The spectral fluctuations in complex quantum systems are modeled in terms of spectra from random matrices. Quantitative agreement with random matrices holds only when the quantum spectrum is a pure sequence, and not a composite spectrum that mixes levels from different symmetry sectors. It was shown earlier that discrete symmetries in composite spectra can be detected using higher-order spacing ratio statistics provided the symmetry sectors are of equal dimensions. The general case is when the symmetry sectors have unequal dimensions, and the symmetries of the Hamiltonian system are unknown or only partially known. In this work, we address how the number of symmetry sectors can be determined from a composite spectrum arising in Hermitian and non-Hermitian settings. A Best Subset Technique is proposed for determining the number of the symmetry sectors by comparing the spectral fluctuations to that of an appropriate random matrix ensemble without requiring de-symmetrization. It is demonstrated using spectra from Gaussian, Wishart and Ginibre ensembles, and also for spectra drawn from many-body quantum systems.
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.
The complex spacing ratio is the complex displacement from a reference eigenvalue to its nearest neighbor divided by the corresponding displacement to its next-to-nearest neighbor. Its statistics provide a useful diagnostic of spectral correlations and nonintegrability in open quantum systems. Here, starting from the exact joint eigenvalue probability densities, we derive finite-
N complex-spacing-ratio distributions for the Gaussian ensembles of non-Hermitian random matrices in classes AI
† and AII
†, realized by complex symmetric and complex self-dual random matrices, respectively. In class AII
†, we obtain an exact algebraic expression for arbitrary
N and explicitly evaluate the distributions and representative moments for
N=3,4,5,6. In class AI
†, although the joint density retains a noncompact integral over nonunitary eigenvector degrees of freedom, we analytically derive a normalized one-dimensional integral representation for
N=3 and determine the asymptotic behavior, including a logarithmic correction to the cubic level repulsion and a nonanalytic contribution to the angular density. We further confirm these analytical results through direct numerical diagonalization of non-Hermitian random matrices.
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)⊃U(Ω)⊗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=Smax. This is further corroborated by analyzing the structure of the ground states in the presence of the exchange interaction
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.