Spectral Properties of the k-Body Embedded Gaussian Ensembles of Random Matrices for Bosons

Abstract

We consider m spinless Bosons distributed over l degenerate singleparticle states and interacting through a k-body random interaction with Gaussian probability distribution (the Bosonic embedded k-body ensembles). We address the cases of orthogonal and unitary symmetry in the limit of infinite matrix dimension, attained either as l → ∞ or as m → ∞. We derive an eigenvalue expansion for the second moment of the many-body matrix elements of these ensembles. Using properties of this expansion, the supersymmetry technique, and the binary correlation method, we show that in the limit l → ∞ the ensembles have nearly the same spectral properties as the corresponding Fermionic embedded ensembles. Novel features specific for Bosons arise in the dense limit defined as m → ∞ with both k and l fixed. Here we show that the ensemble is not ergodic, and that the spectral fluctuations are not of Wigner-Dyson type. We present numerical results for the dense limit using both ensemble unfolding and spectral unfolding. These differ strongly, demonstrating the lack of ergodicity of the ensemble. Spectral unfolding shows a strong tendency towards * Present address:

Statistical properties of spectral fluctuations ofNinteracting bosons in a harmonic trap

“…Whereas at zero temperature the interaction energy (Na s , for N = 3) is very very small compared to the trap energy, which acts as a perturbation and lifts the exact degeneracy of the harmonic trap and leaves quasidegenerate states. It also supports Asaga et al's statement that in an atomic trap, bosonic atoms are in partly degenerate single-particle states [42]. It indicates that there is a strong accumulation of the energy levels and it is reflected in Fig.…”

Spectral fluctuation and1/fαnoise in the energy level statistics of interacting trapped bosons

“…Interacting trapped bosons are a very complex system, and due to the existence of two energy scales it nicely describes chaos to order transition with an increase in the number of energy levels. Our observation of the Shnirelman peak strongly proves the earlier statement of Asaga et al [11]. Our results nicely demonstrate how the degenerate single-particle states of the pure harmonic trap are lifted gradually by increasing the effective interatomic interaction.…”

“…In last few years, interacting bosonic systems have been of special interest due to the experimental observation of Bose-Einstein condensation [7][8][9][10]. The presence of an external harmonic trap makes it more interesting because, as stated by Asaga et al, in an atomic trap, bosonic atoms occupy partly degenerate single-particle states [11]. Although it is argued that the random matrix approach should reveal the generic features of the spectrum, there is neither analytical treatment nor systematic numerical calculations in this direction.…”

Distribution of level spacing ratios using one- plus two-body random matrix ensembles

“…Similarly, for interacting spin-less boson systems, they are denoted by BEGOE(2) [21]. Addition of the mean-field one-body part gives EGOE(1+2) and BEGOE(1+2) for fermion and boson systems, respectively [18,19] and the Hamiltonian H = h(1) + λ{V (2)}.…”