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arXiv:2111.08820

Eigenstate structure in many-body bosonic systems: Analysis using random matrices and $q$-Hermite polynomials

Priyanka Rao, Manan Vyas, N. D. Chavda

Published Nov 16, 2021• 0 citations
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Abstract

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 ($l_H$) 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 $F_k(E|q)$ and the bivariate $q$-Hermite polynomial form for bivariate eigenvalue density $ρ_{biv:q}(E,E_k)$ 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.

DOI
10.48550/arxiv.2111.08820

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Eigenstate structure in many-body bosonic systems: Analysis using random matrices and qqq-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 kkk-body random interactions. The quantities employed are the number of principal components (NPC), the localization length (lHl_HlH​) and the entropy production S(t)S(t)S(t). The numerical results are compared with the analytical formulas obtained using random matrices which are based on bivariate qqq-Hermite polynomials for local density of states Fk(E∣q)F_k(E|q)Fk​(E∣q) and the bivariate qqq-Hermite polynomial form for bivariate eigenvalue density ρbiv:q(E,Ek)ρ_{biv:q}(E,E_k)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.
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 .
For mmm number of bosons, carrying spin ($\cs=\spin$) degree of freedom, in ΩΩΩ 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 (SSS) 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)U(2Ω) \supset U(Ω) \otimes SU(2)U(2Ω)⊃U(Ω)⊗SU(2) with SU(2) generating spin. A method for constructing the ensembles in fixed-(m,Sm,Sm,S) spaces has been developed. Numerical calculations show that for BEGOE(2)-$\cs$, the fixed-(m,S)(m,S)(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 SSS. Covariances in energy centroids and spectral variances are analyzed. Propagation formula is derived for the variance propagator for the fixed-(m,S)(m,S)(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=SmaxS=S_{max}S=Smax​. This is further corroborated by analyzing the structure of the ground states in the presence of the exchange interaction S^2\hat{S}^2S^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 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 (SSS) scalar [called BEGOE(1+2)-S1S1S1]. Embedding algebra for the ensemble, for mmm 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 SSS. 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-(mmm, SSS) 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 SSS for S<Smax/2S<{S}_{\text{max}}/2S<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.
Average-fluctuation separation in energy levels in quantum many-particle systems with kkk-body interactions using qqq-Hermite polynomials
Separation between average and fluctuation parts in the state density in many-particle quantum systems with kkk-body interactions, modeled by the kkk-body embedded Gaussian orthogonal random matrices (EGOE(kkk)), is demonstrated using the method of normal mode decomposition of the spectra and also verified through power spectrum analysis, for both fermions and bosons. The smoothed state density is represented by the qqq-normal distribution (fqNf_{qN}fqN​) (with corrections) which is the weight function for qqq-Hermite polynomials. As the rank of interaction kkk increases, the fluctuations set in with smaller order of corrections in the smooth state density. They are found to be of GOE type, for all kkk values, for both fermion and boson systems.
Random matrix ensembles defined by a mean-field one body plus a chaos generating random two-body interaction [called embedded ensembles of (1+2)-body interactions] predict for wave functions, in the chaotic domain, an essentially one-parameter Gaussian forms for the energy dependence of the number of principal components (NPC) and the localization length l(H) (defined by information entropy), which are two important measures of chaos in finite interacting many-particle systems. Numerical embedded ensemble calculations and nuclear shell-model results, for NPC and l(H), are compared with the theory. These analyses clearly point out that for realistic finite interacting many-particle systems, in the chaotic domain, wave-function structure is given by (1+2)-body embedded random matrix ensembles.
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.