# Pair-density-wave superconductivity of faces, edges and vertices in systems with imbalanced fermions

## TL;DR

This paper investigates boundary effects in FFLO superconductors with imbalanced fermions, revealing a sequence of phase transitions from bulk to edges and vertices, using both analytical and numerical methods.

## Contribution

It extends the Ginzburg-Landau formalism for FFLO systems by including higher order terms and characterizes boundary-induced phase transitions in imbalanced fermionic superconductors.

## Key findings

- Bulk FFLO state with boundary modulation patterns
- Sequential phase transitions: bulk normal, edge superconducting, vertex superconducting
- Numerical confirmation of phase transition sequence

## Abstract

We describe boundary effects in superconducting systems with Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconducting instability, using Bogoliubov-de-Gennes and Ginzburg-Landau (GL) formalisms. First, we show that in dimensions larger than one the standard GL functional formalism for FFLO superconductors is unbounded from below. This is demonstrated by finding solutions with zero Laplacian terms near boundaries. We generalize the GL formalism for these systems by retaining higher order terms. Next, we demonstrate that a cuboid sample of a superconductor with imbalanced fermions at a mean-field level has a sequence of the phase transitions. At low temperatures it forms Larkin-Ovchinnikov state in the bulk but has a different modulation pattern close to the boundaries. When temperature is increased the first phase transition occurs when the bulk of the material becomes normal while the faces remain superconducting. The second transition occurs at higher temperature where the system retains superconductivity on the edges. The third transition is associated with the loss of edge superconductivity while retaining superconducting gap in the vertices. We obtain the same sequence of phase transition by numerically solving the Bogoliubov-de Gennes model.