# Interactions, superconducting $T_c$, and fluctuation magnetization for two coupled dots in the crossover between the Gaussian Orthogonal and Unitary ensembles

## TL;DR

This paper analyzes the effects of crossover regimes between Gaussian Orthogonal and Unitary ensembles on two coupled quantum dots, revealing nonmonotonic behavior of critical temperature and fluctuation magnetization influenced by magnetic flux.

## Contribution

It introduces a diagrammatic approach to describe Green's functions in crossover regimes and investigates the impact on superconducting transition and magnetization in coupled quantum dots.

## Key findings

- Critical temperature exhibits nonmonotonic dependence on magnetic flux.
- Fluctuation magnetization can be diamagnetic or paramagnetic.
- Separate scaling functions describe diffuson and cooperon contributions.

## Abstract

We study a system of two quantum dots connected by a hopping bridge. Both the dots and connecting region are assumed to be in universal crossover regimes between Gaussian Orthogonal and Unitary ensembles. Using a diagrammatic approach appropriate for energy separations much larger than the level spacing we obtain the ensemble-averaged one- and two-particle Green's functions. It turns out that the diffuson and cooperon parts of the two-particle Green's function can be described by separate scaling functions. We then use this information to investigate a model interacting system in which one dot has an attractive s-wave reduced Bardeen-Cooper-Schrieffer interaction, while the other is noninteracting but subject to an orbital magnetic field. We find that the critical temperature is {\it nonmonotonic} in the flux through the second dot in a certain regime of interdot coupling. Likewise, the fluctuation magnetization above the critical temperature is also nonmonotonic in this regime, can be either diamagnetic or paramagnetic, and can be deduced from the cooperon scaling function.