Superconductivity in the attractive Hubbard model: functional renormalization group analysis

Abstract

We present a functional renormalization group analysis of superconductivity in the ground state of the attractive Hubbard model on a square lattice. Spontaneous symmetry breaking is treated in a purely fermionic setting via anomalous propagators and anomalous effective interactions. In addition to the anomalous interactions arising already in the reduced BCS model, effective interactions with three incoming legs and one outgoing leg (and vice versa) occur. We accomplish their integration into the usual diagrammatic formalism by introducing a Nambu matrix for the effective interactions. From a random-phase approximation generalized through use of this matrix we conclude that the impact of the 3 + 1 effective interactions is limited, especially considering the effective interactions important for the determination of the order parameter. The exact hierarchy of flow equations for one-particle irreducible vertex functions is truncated on the two-particle level, with higher-order self-energy corrections included in a scheme proposed by Katanin. Using a parametrization of effective interactions by patches in momentum space, the flow equations can be integrated numerically to the lowest scales without encountering divergences. Momentumshell as well as interaction-flow cutoff functions are used, including a small external field or a large external field and a counterterm, respectively. Both approaches produce momentum-resolved order parameter values directly from the microscopic model. The size of the superconducting gap is in reasonable agreement with expectations from other studies.

Interaction-induced local moments in parallel quantum dots within the functional renormalization group approach

“…To describe the SFL state we decrease the magnetic field continuously during the flow, similarly to a counterterm extension of the fRG [33,34]. To this end, we introduce additional term σχ Λ /2 in the propagator of the quantum dots…”

“…This scheme is similar to a counterterm technique, used earlier to treat the first order quantum phase transitions in lattice fermionic systems [33,34] within fRG approach. This technique was not however applied to description of quantum dot structures.…”

Efficient parametrization of the vertex function,Ωscheme, and thet,t′Hubbard model at van Hove filling

“…Using this modified scheme the exact solution of a mean-field model for superconductivity is obtained in the thermodynamic limit [14,26]. Furthermore, the scheme has been recently applied to the attractive Hubbard model [5]. Due to the emergence of an effective superconducting gap, which regularizes infrared divergencies, all scales can be integrated out.…”

Hierarchical parallelisation of functional renormalisation group calculations — hp-fRG

“…In recent times, functional renormalisation group methods have been established in quantum condensed matter physics as a powerful tool to detect and treat various phenomena which arise in correlated electron systems. In particular, they can be used as an unbiased detector for competing correlations and quite a range of results is available for various models, parameter sets and specifically targeted observables [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]. A pioneering work in this context is given in [12], for a comprehensive review see [13].…”