Spin-orbit Coupling and Multiple Phases in Spin-triplet Superconductor Sr$_2$RuO$_4$

TL;DR

This paper investigates the role of spin-orbit coupling in the multiple superconducting phases of Sr$_2$RuO$_4$, revealing how it influences phase transitions and the rich phase diagram in magnetic fields.

Contribution

It provides a theoretical analysis of spin-orbit coupling effects on multiple SC phases in Sr$_2$RuO$_4$ using a three-orbital Hubbard model and Ginzburg-Landau theory.

Findings

Spin-orbit coupling is small when the $ ext{γ}$-band dominates superconductivity.

Multiple SC transitions occur under magnetic fields, leading to complex phase diagrams.

Experimental indications support the existence of multiple phases in Sr$_2$RuO$_4$.

Abstract

We study the spin-orbit coupling in spin-triplet Cooper pairs and clarify multiple superconducting (SC) phases in Sr$_2$RuO$_4$. Based on the analysis of the three-orbital Hubbard model with atomic LS coupling, we show some selection rules of the spin-orbit coupling in Cooper pairs. The spin-orbit coupling is small when the two-dimensional $\gamma$-band is the main cause of the superconductivity, although the LS coupling is much larger than the SC gap. Considering this case, we investigate multiple SC transitions in the magnetic fields for both H // [001] and H // [100] using the Ginzburg-Landau theory and the quasi-classical theory. Rich phase diagrams are obtained because the spin degree of freedom in Cooper pairs is not quenched by the spin-orbit coupling. Experimental indications for the multiple phases in Sr$_2$RuO$_4$ are discussed.