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.
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.
In lattice models, destructive interference can quench single-particle motion and produce (nearly) dispersionless Bloch bands, known as flat bands. Suppressing kinetic energy enhances the role of interactions and can lead to pairing, constrained dynamics, and localization-like behavior even in translationally invariant settings. This thesis studies attractive Hubbard-type interactions in such flat-band systems using an isolated-band (projected) description, which yields an effective Hamiltonian in a Wannier-orbital basis. The π-flux dice lattice is a convenient focus because its two lowest bands are perfectly flat and degenerate, and their subspace admits compactly localized Wannier functions. This makes the projection straightforward to implement and yields a projected Hamiltonian with a simple, short-range structure. A one-dimensional reference model is used alongside the dice lattice to separate and benchmark the effects of different processes. In Publication I, we construct a class of exact many-body eigenstates of the projected Hamiltonian of generic lattice models with flat bands in which localized unpaired quasiparticles coexist with a background of paired particles in the zero-momentum state. This general result motivates the study of a specific model, the π-flux dice-lattice. Using exact diagonalization and realtime evolution from few-particle initial states, we show that two-body bound states behave ergodically. At the same time, the density and spin distribution retain strong memory of the initial quasiparticle positions at long times. These results demonstrate localization-like dynamics in a translationally invariant flat-band model, where memory arises from projection-induced constraints rather than quenched disorder. In Publication II, we introduce a one-dimensional on-site/bond singlet model whose terms mirror those of the projected dice-lattice Hamiltonian, with tunable couplings controlling their relative strengths. Exact diagonalization shows that, for some couplings, the spectrum and dynamics of the on-site/bond-singlet model closely reproduce those of the projected dice lattice. We then use spectral statistics to probe how the different dynamical regimes manifest in energy-level correlations. In Publication III, we apply spectral statistics to the projected dice lattice using higher-order spacing and gap-ratio diagnostics. We find a robust parity dependence: even particle-number sectors show multi-block GOE behavior within fixed symmetry sectors, while odd particle-number sectors exhibit exact doublets and inter-doublet correlations that are GUE at even orders. The appearance of distinct random-matrix ensembles (GOE and GUE) within the same physical model is an unusual and interesting feature, which calls for a deeper understanding of the hidden symmetry structure of the Hamiltonians obtained by projecting onto the flat-band subspace in the dice lattice and other lattice models with flat bands realized with ultracold gases in optical lattices and in the newly discovered class of moiré materials.
We investigate the spectral properties of all-to-all interacting spin Hamiltonians acting on exactly k spins, whose coupling coefficients are drawn from a normal distribution with mean μ and variance σ2. For μ=0, we demonstrate that the random matrix ensemble -- Gaussian Orthogonal Ensemble (GOE), Gaussian Unitary Ensemble (GUE), or Gaussian Symplectic Ensemble (GSE) -- is determined by the parity of system size L and locality k, following standard time-reversal symmetry classification. For couplings with a nonzero mean, we map the Hamiltonians to deformed random matrix ensembles and analyze conditions for an energy gap between the ground state and the first excited state. For μ<0, we find two distinct regimes: for k≫L, the gap closes at critical disorder σc≈∣μ∣. Near this transition the energy gap Δ exhibits universal quadratic scaling Δ/L∼(σ−σc)2. When k≪L, σc scales with ∣μ∣, but lacks a sharp transition. Our work introduces a semi-solvable model that captures universal features of random-matrix statistics, and spectral gap formation, providing a foundation for systematic extensions to more general many-body systems.
The energy level statistics of the Hubbard model for L\ifmmode\times\else\texttimes\fi{}L square lattices (L=3,4,5,6) at low filling (four electrons) is studied numerically for a wide range of coupling strength. All known symmetries of the model (space, spin, and pseudospin symmetry) have been taken into account explicitly from the beginning of the calculation by projecting into symmetry-invariant subspaces. The details of this group theoretical treatment are presented with special attention to the nongeneric case of L=4, where a particular complicated space group appears. For all the lattices studied, a significant amount of levels within each symmetry invariant subspaces remains degenerated, but except for L=4 the ground state is nondegenerate. We explain the remaining degeneracies, which occur only for very specific interaction-independent states, and we disregard these states in the statistical spectral analysis. The intricate structure of the Hubbard spectra necessitates a careful unfolding procedure, which is thoroughly discussed. Finally, we present our results for the level spacing distribution, the number variance ${\mathrm{\ensuremath{\Sigma}}}^{2}$, and the spectral rigidity ${\mathrm{\ensuremath{\Delta}}}_{3}$, which essentially all are close to the corresponding statistics for random matrices of the Gaussian ensemble independent of the lattice size and the coupling strength. Even very small coupling strengths approaching the integrable zero coupling limit lead to Gaussian ensemble statistics, stressing the nonperturbative nature of the Hubbard model.
For a two-species boson system, it is possible to introduce a fictitious (F) spin for the bosons such that the two projections of F represent the two species. Then, for m bosons the total fictitious spin F takes values m/2, m/2 ? 1,?, 0 or 1/2. For such a system with m number of bosons in ? number of single-particle levels, each doubly degenerate, we introduce and analyze an embedded Gaussian orthogonal ensemble (GOE) of random matrices generated by random two-body interactions that conserve F-spin (BEGOE(1+2)-F); with degenerate single-particle levels, we have BEGOE(2)-F. Embedding algebra for BEGOE(1+2)-F ensemble is U(2?)?U(?)?SU(2) with SU(2) generating F-spin. A method for constructing the ensembles in fixed-(m, F) spaces has been developed. Numerical calculations show that for BEGOE(1+2)-F, the fixed-(m, F) density of states is close to Gaussian and level fluctuations follow the GOE in the dense limit. Similarly, 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 F. Formulas for the fixed-(m, F) space eigenvalue centroids and spectral variances are derived for a given member of the ensemble and also for the variance propagator for the fixed-(m, F) ensemble-averaged spectral variances. Using these, covariances in eigenvalue centroids and spectral variances are analyzed. The variance propagator clearly shows that the BEGOE(2)-F ensemble generates ground states with spin F = Fmax = m/2. Natural F-spin ordering (Fmax, Fmax ? 1, Fmax ? 2, ?, 0 or 1/2) is also observed with random interactions. Going beyond these, we also introduce pairing symmetry in the space defined by BEGOE(1+2)-F. Expectation values of the pairing Hamiltonian show that random interactions generate ground states with a maximum value for the expectation value for a given F and in these it is largest for F = Fmax = m/2.
We construct a class of exact eigenstates of the Hamiltonian obtained by projecting the Hubbard interaction term onto the flat band subspace of a generic lattice model. These exact eigenstates are many-body states in which an arbitrary number of localized fermionic particles coexist with a sea of mobile Cooper pairs with zero momentum. By considering the dice lattice as an example, we provide evidence that these exact eigenstates are, in fact, a manifestation of local integrals of motions of the projected Hamiltonian. In particular, the spin and particle densities retain memory of the initial state for a very long time if localized unpaired particles are present at the beginning of the time evolution. This shows that many-body localization of quasiparticles and superfluidity can coexist even in generic two-dimensional lattice models with flat bands, for which it is not known how to construct local conserved quantities. Our results open interesting perspectives on the old condensed-matter problem of the interplay between superconductivity and localization.
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.