Bogoliubov sum rules and the Knight-shift ellipsoid in noncentrosymmetric superconductors
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866914572534808576 |
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| author | Zhou, Yi |
| author_facet | Zhou, Yi |
| contents | 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 $Π_{μν}=\langle\hat n_μ\hat n_ν\rangle_{\rm FS}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_17030 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bogoliubov sum rules and the Knight-shift ellipsoid in noncentrosymmetric superconductors Zhou, Yi Superconductivity Strongly Correlated Electrons 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 $Π_{μν}=\langle\hat n_μ\hat n_ν\rangle_{\rm FS}$ 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. |
| title | Bogoliubov sum rules and the Knight-shift ellipsoid in noncentrosymmetric superconductors |
| topic | Superconductivity Strongly Correlated Electrons |
| url | https://arxiv.org/abs/2605.17030 |