Department of Theoretical Physics, Institute "Jožef Stefan"
Subgap states: the Richardson-model perspective
Whereas a conventional bulk superconductor is perfectly well described by the mean-field Bardeen-Cooper-Schrieffer theory of superconductivity, in the limit of ultra-small grains of superconducting material a more appropriate description is in terms of the interacting Richardson's (picket-fence) model and its extensions. This charge-conserving Hamiltonian consists of a kinetic energy term and an interacting pairing term between all orbitals, usually with a constant coefficient, in which case the model is integrable and its complete solution can be reduced to numerically solving a system of coupled algebraic equations. In the presence of an impurity carrying a local magnetic moment, the integrability is broken by the exchange scattering that splits the Cooper pairs. We show that the corresponding Hamiltonian admits a compact representation in terms of small matrix product operators, which makes it possible to solve this class of problems accurately and rather efficiently using the density matrix renormalization group (DMRG) implemented using the tensor-network formalism.
I will discuss the pros and cons of the Richardson-model description of systems of coupled superconducting island and quantum dots, the effects of the Coulomb repulsion (charging) term on the superconducting island on the phase diagrams and on the nature of the (subgap) states, the basic properties of the two-channel version of the problem, as well as other possible extensions of the model (level-dependent pairing, spin-orbit coupling, two impurities).