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 $\Pi =⟨{\hat{\mathbf{n}}}_{\mathbf{k}}{\hat{\mathbf{n}}}_{\mathbf{k}}{⟩}_{\mathrm{FS}}$ 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 $F_s(\lambda )=1-\mathrm{asinh}\lambda /[\lambda \sqrt{1+{\lambda }^2}]$, 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.