We study the two-particle spectral functions and collective modes of weakly disordered superconductors using a disordered attractive Hubbard model on a square lattice. We show that the disorder-induced scattering between collective modes leads to a finite subgap spectral weight in the long-wavelength limit. In general, the spectral weight is distributed between the phase and the Higgs channels, but as we move towards half filling the Higgs contribution dominates. The inclusion of the density fluctuations lowers the frequency at which this mode occurs and results in the phase channel gaining a larger contribution to this subgap mode. Near half filling, the proximity of the system to the charge density wave instability leads to strong fluctuations of the effective disorder at the commensurate wave vector ($[\ensuremath{\pi},\ensuremath{\pi}]$). We develop an analytical mode coupling approach where the pure Goldstone mode in the long-wavelength limit couples to the collective mode at $[\ensuremath{\pi},\ensuremath{\pi}]$. This provides insight into the location and distribution of the two-particle spectral weights between the Higgs and the phase channels.
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.
We present a brief review of our studies of disorder influence upon Ginzburg-Landau expansion coefficients in Anderson-Hubbard model with attraction in the framework of the generalized DMFT + Σ approximation. A wide range of attractive potentials U is considered from weak coupling limit, where superconductivity is described by BCS model, to the limit of very strong coupling, where superconducting transition is related to the Bose-Einstein condensation of compact Cooper pairs, which are formed at temperatures significantly higher than the superconducting transition temperature, as well as the wide range of disorders from weak to strong, when the system is in the vicinity of Anderson transition. For the same range of parameters we study in detail the temperature behavior of orbital and paramagnetic upper critical field H c2 ( T ), which demonstrates the anomalies due both to the growth of attractive potential and to the effects of strong disordering.