Spectral properties of embedded Gaussian unitary ensemble of random matrices with Wigner’s SU(4) symmetry

Abstract

For m fermions in Ω number of single particle orbitals, each fourfold degenerate, we introduce and analyze in detail embedded Gaussian unitary ensemble of random matrices generated by random two-body interactions that are SU(4) scalar [EGUE(2)- SU(4)]. Here the SU(4) algebra corresponds to the Wigner’s supermultiplet SU(4) symmetry in nuclei. Embedding algebra for the EGUE(2)- SU(4) ensemble is U(4 Ω) ⊃ U( Ω) ⊗ SU(4). Exploiting the Wigner–Racah algebra of the embedding algebra, analytical expression for the ensemble average of the product of any two m particle Hamiltonian matrix elements is derived. Using this, formulas for a special class of U( Ω) irreducible representations (irreps) {4 r , p}, p = 0, 1, 2, 3 are derived for the ensemble averaged spectral variances and also for the covariances in energy centroids and spectral variances. On the other hand, simplifying the tabulations of Hecht for SU( Ω) Racah coefficients, numerical calculations are carried out for general U( Ω) irreps. Spectral variances clearly show, by applying Jacquod and Stone prescription, that the EGUE(2)- SU(4) ensemble generates ground state structure just as the quadratic Casimir invariant ( C 2) of SU(4). This is further corroborated by the calculation of the expectation values of C 2[ SU(4)] and the four periodicity in the ground state energies. Secondly, it is found that the covariances in energy centroids and spectral variances increase in magnitude considerably as we go from EGUE(2) for spinless fermions to EGUE(2) for fermions with spin to EGUE(2)- SU(4) implying that the differences in ensemble and spectral averages grow with increasing symmetry. Also for EGUE(2)- SU(4) there are, unlike for GUE, non-zero cross-correlations in energy centroids and spectral variances defined over spaces with different particle numbers and/or U( Ω) [equivalently SU(4)] irreps. In the dilute limit defined by Ω → ∞, r ≫ 1 and r/ Ω → 0, for the {4 r , p} irreps, we have derived analytical results for these correlations. All correlations are non-zero for finite Ω and they tend to zero as Ω → ∞.