We introduce a dynamical matrix model where the matrix is interpreted as a Hamiltonian representing interaction of a bosonic system with a single fermion. We show how a system of second-quantized fermions influences the ground state of the whole system by producing a gap between the highest eigenvalue of the occupied single-fermion states and the lowest eigenvalue of the unoccupied single-fermion states. We describe the development of the gap in both the strong and weak coupling regimes, while for the intermediate coupling strength we expect formation of homolumo kinks.
We introduce a bare-bone random matrix quantum impurity model, by hybridizing a localized spinless electronic level with a bath of random fermions in the Gaussian Orthogonal Ensemble (GOE). While stripped out of correlations effects, this model reproduces some salient features of the impurity charge distribution obtained in previous works on interacting disordered impurity models. Computing by numerical sampling the impurity charge distribution in our model, we find a crossover from a Gaussian distribution (centered on half a charge unit) at large hybridization, to a bimodal distribution (centered both on zero and full occupations of the charge) at small hybridization. In the bimodal regime, a universal
(−3/2) power-law is also observed. All these findings are very well accounted for by an analytic surmise computed with a single random electron level in the bath. We also derive an exact functional integral for the general probability distribution function of eigenvalues and eigenstates, that formally captures the statistical behavior of our model for any number
N of fermionic orbitals in the bath. In the Gaussian regime and in the limit
N→∞, we are able to solve exactly the random matrix theory (RMT) for the charge distribution, obtaining perfect agreement with the numerics. Our results could be tested experimentally in mesoscopic devices, for instance by coupling a small quantum dot to a chaotic electronic reservoir, and using a quantum point contact as local charge sensor for the quantum dot occupation.
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.
Embedded random matrix ensembles operating in nuclear shell model spaces, with nucleons occupying a finite set of single particle orbits and interacting via a two-body interaction, form the basis for statistical shell model. With sufficiently strong interaction, the level densities in shell model spaces take close to a Gaussian form and transition strength distributions close to a bivariate Gaussian form. In practice, partitioning via spherical configurations (
m~) and angular momentum
J (also isospin where appropriate) are essential. The resulting statistical spectroscopy or statistical shell model was applied successfully in the past in some studies of nuclear level densities, orbit occupancies,
β-decay matrix elements and so on. Going beyond these, recently it is recognized that embedded ensembles, in a better approximation, generate in-fact
q-normal form (
q=1 gives Gaussian and
q=0 Wigner's semi-circle) for density of eigenvalues, bivariate
q-normal form for transition strengths and conditional
q-normal form for strength functions. These then allow us to develop statistical shell model with
q-normal forms. These new developments in embedded ensembles and statistical shell model are briefly reviewed in this paper. Also described, using some examples, is the role of the
q parameter in generating statistical properties of general quantum many-particle 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.
We investigate the spectral statistics of an interacting fermionic system derived by projecting the Hubbard interaction onto the two lowest-energy, degenerate flat bands of the dice lattice subjected to a
π-flux. Surprisingly, the distributions of level spacings and gap ratios correspond to distinct Gaussian ensembles, depending on the parity of the particle number. For an even number of particles, the spectra conform to the Gaussian Orthogonal Ensemble, as expected for a time-reversal-symmetric Hamiltonian. In stark contrast, the odd-parity sector exhibits exact double degeneracy of all eigenstates even after resolving all known symmetries, and the Gaussian Unitary Ensemble accurately describes the spacing distribution between these doublets. The simultaneous emergence of two different random-matrix ensembles within a single physical system constitutes an unprecedented finding, opening new avenues for both random matrix theory and flat-band physics.