# Vortex core order and field-driven phase coexistence in the attractive Hubbard model

## TL;DR

This paper investigates how magnetic fields induce coexistence of charge density wave order and superconductivity in the attractive Hubbard model, revealing a transition to a supersolid phase and mapping the phase diagram.

## Contribution

It demonstrates the formation and evolution of vortex-induced charge order and identifies a field-driven supersolid phase in the attractive Hubbard model.

## Key findings

- Vortices seed charge density wave regions.
- Charge-ordered vortex cores overlap at high flux.
- Superconductivity transitions to a charge-ordered state.

## Abstract

Superconductivity occurs in the proximity of other competing orders in a wide variety of materials. Such competing phases may reveal themselves when superconductivity is locally suppressed by a magnetic field in the core of a vortex. We explore the competition between superconductivity and charge density wave order in the attractive Hubbard model on a square lattice. Using Bogoliubov-deGennes mean field theory, we study how vortex structures form and evolve as the magnetic flux is tuned. Each vortex seeds a CDW region whose size is determined by the energy cost of the competing phase. The vortices form a lattice whose lattice parameter shrinks with increasing flux. Eventually, their charge-ordered vortex cores overlap, leading to a field-driven coexistence phase exhibiting both macroscopic charge order and superconductivity -- a `supersolid'. Ultimately, superconductivity disappears via a first-order phase transition into a purely charge ordered state. We construct a phase diagram containing these multiple ordered states, using $t'$, the next-nearest neighbour hopping, to tune the competition between phases.