedded Gaussian orthogonal ensemble (GOE) of random matrices generated by random two-body interactions that conserve F-spin (BEGOE(1+2)-F); with degenerate single-particle levels, we have BEGOE(2)-F. Embedding algebra for BEGOE(1+2)-F ensemble is U(2?)?U(?)?SU(2) with SU(2) generating F-spin. A method for constructing the ensembles in fixed-(m, F) spaces has been developed. Numerical calculations show that for BEGOE(1+2)-F, the fixed-(m, F) density of states is close to Gaussian and level fluctuations follow the GOE in the dense limit. Similarly, 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 F. Formulas for the fixed-(m, F) space eigenvalue centroids and spectral variances are derived for a given member of the ensemble and also for the variance propagator for the fixed-(m, F) ensemble-averaged spectral variances. Using these, covariances in eigenvalue centroids and spectral variances are analyzed. The variance propagator clearly shows that the BEGOE(2)-F ensemble generates ground states with spin F = Fmax = m/2. Natural F-spin ordering (Fmax, Fmax ? 1, Fmax ? 2, ?, 0 or 1/2) is also observed with random interactions. Going beyond these, we also introduce pairing symmetry in the space defined by BEGOE(1+2)-F. Expectation values of the pairing Hamiltonian show that random interactions generate ground states with a maximum value for the expectation value for a given F and in these it is largest for F = Fmax = m/2.
Abstract In an m particle quantum system, the rank of interactions and the nature of particles (fermions or bosons) can strongly affect the dynamics of the system. To explore this, we study the nonequilibrium dynamics with particles in a one-body mean-field and quenched by an interaction of body-rank k = 2, 3, …, m . Using fermionic embedded Gaussian orthogonal ensembles (EGOEs) and bosonic embedded Gaussian orthogonal ensembles (BEGOEs) of one plus k -body interactions (also the unitary variants EGUE and BEGUE), it is seen that the short time decay of the survival probability of many-particle systems is given by the Fourier transform of the generating function of the q -Hermite polynomials. Deriving the formulas for q for both fermion and boson systems as a function of m , k and the number of single particle states N , we have verified that the Fourier transform of agrees very well with the numerical ensemble calculations for both fermion and boson systems. These results bridge the gap between the known results for k = 2 and k = m .