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

TL;DR

The paper develops a theory for the spin susceptibility of spin-triplet superconductors, employing a mean-field Hamiltonian with a unitary $d$-vector and a Kubo-bubble formalism to compute the static, uniform susceptibility. It then introduces a semiclassical Doppler-shift treatment of quasiparticles in the vortex state to obtain field-dependent susceptibility components, showing that the finite-field response encodes nodal directions and the momentum-space structure of the $d$-vector. Applying the framework to UTe$_2$ with a two-orbital tight-binding model, the authors predict a distinctive field-dependent susceptibility hierarchy for the $B_{3u}$ order parameter, which can be used to differentiate it from $B_{1u}$ and $B_{2u}$ given the Fermi-surface topology. The work proposes a practical protocol combining Knight-shift and magnetotropic measurements to identify the superconducting order parameter in orthorhombic triplet superconductors at low fields.

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, $\sum_i χ_{ii}(T=0)=2χ^N$, 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.

Figures (7)

• Figure 1: Spin susceptibilities for common triplet states: (a) Isotropic Balian-Werthamer state, (b) two distinct helical states and (c) chiral state. The suppression of the spin susceptibility shows imprints of the direction of the $\vec{d}$-vector on the Fermi surface (insets), but in all cases the three components add up to $2\chi^N$ in the limit $T\rightarrow 0$.
• Figure 2: (a): $\vec{d}(\mathbf{k}) = \hat{x} k_y - \hat{y} k_x$ with anisotropic DOS $m_x=2m_y$. (b) $\vec{d}(\mathbf{k}) = \hat{x} k_x + \hat{y} k_y$ with anisotropic DOS $m_x=2m_y$. The dashed lines indicate isotropic case $m_x=m_y$. DOS anisotropy gives different susceptibility components for states with identical susceptibility components when isotropic, but the sum rule still holds.
• Figure 3: Coordinates show direction of node and direction of increasing $d_i$ for (a)-(c). (a) field direction $j$ that is least affected by Doppler shift. (b) field direction $j_\perp$ where $|\nabla(\Delta\mathbf{v}_F(\mathbf{k})\cdot \mathbf{p}_s)\cdot \nabla d_i(\mathbf{k})|$ minimum. (c) field direction $j_\parallel$ where $|\nabla(\Delta\mathbf{v}_F(\mathbf{k})\cdot \mathbf{p}_s)\cdot \nabla d_i(\mathbf{k})|$ maximum. (d) Visualization of $\Delta v_F$ and gap magnitude around node.
• Figure 4: Quasiparticle excitation region (QP region) for example $B_{1u}$ state when field is perpendicular to nodal direction. Without $\Delta v_F$, the QP region is inside the gray boundary. With $\Delta v_F$, the QP region grows in perpendicular directions for $j_\perp$ (green) and $j_\parallel$ (red) field configurations. The heat map indicates the magnitude of $\Hat{d_i}$.
• Figure 5: Field dependence of susceptibility for $B_{1u}$ state. When $H\rightarrow 0$, sum rule preserved but with finite field $\sum_i\chi_{ii}(H\!\parallel\!i)/\chi^N>2$. $\chi_{xx}$ and $\chi_{yy}$ both follow the field dependence $H\!\parallel\!\mathbf{k}_n<H\!\parallel\! j_\perp\lesssim H\!\parallel\! j_\parallel$. $\Delta\chi_{zz}$ is negative because at first order $d_z=0$ for $B_{1u}$. Here the prefactors are set to $p_1=p_2=p_3=1$, but the qualitative trends of $\chi_{xx}$, $\chi_{yy}$, and $\chi_{zz}$ described above, which determine the gap structure, are insensitive to these prefactors. Although the 3 prefactors are equal, the $d_z=k_x k_y k_z$ has $|\hat{d_z}|^2$ integrates to orders of magnitude smaller over the Fermi surface than the other 2 directions and therefore $\chi_{zz}$ shows almost no reduction from normal state value at $T=0, H\rightarrow 0$.