Exact results for embedded Gaussian unitary ensemble of two-body interactions for fermions with spin [EGUE(2)-s] are derived using the Wigner-Racah algebra of the embedding U(2Ω)⊃U(Ω)⊗SU(2) algebra. Explicit formulas, for systems with m fermions and spin S, are given for the lower order cross correlations, between states with (m,S) and (m′,S′), generated by this ensemble and also, in terms of a SU(Ω) Racah coefficient, for the fourth order shape parameter for the fixed-(m,S) density of states.
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.
Following the earlier studies on embedded unitary ensembles generated by random two-body interactions [EGUE(2)] with spin SU(2) and spin-isospin SU(4) symmetries, developed is a general formulation, for deriving lower order moments of the one- and two-point correlation functions in eigenvalues, that is valid for any EGUE(2) and BEGUE(2) (“B” stands for bosons) with U(Ω)⊗SU(r) embedding and with two-body interactions preserving SU(r) symmetry. Using this formulation with r = 1, we recover the results derived by Asaga et al. [Ann. Phys. (N.Y.) 297, 344 (2002)]10.1006/aphy.2002.6248 for spinless boson systems. Going further, new results are obtained for r = 2 (this corresponds to two species boson systems) and r = 3 (this corresponds to spin 1 boson systems).
Embedded random matrix ensembles are generic models for describing statistical properties of finite isolated quantum many-particle systems. For the simplest spinless fermion (or boson) systems with say m fermions (or bosons) in N single particle states and interacting with say k-body interactions, we have EGUE(k) [embedded GUE of k-body interactions) with GUE embedding and the embedding algebra is U(N). In this paper, using EGUE(k) representation for a Hamiltonian that is fc-body and an independent EGUE(t) representation for a transition operator that is t-body and employing the embedding U(N) algebra, finite-N formulas for moments up to order four are derived, for the first time, for the transition strength densities (transition strengths multiplied by the density of states at the initial and final energies). In the asymptotic limit, these formulas reduce to those derived for the EGOE version and establish that in general bivariate transition strength densities take bivariate Gaussian form for isolated finite quantum systems. Extension of these results for other types of transition operators and EGUE ensembles with further symmetries are discussed.
Embedded random matrix ensembles are generic models for describing statistical properties of finite isolated quantum many-particle systems. For the simplest spinless fermion (or boson) systems with say
m fermions (or bosons) in
N single particle states and interacting with say
k-body interactions, we have EGUE(
k) [embedded GUE of
k-body interactions) with GUE embedding and the embedding algebra is
U(N). In this paper, using EGUE(
k) representation for a Hamiltonian that is
k-body and an independent EGUE(
t) representation for a transition operator that is
t-body and employing the embedding
U(N) algebra, finite-
N formulas for moments up to order four are derived, for the first time, for the transition strength densities (transition strengths multiplied by the density of states at the initial and final energies). In the asymptotic limit, these formulas reduce to those derived for the EGOE version and establish that in general bivariate transition strength densities take bivariate Gaussian form for isolated finite quantum systems. Extension of these results for other types of transition operators and EGUE ensembles with further symmetries are discussed.
For finite quantum many-particle systems, a given system, induced by a transition operator, makes transitions from its states to the states of the same system or to those of another system. Examples are electromagnetic transitions (then the initial and final systems are same), nuclear beta and double beta decay (then the initial and final systems are different), particle addition to or removal from a given system and so on. Working towards developing a complete statistical theory for transition strength densities (transition strengths multiplied by the density of states at the initial and final energies), we have started a program to derive formulas for the lower order bivariate moments of the strength densities generated by a variety of transition operators. In this paper results are presented for a transition operator that removes
k0 number of particle by considering
m spinless fermions in
N single particle states. The Hamiltonian that is
k-body is represented by EGUE(
k) [embedded Gaussian unitary ensemble of
k-body interactions] and similarly the transition operator by an appropriate independent EGUE. Employing the embedding
U(N) algebra, finite-
N formulas for moments up to order four are derived and they show that in general the smoothed (with respect to energy) bivariate transition strength densities take bivariate Gaussian form. Extension of these results to particle addition operator and beta decay type operators are discussed.
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.