Bogoliubov sum rule and the Knight-shift ellipsoid in spin-locked superconductors

Abstract

We establish an exact Bogoliubov sum rule for any Hermitian single-particle operator $O$: at each momentum, its particle-hole and particle-particle matrix-element weights sum to the single-particle trace $\mathrm{Tr}_{s}(O^{2})$. The result follows solely from Hilbert--Schmidt-norm invariance under a canonical Bogoliubov transformation and contains neither excitation-energy denominators nor occupations; physical response identities therefore require additional assumptions. At zero field, a fully gapped helicity-diagonal state with both helicity sheets present and a spin-orbit splitting asymptotically larger than the gap obeys $\chi_{\mu\nu}(0)/\chi_N=\delta_{\mu\nu}-\Pi_{\mu\nu}+o(1)$, where $\chi_N$ is the normal-state Pauli susceptibility and $\Pi=\langle\hat{\mathbf n}_{\mathbf k}\hat{\mathbf n}_{\mathbf k}\rangle_{\rm FS}$ is the Fermi-surface average of the unit spin-locking texture. The eigenvalues of $\Pi$ form a simplex, while the normalized spin Knight-shift tensor defines an ellipsoid whose semi-axes are the residual principal responses. Full cubic invariance of both the superconducting state and locking texture fixes $\Pi=\mathbb I/3$ and hence $\chi(0)/\chi_N=2\mathbb I/3+o(1)$; cubic crystal symmetry alone does not. For zero-field $s$-wave pairing in the reference-Fermi-surface regime, we obtain the exact closed-form kernel $F_s(\lambda)=1-\operatorname{asinh}\lambda/[\lambda\sqrt{1+\lambda^2}]$, valid for arbitrary $\lambda=|\mathbf g|/\Delta$. In a finite Zeeman field, the zero-field helicity reduction generally fails, so the equilibrium magnetization and differential response require a self-consistent BdG calculation rather than a field-dependent locking-tensor substitution. Applied to the $^{75}$As data on K$_2$Cr$_3$As$_3$, the framework identifies a field-dependent axial suppression pattern at $8$--$16$~T.