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

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 $Π{μν}=\langle\hat nμ\hat n_ν\rangle_{\rm FS}$ of the spin-locking direction $\hat n_{\mathbf{k}}$ -- giving the tensor identity $χ{μν}(0)=χ{N}[δ{μν}-Π{μν}]$ independently of pairing symmetry, gap magnitude, and Fermi-surface shape. Because $\mathrm{Tr}\,Π=1$, 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.