In this work, we investigate the Anderson localization problems of the generalized Aubry-André model (Ganeshan-Pixley-Das Sarma's model) with an unbounded quasiperiodic potential where the parameter $\left|\alpha \right|\ge 1$. The Lyapunov exponent $\gamma \left(E\right)$ and the mobility edges $E_c$ are exactly obtained for the unbounded quasiperiodic potential. With the Lyapunov exponent, we find that there exists a critical region in the parameter $\lambda \text{−}E$ plane. The critical region consists of critical states. In comparison with localized and extended states, the fluctuation of spatial extensions of the critical states is much larger. The numerical results show that the scaling exponent of inverse participation ratio (IPR) of critical states $x\simeq 0.5$. Furthermore, it is found that the critical indices of localized length $\nu =1$ for the bounded ($\left|\alpha \right|<1$) case and $\nu =1/2$ for the unbounded ($\left|\alpha \right|\ge 1$) case. The above distinct critical indices can be used to distinguish the localized-extended from localized-critical transitions. At the end, we show that the systems with different $E$ for both cases of $\left|\alpha \right|<1$ and $\left|\alpha \right|\ge 1$ can be classified by the Lyapunov exponent $\gamma \left(E\right)$ and Avila's quantized acceleration $\omega \left(E\right)$.

The superfluid properties of attractive Hubbard model in dice lattice are investigated. It is found that three superfluid order parameters increase as the interaction increases. When the filling factor falls into the flat band, due to the infinite large density of states, the resultant superfluid order parameters are proportional to interaction strength, which is in striking contrast with the exponentially small counterparts in usual superfluid (or superconductor). When the interaction is weak, and the filling factor is near the bottom of the lowest band (or the top of highest band), the superfluid density is determined by the effective mass of the lowest (or highest) single-particle band. When the interaction is strong and filling factor is small, the superfluid density is inversely proportional to interaction strength, which is related to effective mass of tightly bound pairs. In the strong interaction limit and finite filling, the asymptotic behaviors of superfluid density can be captured by a parabolic function of filling factor. Furthermore, when the filling is in flat band, the superfluid density shows a logarithmic singularity as the interaction approaches zero. In addition, there exist three undamped collective modes for strong interactions. The lowest excitation is gapless phonon, which is characterized by the total density oscillations. The two others are gapped Leggett modes, which correspond relative density fluctuations between sublattices. The collective modes are also reflected in the two-particle spectral functions by sharp peaks. Furthermore, it is found that the two-particle spectral functions satisfy an exact sum-rule, which is directly related to the filling factor (or density of particle). The sum-rule of the spectral functions may be useful to distinguish between the hole-doped and particle-doped superfluid (superconductor) in experiments.

The superfluid states of attractive Hubbard model in α–T ₃ lattice are investigated. It is found that one usual needs three non-zero superfluid order parameters to describe the superfluid states due to three sublattices. When two hopping amplitudes are equal, the system has particle–hole symmetry. The flat band plays an important role in superfluid pairing near half filling. For example, when the filling factor falls into the flat band, the large density of states in the flat band favors superfluid pairing and the superfluid order parameters reach relatively large values. When the filling factor is in the gap between the flat band and upper band, the superfluid order parameters take small values due to the vanishing of density of states. The superfluid order parameters show nonmonotonic behaviors with the increase of filling factor. At last, we also investigate the edge states with open boundary conditions. It is shown that there exist some interesting edge states in the middle of quasi-particle bands.

In this work, we investigate the bound states in a one-dimensional spin-1 flat band system with a Coulomb-like potential of type III, which has a unique non-vanishing matrix element in basis ∣1〉. It is found that, for such a kind of potential, there exists infinite bound states. Near the threshold of continuous spectrum, the bound state energy is consistent with the ordinary hydrogen-like atom energy level with Rydberg correction. In addition, the flat band has significant effects on the bound states. For example, there are infinite bound states which are generated from the flat band. Furthermore, when the potential is weak, the bound state energy is proportional to the potential strength α. When the bound state energies are very near the flat band, they are inversely proportional to the natural number n (e.g., E _n ∝ 1/n, n = 1, 2, 3, …). Further we find that the energy spectrum can be well described by quasi-classical approximation (WKB method). Finally, we give a critical potential strength α _c at which the bound state energy reaches the threshold of continuous spectrum. After crossing the threshold, the bound states in the continuum (BIC) would exist in such a flat band system.

In this work, we investigate the bound-state problem in a one-dimensional spin-1 Dirac Hamiltonian with a flat band. It is found that the flat band has significant effects on the bound states. For example, for Dirac delta potential gδ(x), there exists one bound state for both the positive and negative potential strength g. Furthermore, when the potential is weak, the bound-state energy is proportional to the potential strength g. For square well potential, the flat band results in the existence of infinite bound states for arbitrarily weak potential. In addition, when the bound-state energy is very near the flat band, the energy displays a hydrogen atom-like spectrum, i.e. the bound-state energies are inversely proportional to the square of the natural number n (e.g., E _n ∝ 1/n ², n = 1, 2, 3, …). Most of the above nontrivial behaviors can be attributed to the infinitely large density of states of the flat band and its ensuing 1/z singularity of the Green function. The combination of a short-ranged potential and flat band provides a new possibility to get an infinite number of bound states and a hydrogen atom-like energy spectrum. In addition, our findings provide some useful insights and further our understanding of the many-body physics of the flat band.

In our previous work, the concept of critical region in a generalized Aubry–André model (Ganeshan–Pixley–Das Sarma’s model) has been established. In this work, we find that the critical region can be realized in a one-dimensional flat band lattice with a quasi-periodic potential. It is found that the above flat band lattice model can be reduced to an effective Ganeshan–Pixley–Das Sarma’s model. Depending on various parameter ranges, the effective quasi-periodic potential may be bounded or unbounded. In these two cases, the Lyapunov exponent, mobility edge, and critical indices of localized length are obtained exactly. In this flat band model, the localized-extended, localized-critical and critical-extended transitions can coexist. Furthermore, we find that near the transitions between the bound and unbounded cases, the derivative of Lyapunov exponent of localized states with respect to energy is discontinuous. At the end, the localized states in bounded and unbounded cases can be distinguished from each other by Avila’s acceleration.