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arXiv:1207.7225 · Physical Review E

Random matrix ensemble with random two-body interactions in the presence of a mean field for spin-one boson systems

H. N. Deota, N. D. Chavda, V. K. B. Kota, V. Potbhare, Manan Vyas

Published Aug 19, 2013• 11 citations
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Abstract

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 $S<{S}_{\text{max}}/2$. 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.

DOI
10.1103/physreve.88.022130

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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 \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ number of single-particle levels (each triply degenerate), is U(3\ensuremathΩ)\ensuremath⊃G\ensuremath⊃G1\ensuremath⨂SO(3)\mathrm{U}(3\ensuremath{\Omega})\ensuremath{\supset}\mathrm{G}\ensuremath{\supset}\mathrm{G}1\ensuremath{\bigotimes}\mathrm{SO}(3)U(3\ensuremathΩ)\ensuremath⊃G\ensuremath⊃G1\ensuremath⨂SO(3) with SO(3) generating the spin SSS. A method for constructing the ensemble for a given (\ensuremathΩ,m,S)(\ensuremath{\Omega},m,S)(\ensuremathΩ,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 BEGOE(1+2)\ensuremath−S1\text{BEGOE}(1+2)\ensuremath{-}S1BEGOE(1+2)\ensuremath−S1 and numerical results show that random interactions generate ground states with maximal value for the pair expectation value.
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 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 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.
Random matrix ensembles for a system of m number of fermions or bosons in Ω number of single particle levels each r-fold degenerate and interacting with two-body forces are considered. The spectrum generating algebra for these systems is U(rΩ) and a subalgebra of interest is U(rΩ) ⊃ U(Ω) ⊗ SU(r) algebra. Now, for random two-body interactions preserving SU(r) symmetry, one can introduce embedded Gaussian unitary ensemble of random matrices with U(Ω)⊗SU(r) embedding and this class of ensembles are denoted by EGUE(2)-SU(r). Ensembles with r = 1,2 and 4 for fermions correspond to spinless fermions, fermions with spin and fermions with Wigner's spin-isospin SU(4) symmetry respectively. Similarly, for bosons r = 1, 2 and 3 correspond to spinless bosons, two species boson systems and bosons with spin one respectively. The distinction between fermions and bosons is in the U(Ω) irreducible representations. General formulation based on Wigner-Racah algebra for lower order moments of the one- and two-point functions in eigenvalues generated by EGUE(2)- SU(r) is briefly reviewed. The final formulas for the moments involve only SU(Ω) Racah coefficients. For the fourth moment of the one-point function for r > 1 and for the higher order (> 4) bivariate moments of the two-point function for r ≥ 1, formulas are not available for the SU(Ω) Racah coefficients that are needed. It is necessary to derive analytical formulas for these or develop methods that give asymptotic results (an example for this is given in the paper) or develop methods that allow for their numerical evaluation. This important open problem is discussed in some detail.
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.
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.