# Phase stiffness in an antiferromagnetic superconductor

## TL;DR

This paper investigates how antiferromagnetic order suppresses phase stiffness in superconductors, deriving formulas and performing numerical simulations to understand the effects of magnetic order on superconducting properties.

## Contribution

It provides a general expression for phase stiffness in coexisting superconductivity and spiral magnetism, including a simple formula near half-filling and numerical results for the Hubbard model.

## Key findings

- Phase stiffness is influenced by magnetic quasi-particle bands.
- Electron-hole asymmetry affects phase stiffness magnitude.
- Ground state pairing gap determines T_c^{KT} in hole-doped regime.

## Abstract

We analyze the suppression of the phase stiffness in a superconductor by antiferromagnetic order. The analysis is based on a general expression for the phase stiffness in a mean-field state with coexisting spin-singlet superconductivity and spiral magnetism. Neel order is included as a special case. Close to half-filling, where the pairing gap is much smaller than the magnetic gap, a simple formula for the phase stiffness in terms of magnetic quasi-particle bands is derived. The phase stiffness is determined by charge carriers in small electron or hole pockets in this regime. The general analysis is complemented by a numerical calculation for the two-dimensional Hubbard model with nearest and next-to-nearest neighbor hopping amplitudes at a moderate interaction strength. The resulting phase stiffness exhibits a striking electron-hole asymmetry. In the ground state, it is larger than the pairing gap on the hole-doped side, and smaller for electron doping. Hence, in the hole-doped regime near half-filling the ground state pairing gap sets the scale for the Kosterlitz-Thouless temperature T_c^{KT}, while in the slightly electron-doped regime T_c^{KT} is determined essentially by the ground state phase stiffness.