Vakaryuk and Leggett [Phys. Rev. Lett. 103, 057003 (2009)] predicted that a half-quantum vortex (HQV) in an equal-spin-pairing (ESP) superfluid — a topological defect whose order-parameter phase winds by π\pi with a compensating rotation of the d\boldsymbol{d} vector, trapping half a flux quantum — must carry, in thermal equilibrium, a nonzero spin polarization generated by an effective Zeeman field BeffB_{\mathrm{eff}} of order Φ0/πR2\Phi_0/\pi R^2 (about 10 G for a 1 μm Sr2_2RuO4_4 ring), additive to the Zeeman polarization and absent in ordinary vortices, hence a clean HQV signature for NMR-based detection. The prediction is quantitatively incomplete for Sr2_2RuO4_4: the Fermi-liquid Landau parameters F1F_1 and Z1Z_1 — equivalently the zero-temperature ratio ρsp/ρs\rho_{\mathrm{sp}}/\rho_{\mathrm{s}} of spin to charge superfluid density — and the constants gSg_S, m∗m^*, χ\chi entering BeffB_{\mathrm{eff}} are unknown; the calculation idealizes the sample as a core-free thin annulus with in-plane d\boldsymbol{d} vector and omits spin-orbit coupling; and the self-inductance stability constraint has not been mapped over the experimentally relevant (R,d,λL,ρsp/ρs)(R, d, \lambda_L, \rho_{\mathrm{sp}}/\rho_{\mathrm{s}}) space. This problem asks for the quantitative completion: determine the missing material and geometrical inputs and compute, with controlled uncertainties, the HQV stability diagram and the magnitude, sign, and flux dependence of BeffB_{\mathrm{eff}} for realistic mesoscopic Sr2_2RuO4_4 samples, so that the spin-polarization signature becomes a decisive experimental test of ESP pairing.

Background

In a spin-triplet superconductor, Cooper pairs carry spin one and the order parameter is described by a vector d\boldsymbol{d} whose direction fixes the spin quantization axis of the pairs. In an equal-spin-pairing (ESP) state the condensate is a superposition of pairs with both spins up and pairs with both spins down (along the axis perpendicular to d\boldsymbol{d}), so a magnetic field does not break pairs; superfluid 3^3He-A is the established realization, and the layered perovskite Sr2_2RuO4_4 (Tc≈1.5T_c \approx 1.5 K) was identified from the late 1990s as the leading candidate ESP superconductor (muon-spin-rotation evidence for broken time-reversal symmetry and early oxygen-17 Knight-shift data — muon-spin rotation being a local probe of internal magnetization, and the Knight shift the nuclear magnetic resonance shift that tracks the electronic spin susceptibility), although this identification is now contested (see below). In a doubly connected superconductor such as a ring, the magnetic flux threading the hole is quantized in units of the flux quantum Φ0=hc/2e\Phi_0 = hc/2e; the combined object of trapped flux plus circulating supercurrent is called a fluxoid. Because the ESP order parameter is a product of a phase and a d\boldsymbol{d}-vector orientation, a half-quantum vortex (HQV) is possible: the phase winds by π\pi around the hole while d\boldsymbol{d} rotates by π\pi, keeping the triplet order parameter single-valued; the HQV traps half a flux quantum and carries a circulating spin current. Such objects were proposed in the 1970s for superfluid 3^3He-A and are of special interest because HQV cores in chiral states can bind Majorana zero modes relevant to topological quantum computation. The stability problem is the central obstacle. Electromagnetic currents are screened over the London penetration depth λL\lambda_L, but the spin current of an isolated HQV is not screened and has a logarithmically divergent energy cost in the bulk. Chung, Bluhm and Kim showed that mesoscopic annuli (radius RR, wall thickness dd) can stabilize single HQVs subject to the self-inductance constraint ρsp/ρs<(1+Rd/2λL2)−1\rho_{\mathrm{sp}}/\rho_{\mathrm{s}} < (1 + Rd/2\lambda_L^2)^{-1}, where ρsp\rho_{\mathrm{sp}} is the spin superfluid density (the stiffness for counterflow of the two spin components) and ρs\rho_{\mathrm{s}} the ordinary (charge) superfluid density. In Landau Fermi-liquid theory — the standard description of interaction corrections in a degenerate Fermi system — the zero-temperature ratio is ρsp/ρs=(1+Z1/12)/(1+F1/3)\rho_{\mathrm{sp}}/\rho_{\mathrm{s}} = (1 + Z_1/12)/(1 + F_1/3), where the Landau parameters F1F_1 and Z1Z_1 renormalize charge and spin currents respectively (m∗/m=1+F1/3m^*/m = 1 + F_1/3). In 3^3He-A this ratio is known to lie well below 1; in Sr2_2RuO4_4 it is unknown. Vakaryuk and Leggett (2009) treated the HQV in an annular ESP superconductor by a variational BCS-like wave function with a spin-dependent boost (the two spin species acquire different winding numbers, ℓ↑≠ℓ↓\ell_\uparrow \ne \ell_\downarrow). The weak-coupling energy alone leaves the HQV at best degenerate with a full vortex; Fermi-liquid corrections stabilize it when ρsp/ρs<1\rho_{\mathrm{sp}}/\rho_{\mathrm{s}} < 1. Their new result is that the velocity mismatch between the spin species shifts the up- and down-spin chemical potentials, so that in thermal equilibrium the HQV carries a nonzero spin polarization, equivalent to an effective Zeeman field

