Magnetic Susceptibility of Andreev Bound States in Superfluid $^{3}\mathrm{He}\text{\ensuremath{-}}\mathrm{B}$

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Magnetic Susceptibility of Andreev Bound States in Superfluid ${}^{3}{He} − B$

J. W. Scott, M. D. Nguyen, D. Park, and W. P. Halperin

Phys. Rev. Lett. 131, 046001 – Published 27 July 2023

Abstract

Nuclear magnetic resonance measurements of the magnetic susceptibility of superfluid ${}^{3}{He}$ imbibed in anisotropic aerogel reveal anomalous behavior at low temperatures. Although the frequency shift clearly identifies a low-temperature phase as the $B$ phase, the magnetic susceptibility does not display the expected decrease associated with the formation of the opposite-spin Cooper pairs. This susceptibility anomaly appears to be the predicted high-field behavior corresponding to the Ising-like magnetic character of surface Andreev bound states within the planar aerogel structures.

Received 3 February 2023
Revised 23 May 2023
Accepted 12 June 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.046001

Physics Subject Headings (PhySH)

Impurities

Aerogel Helium-3 superfluids Superfluids

Nuclear magnetic resonance

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

J. W. Scott ^*, M. D. Nguyen , D. Park, and W. P. Halperin ^†

Northwestern University, Evanston, Illinois 60208, USA

^*johnscott2025@u.northwestern.edu
^†w-halperin@northwestern.edu

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Vol. 131, Iss. 4 — 28 July 2023

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Figure 1
Temperature dependence and tip angle dependence of NMR absorption spectra of ${}^{3}{He}$ imbibed in 16% stretched aerogel compared with pure ${}^{3}{He}$ . In (a)–(d) the absorption spectrum is shown at each temperature and tip angle as a function of frequency by a color scale representing the spectral amplitude normalized to the maximum absorption amplitude in each panel. (a) Temperature dependence of spectra at tip angle, $β = 8 °$ , $P = 26.6$ bar, $H_{0} = 74.5 mT$ . The configurations of the order parameter in order on warming; below the orbital flop transition $T_{x} = 0.77 T_{c}$ , $ℓ ⊥ H_{0}$ [22]; above $T_{x} = 0.77 T_{c}$ , $ℓ ∥ H_{0}$ ; above $T_{a b} = 0.93 T_{c}$ the tip angle dependence is characteristic of the $A$ phase with $ℓ ∥ H_{0}$ . (b)–(d) Tip angle dependence of spectra in (a). (e)–(g) Theoretical tip angle dependence of spectra for pure ${}^{3}{He}$ corresponding to (b)–(d) [28].
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Figure 2
Magnetic susceptibility $χ / χ_{N}$ (green circle) over a range of fields at $P = 26.6 bar$ . In contrast to the behavior expected, the magnetic susceptibility is temperature independent in the $B$ phase (blue background) for all fields, different pressures, temperatures below $T_{a b}$ , and different orbital configurations. For comparison, $χ / χ_{N}$ of the pure superfluid at $P = 27.0 bar$ is shown in the lowest panel (purple circle), taken from Ref. [36]. The blue solid (dashed) lines are the Born (unitary), Ginzburg-Landau results from the homogeneous isotropic scattering model for ${}^{3}{He}$ in aerogel [9].
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Figure 3
Measured order parameter distortion. The order parameter amplitude distortion in the direction of the magnetic field, $1 - Δ_{∥} / Δ_{⊥}$ at $H_{0} = 74.5 mT$ for the temperature range between $T_{a b}$ and $T_{x} = 0.77 T_{c}$ . The degree of order parameter distortion $\approx 0.34$ close to $T_{a b}$ is comparable to that seen in nanoconfined planar slabs [37]. At lower temperatures the distortion we measure $\approx 0.09$ approaches the value for bulk, pure ${}^{3}{He} \approx 0.005$ in a 112 mT magnetic field [38].
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Figure 4
Magnetic phase diagram at $P = 26.6 bar$ . The green trace (green shaded triangle) denotes the experimentally observed $T_{a b}$ in the aerogel. A linear fit shows the extrapolated $T_{a b}$ at zero field to be $T_{a b} = 0.93 T_{c}$ . The purple trace (purple shaded triangle) denotes the $A - B$ transition in pure bulk superfluid at $P = 27.0 bar$ [36, 44] for comparison. Inset: the pressure-temperature phase diagram at $H_{0} = 195 mT$ , with critical temperature $T_{c}$ (green open circle), $A - B$ transition temperature $T_{a b}$ (green open triangle). The black curve is a fit for the observed $T_{c}$ based on an impurity model [7, 45, 46]. The purple line denotes the pure liquid $T_{c}$ , the dashed black line is a guide to the eye.
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Physical Review Letters

Magnetic Susceptibility of Andreev Bound States in Superfluid He3−B

J. W. Scott, M. D. Nguyen, D. Park, and W. P. Halperin

Phys. Rev. Lett. 131, 046001 – Published 27 July 2023

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Magnetic Susceptibility of Andreev Bound States in Superfluid ${}^{3}{He} − B$