Self-consistent-field ensembles of disordered Hamiltonians: Efficient solver and application to superconducting films

The general topic of this thesis is the study of interaction effects in disordered metals. A major focus was placed on phase transitions that these systems undergo. Methodologically we employ a mean-field framework. Within such an approach self-consistent-field(scf) ensembles of random Hamiltonians naturally arise. These ensembles may exhibit novel, as of yet unstudied critical behavior and their study can thus lead to a more complete understanding of universality classes in disordered systems. The investigation of scf-ensembles is a very challenging endeavor. The difficulty is that each disorder configuration requires to find its own self-consistent fields. The solution of the scf-cycle is very difficult to do with analytical techniques. But also numerically it is demanding already at moderate systemsizes of a few thousand sites. Presumably, this is the main reason why numerical studies of scf-ensembles have been performed infrequently in the past, despite of their obvious fundamental relevance. Thus motivated, we have developed a state-of-the-art implementation of the scf-problem following a Kernel-Polynomial-Method(KPM) approach. With it the system sizes that we can address at an affordable numerical cost exceed the ones of prior studies by two orders of magnitude. The interplay of disorder induced quantum-interference and mean-field interactions can be studied on length scales that exceed the lattice constant by two orders of magnitude. Throughout this thesis we apply our code to disordered superconductors with screened Coulomb interaction. We investigate the Bogoliubov-deGennes(BdG) theory of inhomogeneous superconductors focussing on s-wave pairing in thin films. To understand how superconductors are affected by disorder it is instructive to first study the effect of a single impurity. Furthermore we are motivated by a collaboration with the experimental group of Wulf Wulfhekel at the Karlsruhe Institute of Technology to study such a system. In this group scanning tunneling microscopy(STM) measurements of superconducting bulk Al(111) around a Fe impurity have been conducted. The self-consistency requirement complicates the description of the response to an impurity at the surface of a bulk superconductor. An analytical formalism remains unknown. With our numerical simulations we find an excellent agreement of the response to the impurity with the experimental findings. The main part of the thesis focusses on homogeneously disordered s-wave superconductors with screened Coulomb interaction. These systems may exhibit a direct Superconductor-Insulator-Transition (SIT). Despite considerable effort in the study of these systems, the current situation is not fully satisfying: On the one hand, computational mean-field studies of the Hubbard model have been limited to system sizes L that do not allow to study the most interesting regime of length scales where the coherence length strongly exceeds the lattice spacing. While analytical approaches, on the other hand, operate in this regime, they rely on partial self-consistency in order to become tractable. Our computational machinery allows us to cover the full parameter space from the extreme regimes, which have been addressed computationally before, to the analytically tractable weak coupling limit. Our software package allows us to study disordered superconductors numerically for the first time on mesoscopic scales considerably exceeding the lattice constant. For example (i)we observe a non-monotonic behavior with disorder strength of the superconducting correlation length already at intermediate interaction strengths; (ii)we for the first time investigate the fluctuations of the LDoS in regimes from weak to strong disorder, where the fluctuations are particularily pronounced. Furthermore in a collaboration with Igor Burmistrov of the Landau institute, we determine a low disorder regime, in which a quantitative agreement with an analytical description of the LDoS fluctuations is found; (iii) we for the first time identify a regime, where the gap is strongly enhanced (up to ∼ 20%) by disorder. We also pay a special attention to the sensitivity of the behavior of observables to approximations made in the self-consistency procedure. For instance we find that island formation when observed in moderate parameter regions is a characteristic hallmark of full self-consistency. It escapes partial (“energy-only”) self-consistent schemes.