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
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
Finite interacting Fermi systems with a mean-field and a chaos generating two-body interaction are modeled by one plus two-body embedded Gaussian orthogonal ensemble of random matrices with spin degree of freedom [called EGOE(1+2)-s]. Numerical calculations are used to demonstrate that, as lambda , the strength of the interaction (measured in the units of the average spacing of the single-particle levels defining the mean-field), increases, generically there is Poisson to GOE transition in level fluctuations, Breit-Wigner to Gaussian transition in strength functions (also called local density of states) and also a duality region where information entropy will be the same in both the mean-field and interaction defined basis. Spin dependence of the transition points lambda_{c} , lambdaF, and lambdad , respectively, is described using the propagator for the spectral variances and the formula for the propagator is derived. We further establish that the duality region corresponds to a region of thermalization. For this purpose we compared the single-particle entropy defined by the occupancies of the single-particle orbitals with thermodynamic entropy and information entropy for various lambda values and they are very close to each other at lambda=lambdad.
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)⊃U(Ω)⊗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=Smax. This is further corroborated by analyzing the structure of the ground states in the presence of the exchange interaction 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.
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.
Eigenstate structure in many-body bosonic systems: Analysis using random matrices and q-Hermite polynomials
We analyze the structure of eigenstates in many-body bosonic systems by modeling the Hamiltonian of these complex systems using Bosonic Embedded Gaussian Orthogonal Ensembles (BEGOE) defined by a mean-field plus k-body random interactions. The quantities employed are the number of principal components (NPC), the localization length (lH) and the entropy production S(t). The numerical results are compared with the analytical formulas obtained using random matrices which are based on bivariate q-Hermite polynomials for local density of states Fk(E∣q) and the bivariate q-Hermite polynomial form for bivariate eigenvalue density ρbiv:q(E,Ek) that are valid in the strong interaction domain. We also compare transport efficiency in many-body bosonic systems using BEGOE in absence and presence of centrosymmetry. It is seen that the centrosymmetry enhances quantum efficiency.
For bosons carrying spin-one degree of freedom, we introduce an embedded Gaussian orthogonal ensemble of random matrices generated by random two-body interactions in the presence of a mean field that is spin (S) scalar [called BEGOE(1+2)-S1]. Embedding algebra for the ensemble, for m bosons in $\ensuremath{\Omega}$ number of single-particle levels (each triply degenerate), is $\mathrm{U}(3\ensuremath{\Omega})\ensuremath{\supset}\mathrm{G}\ensuremath{\supset}\mathrm{G}1\ensuremath{\bigotimes}\mathrm{SO}(3)$ with SO(3) generating the spin S. A method for constructing the ensemble for a given $(\ensuremath{\Omega},m,S)$ has been developed. Numerical calculations show that (i) the form of the fixed-(m, S) density of states is close to a Gaussian; (ii) for a strong enough interaction, level fluctuations follow GOE; (iii) fluctuation in energy centroids is large; and (iv) spectral widths are nearly constant with respect to S for S<Smax/2. Moreover, we identify two different pairing symmetry algebras in the space defined by $\text{BEGOE}(1+2)\ensuremath{-}S1$ and numerical results show that random interactions generate ground states with maximal value for the pair expectation value.
Random matrix ensembles defined by a mean-field one-body plus a chaos generating random two-body interaction (called embedded Gaussian orthogonal ensembles of (1+2)-body interactions[EGOE(1+2)]) predict for the entropy defined by the occupation numbers of single-particle states, in the chaotic domain, an essentially one parameter Gaussian form for their energy dependence. Numerical embedded ensemble calculations are compared with the theory. In addition, it is shown that the single-particle entropy, thermodynamic entropy defined by the state density and information entropy defined by wave functions in the mean-field basis for EGOE(1+2) describe the results known for interacting Fermi systems such as those obtained from nuclear shell model.