We investigate the competition between superconductivity (SC) and charge density wave (CDW) under a time-dependent periodic field in the attractive Hubbard model. By employing the time-dependent exact diagonalization method, we show that the driving frequency and amplitude of the external field can control the enhancement of either the superconducting pair or the CDW correlation in the system, for which SC and CDW are degenerate in the ground state of the half-filled attractive Hubbard model in the absence of the field. In the strong-coupling limit of the attractive Hubbard interaction, the controllability is characterized by the anisotropic interaction of the effective model. The anisotropy is induced by the external field and lifts the degeneracy of SC and CDW. We find that the enhancement or suppression of the superconducting pair and CDW correlations in the periodically driven attractive Hubbard model can be well interpreted by the quench dynamics of the effective model derived in the strong-coupling limit.
We have studied in terms of the effective T-matrix the self-energy part for both the attractive (U<0) and repulsive (U>0) Hubbard models. It is shown that the fluctuation pairing for strong coupling at T>T c for the attractive Hubbard model does not cause any pseudogap behavior in the single-particle density of states. We have also identified the origin of the false pseudogap for that case. Meanwhile, the fluctuation pairing significantly affect the density–density and the current–current correlation functions. As a result, there exists a pseudogap in the charge (and current) response for the attractive Hubbard model. In contrast, the pseudogap in the spin response appears to exist for the repulsive Hubbard model only. It is also shown that the contribution from the Hartree–"Cooper" channel gives an anomalous Fermi liquid term, Im χ c ≈-ω/T, to the charge response of the nearly antiferromagnetic Fermi liquid.
We study the two-dimensional attractive Hubbard model using the mapping onto the half-filled repulsive Hubbard model in a uniform magnetic field coupled to the fermion spins. The low-energy effective action for charge and pairing fluctuations is obtained in the hydrodynamic regime. We recover the action of a Bose superfluid where half the fermion density is identified as the conjugate variable of the phase of the superconducting order parameter. By integrating out charge fluctuations, we obtain a phase-only action. In the zero-temperature superconducting state, this action describes a collective phase mode smoothly evolving from the Anderson-Bogoliubov mode at weak coupling to the Bogoliubov mode of a Bose superfluid at strong coupling. At finite temperature, the phase-only action can be used to extract an effective
XY model and thus obtain the Berezinskii-Kosterlitz-Thouless (BKT) phase transition temperature. We also identify a renormalized classical regime of superconducting fluctuations above the BKT phase transition, and a regime of incoherent pairs at higher temperature. Special care is devoted to the nearly half-filled case where the symmetry of the order parameter is enlarged to SO(3) due to strong $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$ charge fluctuations. The low-energy effective action is then an SO(3) nonlinear $\ensuremath{\sigma}$ model with a (symmetry breaking) magnetic field proportional to the doping. In the strong-coupling limit, the attractive Hubbard model can be mapped onto the Heisenberg model in a magnetic field, which reduces to the quantum
XY model (except for a weak magnetic field, i.e., in the low-density limit of the attractive model). In the low-density limit, the Heisenberg model allows one to recover the action of a Bose superfluid, including the ${(\mathbf{\ensuremath{\nabla}}\ensuremath{\rho})}^{2}$ term (with $\ensuremath{\rho}$ the density), and in turn the Gross-Pitaevskii equation.