Self-consistent treatment of disorder-induced interacting criticality

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The project goal is to achieve an improved understanding of self-consistent field (scf) theories of disorder-induced interacting criticality. The corresponding ensembles of scf-random Hamiltonians can be categorised acording to the Altland-Zirnbauer symmetry classification.Remarkably, the scf-condition can induce extra correlations among single-particle states of the scf-Hamiltonian that can result in new phase diagram. Physically, scf-ensembles are important because they provide the reference point for a perturbative analysis of interaction effects in disordered fermion systems. Also they are important because understanding the mean-field behavior is a prerequisite for revealing the effect of interaction-mediated quantum fluctuations. Finally, scf-random Hamiltonians provide a rich and largely unexplored field for fundamental research in mathematical physics.The focus of this project is on the Hartree-Fock theory and the Boguluibov-deGennes theory of disordered electrons in the presence of repulsive or pairing interactions. We will solve the corresponding problems by combined numerical and theoretical efforts. On the one hand, we will implement and improve numerical codes for these problems employing the kernel-polynomial-method. Thus, extensive numerical data can be generated for spin-full and spin-less models of very large system sizes. On the other hand, we will develop the analytical theory (of the nonlinear sigma model type) for scf-Hamiltonians in the weak disorder regime. In a concerted effort the simulation data will be contrasted against and analysed with the help of the theoretical results. Both approaches, numerics and field theory, complement each other ideally. Numerics allows to treat even strong-disorder effects exactly, but can only generate a limited understanding per se, because for this analytical formulae are required. Such formulae can be generated, e.g., with field-theoretic approaches; hower, with good control this can be achieved only in a relatively small sector parameter space. By combining both approaches, and only by combining, a deeper understanding of the physics in the full parameter space can be achieved. Both project teams have already successfully collaborated in this way before and the experiences made in this way are very encouraging.