Localization‐delocalization transitions in bosonic random matrix ensembles

Abstract

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 λ > λ c ) after the onset of Poisson to GOE transition in energy levels, the strength functions exhibit Breit-Wigner to Gaussian transition for λ > λ F k > λ c . Further, analyzing information entropy and participation ratio, it is established that there is a region defined by λ ∼ λ t where the system exhibits thermalization. The F -spin dependence of the transition markers λ F k and λ t follow from the propagator for the spectral variances. These results, well tested near the center of the spectrum and extend to the region within ±2σ to ±3σ 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.

Temperature of a single chaotic eigenstate

“…They were introduced in [16] and extensively studied for fermions [17]. On the other hand the case of Bose particles has been less investigated and only few results are known and typically for the dense limit, N ≫ M [18,19].…”

Random k-Body Ensembles for Chaos and Thermalization in Isolated Systems

“…This is clearly seen from the results, as shown in Figure 2 , for the chaos measure number of principal components (NPC or ) and the information entropy ( ) both are defined in terms of , In the thermalization region, , and the spreading produced by and will be equal, generating maximum mixing with Gaussian strength functions and GOE fluctuations. These results have been well verified numerically (also the parametric forms of , and are well understood) using EGOE and BEGOE for spinless fermion and boson systems and also for fermion and boson systems with the spin, , degree of freedom (see [ 5 , 20 ] for details). Figure 2 shows some numerical results from EE, and it can be seen from these results that in isolated finite systems, interactions act as the heat bath [ 4 ], and there is the phenomena of localized thermal states.…”

“… ( a ) Example of the number of principal components (NPC) vs. the normalized energy, , for a bosonic BEGOE - F ensemble with bosons in sp orbits, each doubly degenerate with a fictitious spin, . The numerical ensemble averaged results are represented by filled circles, while the continuous curve is from the formula for NPC (for further detail, see reference [ 20 ]). Note that the NPC value for GOE is 1 in the graph.…”

“…More interestingly, there are many other random matrix ensembles that are closely related to EE, and they will be described in the following text. Focusing on thermalization, many concepts, such as quantum chaos, de-localization transitions, the eigenstate thermalization hypothesis (ETH), spreading of an eigenstate over an energy shell, fidelity decay, the emergence of Gibbs ensemble, canonical typicality and so on have been studied using EE in some selective investigations in the last decade [ 6 , 18 , 19 , 20 , 21 , 22 , 23 , 24 ]. In addition, with recent developments in carrying out controlled experiments with cold atoms, new theoretical studies of trapped boson systems applying RMT have emerged [ 25 , 26 , 27 , 28 , 29 ].…”

Random matrix theory for transition strengths: Applications and open questions

“…Here with λ ∼ λ t , the spreading produced by h (1) and V(2) will be equal and thus generate maximum mixing with strength functions Gaussian and fluctuations GOE. These results are well verified numerically (also the parametric forms of λ c , λ F and λ t are well understood) using EGOE(1+2) and BEGOE(1+2) for spinless fermion and boson systems and also for fermion and boson systems with spin 1 2 degree of freedom [2,9,10,11,12]. There is good evidence that nuclear effective interactions are such that complex nuclei are in general in the region of thermalization [1,2,13,14,15].…”