Observation of Half-Quantum Vortices in Topological Superfluid $^{3}\mathrm{He}$

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Observation of Half-Quantum Vortices in Topological Superfluid ${}^{3}{He}$

S. Autti, V. V. Dmitriev, J. T. Mäkinen, A. A. Soldatov, G. E. Volovik, A. N. Yudin, V. V. Zavjalov, and V. B. Eltsov

Phys. Rev. Lett. 117, 255301 – Published 14 December 2016

See Viewpoint: Half-Quantum Vortices in Superfluid Helium

Abstract

One of the most sought-after objects in topological quantum-matter systems is a vortex carrying half a quantum of circulation. They were originally predicted to exist in superfluid ${}^{3}{He} − A$ but have never been resolved there. Here we report an observation of half-quantum vortices (HQVs) in the polar phase of superfluid ${}^{3}{He}$ . The vortices are created with rotation or by the Kibble-Zurek mechanism and identified based on their nuclear magnetic resonance signature. This discovery provides a pathway for studies of unpaired Majorana modes bound to the HQV cores in the polar-distorted $A$ phase.

Received 24 August 2016

DOI:https://doi.org/10.1103/PhysRevLett.117.255301

Physics Subject Headings (PhySH)

Majorana fermions Solitons Spin waves Topological phases of matter Vortices in superfluids

Helium-3 superfluids Node-line semimetals

Nuclear magnetic resonance

Condensed Matter, Materials & Applied PhysicsInterdisciplinary Physics

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Half-Quantum Vortices in Superfluid Helium

Published 14 December 2016

Researchers working with superfluid helium in a porous medium have generated vortices with half the quantum unit of fluid flow.

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Authors & Affiliations

S. Autti^1,*, V. V. Dmitriev², J. T. Mäkinen¹, A. A. Soldatov^2,3, G. E. Volovik^1,4, A. N. Yudin², V. V. Zavjalov¹, and V. B. Eltsov¹

¹Low Temperature Laboratory, Department of Applied Physics, Aalto University, P.O. Box 15100, FI-00076 AALTO, Finland
²P. L. Kapitza Institute for Physical Problems of RAS, 119334 Moscow, Russia
³Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Russia
⁴Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia

^*samuli.autti@aalto.fi

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Vol. 117, Iss. 25 — 16 December 2016

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Figure 1
(a) Sketch of the sample: Cubic container made from epoxy (light blue) is filled with the nafen (blue cube). The nafen strands are oriented along the vertical $\hat{z}$ direction, their diameter is $d_{1} \approx 9 nm$ , and their average separation $d_{2} \approx 35 nm$ . The space between strands is filled with liquid ${}^{3}{He}$ . The sample is surrounded by nuclear magnetic resonance (NMR) coils made of copper wire (brown rectangles). The magnetic field $H$ is applied transverse to the NMR coil axis at an arbitrary angle $μ$ from the direction of the orbital anisotropy vector $\hat{m}$ , which is pinned along the nafen strands. The sample can be rotated around the vertical axis with the angular velocity $Ω$ up to $3 rad / s$ . (b) Vortex types in the polar phase: The order-parameter phase $ϕ$ is shown by the background colour. The spin vector $\hat{d}$ is locked to the plane (green disks) perpendicular to the magnetic field $H$ . Within this plane, $\hat{d}$ rotates by $π$ around the HQV core and by $2 π$ around the SCV core. In a tilted magnetic field ( $μ > 0$ ), this winding is concentrated in one $\hat{d}$ soliton terminating at the HQV core or in two solitons terminating at the SCV core. The approximate soliton extents are illustrated by dashed lines. The nafen strands, the $\hat{m}$ field, and the vortex lines are orthogonal to the plane of the picture. The SQV and HQV have hard cores (red disks) of the size of coherence length $\sim 40 nm$ , while the SCV has only a soft core (blue disk) of much larger size $\sim 10 μ m$ .
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Figure 2
NMR spectra in the polar phase with HQVs. (a) Normalized spectra measured in transverse field $μ = π / 2$ (blue thick line) show the HQV satellite at the negative frequency shift $Δ ω_{sat}$ and the main line at zero frequency shift. In the axial field $μ = 0$ (red thin line), only the main line at positive shift $Δ ω_{main}$ is seen. This spectrum is not sensitive to presence of vortices. (b) Temperature dependences of the satellite position in the transverse field $| Δ ω_{sat} (μ = π / 2) |$ (blue circles) and the main line position in the axial field $Δ ω_{main} (μ = 0)$ (red squares) closely match as expected for HQVs. The example error bars show full width at half maximum of non-Lorentzian main line as an estimate of possible systematic error.
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Figure 3
Frequency shift of the HQV satellite in the NMR spectrum. (a) Measured values of the dimensionless frequency shift $λ$ as a function of the field tilt angle $μ$ (symbols) and numerical calculations for the uniform polar phase (solid line) using theoretical value of $ξ_{D} = 17 μ m$ [19]. Leggett frequency $Ω_{P}$ is determined from a separate measurement at $μ = 0$ . As expected, the deviation from the infinitely long $\hat{d}$ soliton value $λ = 1$ increases towards small $μ$ . The disagreement between the experiment and calculations probably originates from disorder in the nafen strand orientation [57], which leads to fluctuations of spin-orbit interaction energy within the solitons. (b) Values of $λ$ are found from positions of the HQV satellite $Δ ω_{sat}$ (blue circles) and of the main line $Δ ω_{main}$ (red squares). The red and blue solid lines show results of Eqs. (3) and (4), respectively, for $λ = 1$ . (c) Main line and satellite line positions are determined using Lorentzian fits (thick black line) of the dispersion signal (red line). For clarity, the lines have been shifted vertically. Values of $μ$ are marked above the lines. Similar fits are used in extracting the satellite line intensity in Fig. 4. In panels (a) and (b), the bars show full width at half maximum of the spectral lines.
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Figure 4
Intensity $I_{sat}$ of the HQV satellite in the NMR spectrum. (a) After cooldown in zero field (red circles), $I_{sat}$ follows expected dependence for HQVs $I_{sat} \propto Ω^{1 / 2}$ . The solid line is the theoretical prediction for the equilibrium array of HQVs ignoring Kibble-Zurek vortices and with no fitted parameters. For cooldowns in the transverse field (cyan squares), formation of HQVs is suppressed. Residual intensity is attributed to the possible formation of the soliton sheets terminating at the sample walls. The cooldown rate here is $τ_{Q}^{- 1} = 5 \times 10^{- 6} s^{- 1}$ . (b) For cooldowns at $Ω = 0 rad / s$ , the satellite intensity (symbols) follows dependence $I_{sat} \propto τ_{Q}^{- 1 / 4}$ (solid line) expected for vortices created by the Kibble-Zurek mechanism, where the initial vortex density agrees with earlier measurements in ${}^{3}{He} − B$ [58]. In all cases, spectra are measured at $μ = π / 2$ , and $I_{sat}$ is determined as the area of the satellite normalized to the total area of the spectrum.
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Physical Review Letters

Observation of Half-Quantum Vortices in Topological Superfluid He3

S. Autti, V. V. Dmitriev, J. T. Mäkinen, A. A. Soldatov, G. E. Volovik, A. N. Yudin, V. V. Zavjalov, and V. B. Eltsov

Phys. Rev. Lett. 117, 255301 – Published 14 December 2016

See Viewpoint: Half-Quantum Vortices in Superfluid Helium

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Half-Quantum Vortices in Superfluid Helium

Published 14 December 2016

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Observation of Half-Quantum Vortices in Topological Superfluid ${}^{3}{He}$