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arXiv:1010.6054

Spectral properties of two-body random matrix ensembles for boson systems with spin

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

Published Oct 28, 2010• 0 citations
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Abstract

For $m$ number of bosons, carrying spin ($\cs=\spin$) degree of freedom, in $Ω$ 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 ($S$) 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 $U(2Ω) \supset U(Ω) \otimes SU(2)$ with SU(2) generating spin. A method for constructing the ensembles in fixed-($m,S$) spaces has been developed. Numerical calculations show that for BEGOE(2)-$\cs$, the fixed-$(m,S)$ 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 $S$. Covariances in energy centroids and spectral variances are analyzed. Propagation formula is derived for the variance propagator for the fixed-$(m,S)$ 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 $S=S_{max}$. This is further corroborated by analyzing the structure of the ground states in the presence of the exchange interaction $\hat{S}^2$ 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.

DOI
10.48550/arxiv.1010.6054

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We analyze several ground state related properties of mesoscopic systems using the random interaction matrix model EGOE(1+2)-$\cs$ (or RIMM) for many fermion systems with spin degree of freedom and the Hamiltonian containing pairing and exchange interactions in addition to the mean-field one-body and random two-body parts. RIMM reproduces the essential features of various properties: odd-even staggering in ground state energies as a function of particle number, delay in ground state magnetization and conductance peak spacing distributions. The analytical formula, we have derived, for the ensemble averaged spectral variances provides a simple understanding of some of these properties.
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.