We investigate the effect of site disorder on the superconducting state in the attractive Hubbard model within the framework of dynamical mean field theory. For a fixed interaction strength (U), the superconducting order parameter decreases monotonically with increasing disorder (x), while the single-particle spectral gap decreases for small x, reaches a minimum and keeps increasing for larger x. Thus, the system remains gapped beyond the destruction of the superconducting state, indicating a disorder-driven superconductor-insulator transition. We investigate this transition in depth considering the effects of weak and strong disorder for a range of interaction strengths. In the clean case, the order parameter is known to increase monotonically with increasing interaction, saturating at a finite value asymptotically for U→∞. The presence of disorder results in destruction of superconductivity at large U, thus drastically modifying the clean case behaviour. A physical understanding of our findings is obtained by invoking particle-hole asymmetry and the probability distributions of the order parameter and spectral gap.
I discuss the mean-field theory of superconductivity in a strongly disordered system of fermions with short-range attraction. It is argued that in this limit the effective theory at low energies is equivalent to the disordered Bose–Hubbard model, and I consider both the infinite-range and the "nearest-neighbor" hopping of bosons between the localized states. In the infinite-range case the mean-field theory is exact, and the superconducting gap is uniform in space, while in the latter case, the gap becomes highly non-uniform in space, but, surprisingly, is finite everywhere at T < T MF . I find that the mean-field transition temperature T MF > 0 always, and argue that the superconductor–insulator transition at T = 0 in models with net attraction between fermions is in the universality class of "dirty-bosons".
We have considered the half-filled disordered attractive Hubbard model on a square lattice, in which the on-site attraction is switched off on a fraction
f of sites, while keeping a finite
U on the remaining ones. Through quantum Monte Carlo simulations for several values of
f and
U and for system sizes ranging from $8\ifmmode\times\else\texttimes\fi{}8$ to $16\ifmmode\times\else\texttimes\fi{}16$, we have calculated the configurational averages of the equal-time pair structure factor
Ps and, for a more restricted set of variables, the helicity modulus ${\ensuremath{\rho}}_{s}$, as functions of temperature. Two finite-size scaling Ans\"atze for
Ps have been used: one for zero temperature and the other for finite temperatures. We have found that the system sustains superconductivity in the ground state up to a critical impurity concentration
fc, which increases with
U, at least up to
U=4 (in units of the hopping energy). Also, the normalized zero-temperature gap as a function of
f shows a maximum near $f\ensuremath{\sim}0.07$ for $2\ensuremath{\lesssim}U\ensuremath{\lesssim}6$. Analyses of the helicity modulus and of the pair structure factor led to the determination of the critical temperature as a function of
f for
U=3, 4, and 6: they also show maxima near $f\ensuremath{\sim}0.07$, with the highest
Tc increasing with
U in this range. We argue that, overall, the observed behavior results from both the breakdown of charge-density-wave-superconductivity degeneracy and the fact that free sites tend to ``push'' electrons toward attractive sites; the latter effect being more drastic at weak couplings.
We determine the effect of a random potential on the density of states $N(\ensuremath{\omega})$ in the attractive Hubbard model. At weak coupling, disorder closes the gap concurrently with the destruction of superconductivity. At larger, but still intermediate coupling, a pseudogap in $N(\ensuremath{\omega})$ remains beyond the point at which off-diagonal long range order vanishes. This change in the elementary excitations of the insulating phase corresponds to a crossover between Fermi and Bose insulators.
We derive the disorder vs doping phase diagram of the doped Hubbard model via dynamical mean-field theory combined with typical medium theory, which allows the description of both Mott (correlation driven) and Anderson (disorder driven) metal-insulator transitions. We observe a transition from a metal to an Anderson-Mott insulator for increasing disorder strength at all interactions. In the weak correlation regime and rather small doping, the Anderson-Mott insulator displays properties which are similar to the ones found at half filling. In particular, this phase is characterized by the presence of empty sites. If we further increase either the doping or the correlation, however, an Anderson-Mott phase of a different kind arises for sharply weaker disorder strength. This phase occupies the largest part of the phase diagram in the strong-correlation regime and is characterized by the absence of the empty sites.
The finite-temperature phase diagram for the two-dimensional attractive fermion Hubbard model with spin-dependent disorder is considered within Bogoliubov--de Gennes mean-field theory. Three types of disorder are studied. In the first, only one species is coupled to a random site energy; in the second, the two species both move in random site energy landscapes which are of the same amplitude but different realizations; and finally, in the third, the disorder is in the hopping rather than the site energy. For all three cases, we find that, unlike the case of spin-symmetric randomness, where the energy gap and average order parameter do not vanish as the disorder strength increases, a critical disorder strength exists separating distinct phases. In fact, the energy gap and the average order parameter vanish at distinct transitions,
Vcgap and
Vcop, allowing for a gapless superconducting (gSC) phase. The gSC phase becomes smaller with increasing temperature until it vanishes at a temperature
T∗.
We investigate the disorder-driven superconductor to insulator quantum phase transition (SIT) in an interacting fermion model using determinantal quantum Monte Carlo (QMC) methods. The disordered superconductor is modeled by an attractive Hubbard model with site disorder chosen randomly from a uniform distribution. The superconducting state which exists for small disorder is shown to evolve into an insulating phase beyond a critical disorder. The transition is tracked by the vanishing of (a) the superfluid stiffness, and (b) the charge stiffness or the delta function peak in the optical conductivity at zero frequency. We also show the behavior of the charge, spin, pair, and current correlations in the presence of disorder. Results for the temperature dependence of the dc conductivity, obtained by an approximate analytic continuation technique, are also presented both in the metallic phase above
Tc and the insulating phase. We discuss some of the complications in extracting the resistance at the transition point.