Bogoliubov sum rules and the Knight-shift ellipsoid in noncentrosymmetric superconductors

TL;DR

This paper derives a universal identity for the residual Knight shift in noncentrosymmetric superconductors, linking it to Fermi-surface averages and classifying pairing textures via a geometric ellipsoid.

Contribution

It introduces a Bogoliubov sum rule-based identity that determines the Knight shift tensor independently of pairing symmetry and Fermi-surface shape, and develops experimental protocols.

Findings

The residual Knight shift tensor is fully determined by a single Fermi-surface average.

The Knight-shift ellipsoid classifies all pairing textures in noncentrosymmetric superconductors.

Application to experimental data reveals specific spin-locking textures and spin-fluctuation signatures.

Abstract

We show that the residual $T=0$ Knight shift of a noncentrosymmetric superconductor in the strong-locking regime is completely determined by a single Fermi-surface average -- the projector $\Pi_{\mu\nu}=\langle\hat n_{\mu}\hat n_{\nu}\rangle_{\rm FS}$ of the spin-locking direction $\hat n_{\mathbf{k}}$ -- giving the tensor identity $\chi_{\mu\nu}(0)=\chi_{N}[\delta_{\mu\nu}-\Pi_{\mu\nu}]$ independently of pairing symmetry, gap magnitude, and Fermi-surface shape. Because $\mathrm{Tr}\,\Pi=1$, the three principal Knight shifts at $T=0$ lie on a two-dimensional simplex of locking textures, the \emph{Knight-shift ellipsoid}, whose vertices, edges, and interior classify every canonical pairing class. The identity follows from a Bogoliubov sum rule, $\sum|M_{ph,O}|^{2}+\sum|M_{pp,O}|^{2}=\mathrm{Tr}_{s}(O^{2})$, valid at every momentum for every Hermitian single-particle operator $O$ as the BdG-doubled form of unitary invariance. Around the central theorem we develop controlled departures (a closed-form $s$-wave SOC interpolation, a finite-field strong-locking identity), a dynamical counterpart (a spin Ferrell--Glover--Tinkham sum rule and a rigorous vanishing-projection theorem for $1/T_{1}$), and a decoupled-pocket multiband baseline, packaged into six experimental protocols. Applied to the $^{75}$As NMR data on K$_{2}$Cr$_{3}$As$_{3}$, the observed ellipsoid sits at the oblate-axial vertex $(0,0,1)$ and saturates the trace bound; the decoupled-pocket SOC-texture baseline is excluded by $\sim 0.5$ in normalized units, requiring a common $\hat c$-axis locking on all three pockets, and the suppression of $1/T_{1}\parallel\hat c$ is identified, via the vanishing-projection theorem, as a fingerprint of finite-$\mathbf{q}$ ferromagnetic spin-fluctuation gap formation.