Spin susceptibility for orbital-singlet Cooper pair in the three-dimensional Sr$_2$RuO$_4$ superconductor

TL;DR

This study investigates the spin susceptibility behavior in three-dimensional Sr$_2$RuO$_4$ superconductors with orbital-singlet pairing, revealing how temperature and spin-orbit coupling influence susceptibility and aiding in identifying pairing symmetry.

Contribution

It provides a detailed analysis of spin susceptibility in orbital-singlet pairings of Sr$_2$RuO$_4$, highlighting the effects of spin-orbit coupling and temperature on pseudospin structures.

Findings

Spin susceptibility decreases with temperature regardless of magnetic field direction.

Susceptibility reduction is 5-10% along the z and x axes due to spin-orbit coupling.

Results can help determine the pairing symmetry via NMR measurements.

Abstract

We study the spin susceptibility of the orbital-singlet pairings, including the spin-triplet/orbital-singlet/$s$-wave $E_g$ representation proposed by Suh \textit{et al}., [H.\ G.\ Suh \textit{et al}., Phys.\ Rev.\ Research \textbf{2}, 032023 (2020)], for a three-orbital model of superconducting Sr$_2$RuO$_4$ in three dimensions. For the pseudospin-singlet states represented in the band basis, the spin susceptibility decreases when reducing the temperature, irrespective of the direction of the applied magnetic fields, even if they are spin-triplet/orbital-singlet pairings in the spin-orbital space. However, because the pseudospin-triplet \textbf{d}-vector in the band basis is not completely aligned in the $xy$-plane (along $z$-axis) owing to the strong atomic spin-orbit coupling, the spin susceptibility for spin-singlet/orbital-singlet/odd-parity pairings is reduced around $5$-$10$ percent with the decrease of the temperature along the $z$ ($x$) axis. We can determine the symmetry of the pseudospin structure of the Cooper pair by the temperature dependence of the spin susceptibility measured by nuclear magnetic resonance experiments. Our obtained results serve as a guide to determine the pairing symmetry of Sr$_2$RuO$_4$.