# The Order Parameter for the Superconducting Phases of UPt$_3$

## TL;DR

This paper reviews theories for UPt3's superconducting phases, highlighting a viable model involving an odd-parity E2u order parameter coupled to antiferromagnetism, explaining key experimental observations.

## Contribution

Proposes a model with an E2u order parameter coupled to antiferromagnetism that explains UPt3's phase diagram and properties, refining previous theories.

## Key findings

- The phase diagram and critical field anisotropy are explained by the E2u model.
- The coupling to antiferromagnetism accounts for the tetracritical point.
- Low-temperature power laws match predictions of the E2u order parameter.

## Abstract

I review the principal theories that have been proposed for the superconducting phases of UPt$_3$. The detailed H-T phase diagram places constraints on any theory for the multiple superconducting phases. I comment on the limitations of the models proposed so far for the superconducting phases of UPt$_3$. I also find that a theory in which the order parameter belongs to an orbital 2D representation coupled to a SBF is a viable model for the phases of UPt$_3$, based on the existing body of experimental data. Specifically, I show that (1) the existing phase diagram (including an apparent tetracritical point for all field orientations), (2) the anisotropy of the upper critical field over the full temperature range, (3) the correlation between superconductivity and basal plane antiferromagnetism and (4) low-temperature power laws in the transport and thermodynamic properties can be explained qualitatively, and in many respects quantitatively, by an odd-parity, E$_{2u}$ order parameter with a pair spin projection of zero along the ${\bf c}$-axis. The coupling of an AFM moment to the superconducting order parameter acts as a symmetry breaking field (SBF) which is responsible for the apparent tetracritical point, in addition to the zero-field double transition. The results presented here for the E$_{2u}$ representation are based on an analysis of the material parameters calculated within BCS theory for the 2D representations, and a refinement of the SBF model of Hess, et al. [J. Phys. Condens. Matter, 1, 8135 (1989)]. I also discuss possible experiments to test the symmetry of the order parameter.