Orbital-Flop Transition of Angular Momentum in a Topological Superfluid

A. M. Zimmerman, J. I. A. Li, M. D. Nguyen, and W. P. Halperin

Phys. Rev. Lett. 121, 255303 – Published 18 December 2018

Abstract

The direction of the orbital angular momentum of the $B$ phase of superfluid ${}^{3}{He}$ can be controlled by engineering the anisotropy of the silica aerogel framework within which it is imbibed. In this work, we report our discovery of an unusual and abrupt “orbital-flop” transition of the superfluid angular momentum between orientations perpendicular and parallel to the anisotropy axis. The transition has no hysteresis, warming or cooling, as expected for a continuous thermodynamic transition, and is not the result of a competition between strain and magnetic field. This demonstrates the spontaneous reorientation of the order parameter of an unconventional BCS condensate.

Received 19 September 2018

DOI:https://doi.org/10.1103/PhysRevLett.121.255303

Physics Subject Headings (PhySH)

Impurities Impurities in superconductors Order parameters

Helium-3 superfluids Superfluids

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

A. M. Zimmerman^*, J. I. A. Li^‡, M. D. Nguyen, and W. P. Halperin^†

Northwestern University, Evanston, Illinois 60208, USA

^*andrewzimmerman2016@u.northwestern.edu
^†w-halperin@northwestern.edu
^‡Present address: Department of Physics, Brown University, Providence, Rhode Island 02912, USA.

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Vol. 121, Iss. 25 — 21 December 2018

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Images

Figure 1
Temperature dependence of the NMR spectra at small tip angle $β \sim 10 °$ showing an orbital reorientation transition at $T_{x}$ . Aerogel samples with $ϵ \sim - 20 %$ strain at $H_{0} = 0.096 T$ ; (a) $ϵ \sim - 23 %$ , $ϵ ⊥ H_{0}$ , 26 bar; (b) $ϵ \sim - 19 %$ , $ϵ ∥ H_{0}$ , 26 bar; both (a) and b) without ${}^{4}{He}$ preplating; and (c) the same sample as (b) but with ${}^{4}{He}$ preplating at 27 bar. The superfluid transition temperatures are $T_{c} = 2.19$ , 2.19, and 2.24 mK for (a)–(c), respectively. The large frequency shift below $T_{x}$ in (a) continues smoothly from that above $T_{x}$ in (b). The ripples at high frequency shift are from small bulk superfluid components in the $A$ phase.
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Figure 2
Pressure-temperature phase diagram at low magnetic field, extrapolated to zero, for an aerogel compressed by $\sim 20 %$ from an isotropic sample reported upon earlier [11]. The green squares correspond to the superfluid phase transition $T_{c}$ in aerogel, and the solid red circles to the reorientation transition $T_{x}$ without ${}^{4}{He}$ preplating. The dashed lines are guides to the eye. The solid red lines are the phase diagram of pure superfluid ${}^{3}{He}$ .
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Figure 3
(a) NMR frequency shift $Δ ω$ as a function of temperature with $ϵ ∥ H_{0}$ and ${}^{4}{He}$ preplating at $H_{0} = 0.096 T$ and $P = 27$ bar. These data correspond to the spectra in Fig. 1. The large $Δ ω$ for $T > T_{x}$ indicates $l_{B} ⊥ H_{0}$ and the small $Δ ω$ at low temperatures $l_{B} ∥ H_{0}$ . There is a small window of the superfluid $A$ phase close to $T_{c}$ . (b) (Traces are offset vertically for clarity.) Tracking $Δ ω$ back and forth across $T_{x}$ in directions indicated by arrows, shows no hysteresis within the reproducibility of the measurement of temperature $\sim 15 μ K$ .
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Figure 4
Tipping angle dependence of the frequency shift of data above $T_{x}$ at $T = 2.03 mK$ (blue circles) for the sample in Fig. 1, and below $T_{x}$ at $T = 1.76 mK$ (green triangles), scaled to 2.03 mK. Measurements were taken with $ϵ ∥ H_{0}$ and ${}^{4}{He}$ preplating at $H_{0} = 0.05 T$ . The solid (dashed) curve shows the theoretical dependence for $Δ ω_{⊥}$ ( $Δ ω_{∥}$ ) scaled to the frequency shift of isotropic aerogel [11]. The $B$ phase is distorted by strain resulting in a larger energy gap and frequency shift compared to the isotropic case [13].
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Figure 5
(a) Magnetic field independence of $T_{x} / T_{c}$ for different pressures without preplating. At $P = 26$ bar with $\sim - 19 %$ strain, $ϵ ∥ H_{0}$ (open circles), and with $\sim - 23 %$ , $ϵ ⊥ H_{0}$ (closed circles). (b) Strain dependence is very small between $\sim - 20 %$ and $\sim - 30 %$ at 27 bar with $ϵ ⊥ H_{0}$ .
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Figure 6
Transition temperatures for $T_{c}$ and $T_{x}$ scaled to the pure transition temperature as a function of the pure superfluid coherence length. Closed (open) circles indicate measurements without (with) ${}^{4}{He}$ preplating. The linear dependence suggests that both transitions are driven by impurity and can be characterized by a length scale in the aerogel. Without preplating, the extracted length scales are $λ = 385$ and $λ_{x} = 150 nm$ . With pre-plating, they are $λ = 550$ and $λ_{x} = 347 nm$ . For all data, we find $ξ_{a} \sim 3 - 6 nm$ . These should be compared with estimates for the length scales for 98% porosity isotropic aerogel, $λ \approx 150$ and $ξ_{a} \approx 40 nm$ [24].
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