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 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 Ω 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(2Ω)⊃U(Ω)⊗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.

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Probability distribution for the ratio (r) of consecutive level spacings of the eigenvalues of a Poisson (generating regular spectra) spectrum and that of a GOE random matrix ensemble are given recently. Going beyond these, for the ensemble generated by the Hamiltonian H λ = (H0 + λV )/ √ 1 + λ 2 interpolating Poisson (λ = 0) and GOE (λ → ∞) we have analyzed the transition curves for r and r as λ changes from 0 to ∞;r = min(r, 1/r). Here, V is a GOE ensemble of real symmetric d × d matrices and H0 is a diagonal matrix with a Gaussian distribution (with mean equal to zero) for the diagonal matrix elements; spectral variance generated by H0 is assumed to be same as the one generated by V . Varying d from 300 to 1000, it is shown that the transition parameter is Λ ∼ λ 2 d, i.e. the r vs λ (similarly for r vs λ) curves for different d's merge to a single curve when this is considered as a function of Λ. Numerically, it is also found that this transition curve generates a mapping to a 3 × 3 Poisson to GOE random matrix ensemble. Example for Poisson to GOE transition from a one dimensional interacting spin-1/2 chain is presented.

Embedded random matrix ensembles generated by random interactions (of low body rank and usually two-body) in the presence of a one-body mean field, introduced in nuclear structure physics, are now established to be indispensable in describing statistical properties of a large number of isolated finite quantum many-particle systems. Lie algebra symmetries of the interactions, as identified from nuclear shell model and the interacting boson model, led to the introduction of a variety of embedded ensembles (EEs). These ensembles with a mean field and chaos generating two-body interaction generate in three different stages, delocalization of wave functions in the Fock space of the mean-field basis states. The last stage corresponds to what one may call thermalization and complex nuclei, as seen from many shell model calculations, lie in this region. Besides briefly describing them, their recent applications to nuclear structure are presented and they are (i) nuclear level densities with interactions; (ii) orbit occupancies; (iii) neutrinoless double beta decay nuclear transition matrix elements as transition strengths. In addition, their applications are also presented briefly that go beyond nuclear structure and they are (i) fidelity, decoherence, entanglement and thermalization in isolated finite quantum systems with interactions; (ii) quantum transport in disordered networks connected by many-body interactions with centrosymmetry; (iii) semicircle to Gaussian transition in eigenvalue densities with [Formula: see text]-body random interactions and its relation to the Sachdev–Ye–Kitaev (SYK) model for majorana fermions.

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