Beff=−ℏ22m∗R2(gSμB)−1 ℓspℓsΦ(1+F13+Z112),B_{\mathrm{eff}} = -\frac{\hbar^2}{2m^*R^2}(g_S\mu_B)^{-1}\,\ell_{\mathrm{sp}}\ell_{s\Phi}\left(1+\frac{F_1}{3}+\frac{Z_1}{12}\right),

where gSg_S is the gyromagnetic ratio, m∗m^* the effective mass, μB\mu_B the Bohr magneton, ℓsp=(ℓ↑−ℓ↓)/2\ell_{\mathrm{sp}} = (\ell_\uparrow-\ell_\downarrow)/2, and ℓsΦ=(ℓ↑+ℓ↓)/2−Φ/Φ0\ell_{s\Phi} = (\ell_\uparrow+\ell_\downarrow)/2 - \Phi/\Phi_0. This field is a sawtooth-like periodic function of flux with period Φ0\Phi_0, changes sign at half-integer flux, is absent in the ordinary vortex state, and is of order Φ0/πR2\Phi_0/\pi R^2 — about 10 G for a 1 μm Sr2_2RuO4_4 ring — so that with a molar spin susceptibility of order 10−310^{-3} emu/mol the associated polarization should be visible in NMR spin-lattice relaxation (T1T_1) or even Knight-shift measurements. The paper explicitly flags what is missing to make this quantitative for Sr2_2RuO4_4: the values of F1F_1, Z1Z_1 (hence ρsp/ρs\rho_{\mathrm{sp}}/\rho_{\mathrm{s}}) and of gSg_S, m∗m^*, χ\chi; the effects of departures from the thin-annulus geometry, of vortex-core structure, and of spin-orbit coupling (which may prefer d\boldsymbol{d} orientations other than in-plane); and a survey of the (R,d,λL,ρsp/ρs)(R, d, \lambda_L, \rho_{\mathrm{sp}}/\rho_{\mathrm{s}}) space under the self-inductance constraint. The experimental context: cantilever magnetometry (mechanical-torque magnetometry on a micron-scale cantilever) of micron-scale Sr2_2RuO4_4 annuli found half-height magnetization steps consistent with HQV fluxoid transitions (Jang et al. 2011), whose stability wedge grows with applied in-plane field — interpreted via the coupling of the in-plane field to this same (so-called kinematic) spin polarization; Little-Parks-type (fluxoid-driven resistance) and magnetoresistance oscillations in microrings and cylinders support half-integer fluxoid states, again controlled by an in-plane field (Yasui et al. 2017; Cai et al. 2013, 2022). Since 2019, however, oxygen-17 NMR constraints (Pustogow et al. 2019) have challenged the bulk ESP identification itself; Leggett and Liu conclude that the newest Knight-shift and polarized-neutron results exclude the bulk chiral p-wave state while leaving odd-parity helical states possible, and that the HQV experiments argue against even-parity singlet pairing in mesoscopic samples — an unresolved tension. Determining the missing parameters of the 2009 prediction is therefore intertwined with the still-open pairing-symmetry question, while the ESP theory itself remains well posed.

Problem Statement

Within the ESP framework of Vakaryuk and Leggett, replace the order-of-magnitude prediction of HQV spin polarization in Sr2_2RuO4_4 by a quantitative, material-specific theory with a decisive experimental test. This requires four jointly needed results. (a) Material parameters. Determine, with uncertainties, the values of the Landau parameters F1F_1 and Z1Z_1 in the superconducting state of Sr2_2RuO4_4 — equivalently the zero-temperature ratio ρsp/ρs=(1+Z1/12)/(1+F1/3)\rho_{\mathrm{sp}}/\rho_{\mathrm{s}} = (1+Z_1/12)/(1+F_1/3) — together with the gyromagnetic ratio gSg_S, the effective mass m∗m^*, and the superconducting-state spin susceptibility χ\chi entering the effective Zeeman field

Beff=−ℏ22m∗R2(gSμB)−1 ℓspℓsΦ(1+F13+Z112),B_{\mathrm{eff}}=-\frac{\hbar^2}{2m^*R^2}(g_S\mu_B)^{-1}\,\ell_{\mathrm{sp}}\ell_{s\Phi}\left(1+\frac{F_1}{3}+\frac{Z_1}{12}\right),

whether from microscopic (multi-band, anisotropic) Fermi-liquid theory or from experiment. (b) Realistic-geometry theory. Extend the thin-annulus, in-plane-d\boldsymbol{d}, core-free variational treatment to include finite wall thickness, explicit vortex-core energetics, and spin-orbit coupling (including d\boldsymbol{d} orientations other than in-plane and their surface reorientation), and establish their consequences both for the stability criterion — in particular the self-inductance constraint ρsp/ρs<(1+Rd/2λL2)−1\rho_{\mathrm{sp}}/\rho_{\mathrm{s}} < (1+Rd/2\lambda_L^2)^{-1} — and for the magnitude, sign, and flux dependence of BeffB_{\mathrm{eff}} and of the associated equilibrium spin polarization. (c) Stability map. Produce the quantitative survey of the (R,d,λL,ρsp/ρs)(R, d, \lambda_L, \rho_{\mathrm{sp}}/\rho_{\mathrm{s}}) parameter space identifying the regions in which HQVs are thermodynamically stable in Sr2_2RuO4_4 samples realizable in practice (micron and sub-micron doubly connected geometries), including whether and how an applied in-plane magnetic field affects the stability region through its coupling to the spin polarization. (d) Observable prediction. Derive the flux-periodic (period Φ0\Phi_0) spin polarization for realistic samples with error bands, and specify the experimental protocol by which the effect can be detected and distinguished from the polarization of ordinary integer-fluxoid states. These deliverables are the missing quantitative inputs of one prediction, not independent goals: (a) supplies the constants, (b) supplies the geometry and interaction corrections, (c) combines them into the stability phase diagram, and (d) turns the corrected prediction into a test.

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