Possible Pairing Symmetry of Three-dimensional Superconductor UPt$_3$ -- Analysis Based on a Microscopic Calculation --

TL;DR

This study investigates possible pairing symmetries in UPt$_3$ using microscopic calculations, finding that 3D dispersion suppresses ferromagnetic fluctuation-mediated states and stabilizes certain p-wave pairings consistent with experimental data.

Contribution

It provides a detailed analysis of pairing symmetries in UPt$_3$ through third order perturbation theory, highlighting the impact of 3D dispersion on pairing stability.

Findings

3D dispersion destroys f-wave pairing caused by ferromagnetic fluctuations.

p_z-wave pairing is stabilized by large c-axis hopping, nearly degenerate with p_x/p_y states.

Vertex corrections are crucial for stabilizing p-wave states far from half-filling.

Abstract

Stimulated by the anomalous superconducting properties of UPt$_3$, we investigate the pairing symmetry and the transition temperature in the two-dimensional(2D) and three-dimensional(3D) hexagonal Hubbard model. We solve the Eliashberg equation using the third order perturbation theory with respect to the on-site repulsion $U$. As results of the 2D calculation, we obtain distinct two types of stable spin-triplet pairing states. One is the $f$-wave(B$_1$) pairing around $n = 1.2$ and in a small $U$ region, which is caused by the ferromagnetic fluctuation. Then, the other is the $p_x$(or $p_y$)-wave(E$_1$) pairing in large $U$ region far from the half-filling ($n = 1$) which is caused by the vertex corrections only. However, we find that the former $f$-wave pairing is destroyed by introduced 3D dispersion. This is because the 3D dispersion breaks the favorable structures for the $f$-wave pairing such as the van Hove singularities and the small pocket structures. Thus, we conclude that the ferromagnetic fluctuation mediated spin-triplet state can not explain the superconductivity of UPt$_3$. We also study the case of the pairing symmetry with a polar gap. This $p_z$-wave(A$_1$) is stabilized by the large hopping integral along c-axis $t_z$. It is nearly degenerate with the suppressed $p_x$(or $p_y$)-wave(E$_1$) in the best fitting parameter region to UPt$_3$ ($1.3 \le t_z \le 1.5$). These two p-wave pairing states exist in the region far from the half-filling, in which the vertex correction terms play crucial roles like the case in Sr$_2$RuO$_4$.