Determination of gap structure of triplet superconductors from field-dependent Knight shift measurements

Abstract

We analyze the spin susceptibility of spin-triplet superconductors from the zero-field to finite-field regimes, with emphasis on its implications for Knight-shift measurements. In the zero-field limit, we review the general expression for the static spin susceptibility and highlight the universal zero-temperature sum rule, $\sum_i \chi_{ii}(T=0)=2\chi^N$, which constrains the residual susceptibility components for any triplet state. Using representative isotropic, helical, and chiral $\vec{d}$-vectors, we illustrate how the Knight shift encodes the spin configuration of the order parameter and show that the sum rule remains robust even for anisotropic Fermi surfaces. We then incorporate magnetic field effects through a semiclassical Doppler shift of quasiparticle energies in the vortex state. The resulting field dependence of the susceptibility - including both longitudinal (Knight-shift) and transverse magnetic susceptibility components - provides a sensitive probe of nodal directions and the momentum dependence of the $\vec{d}$-vector. Applying this framework to UTe$_2$, we demonstrate how the distinct irreducible representations allowed by orthorhombic symmetry can be differentiated by their field-dependent susceptibility.