Exceeding the Landau speed limit with topological Bogoliubov Fermi surfaces

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Exceeding the Landau speed limit with topological Bogoliubov Fermi surfaces

S. Autti, J. T. Mäkinen, J. Rysti, G. E. Volovik, V. V. Zavjalov, and V. B. Eltsov

Phys. Rev. Research 2, 033013 – Published 2 July 2020

Abstract

A common property of topological systems is the appearance of topologically protected zero-energy excitations. In a superconductor or superfluid, such states set the critical velocity of dissipationless flow $v_{cL}$ , proposed by Landau, to zero. We check experimentally whether stable superflow is nevertheless possible in the polar phase of $p$ -wave superfluid ${}^{3}{He}$ , which features a Dirac node line in the energy spectrum of Bogoliubov quasiparticles. The fluid is driven by rotation of the whole cryostat, and superflow breakdown is seen as the appearance of single- or half-quantum vortices. Vortices are detected using the relaxation rate of a Bose-Einstein condensate of magnons, created within the fluid. The superflow in the polar phase is found to be stable up to a finite critical velocity $v_{c} \approx 0.2 cm$ /s, despite the zero value of the Landau critical velocity. We suggest that the stability of the superflow above $v_{cL}$ but below $v_{c}$ is provided by the accumulation of the flow-induced quasiparticles into pockets in the momentum space, bounded by Bogoliubov Fermi surfaces. In the polar phase, this surface has nontrivial topology which includes two pseudo-Weyl points. Vortices forming above the critical velocity are strongly pinned in the confining matrix, used to stabilize the polar phase, and hence stable macroscopic superflow can be maintained even when the external drive is brought to zero.

Received 27 February 2020
Revised 16 June 2020
Accepted 16 June 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.033013

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Superconducting gap Superfluid density Topological phases of matter Vortices in superfluids

Helium-3 superfluids

BCS theory Nuclear magnetic resonance

Condensed Matter, Materials & Applied PhysicsFluid Dynamics

Authors & Affiliations

S. Autti ^1,2, J. T. Mäkinen ^1,3,4, J. Rysti¹, G. E. Volovik^1,5, V. V. Zavjalov^1,2, and V. B. Eltsov ^1,*

¹Department of Applied Physics, Aalto University, P.O. Box 15100, FI-00076 AALTO, Finland
²Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom
³Department of Physics, Yale University, New Haven, Connecticut 06520, USA
⁴Yale Quantum Institute, Yale University, New Haven, Connecticut 06520, USA
⁵L.D. Landau Institute for Theoretical Physics, Moscow, Russia

^*vladimir.eltsov@aalto.fi

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Vol. 2, Iss. 3 — July - September 2020

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Images

Figure 1
The polar phase of superfluid ${}^{3}{He}$ engineered with nanostructured confinement. (a) The confining matrix is a set of parallel solid strands, realized using commercial NAFEN material with $d \approx 9 nm$ and $D \approx 35 nm$ [25]. In the stationary polar phase, the energy spectrum of Bogoliubov quasiparticles includes a Dirac node line in the plane perpendicular to strands. (b) In the presence of superflow $v_{s}$ , the node line transforms to the Bogoliubov Fermi surface, consisting of two Fermi pockets which touch each other. Here the superflow is applied along the $x$ axis, and touching points at $p = \pm p_{F} \hat{y}$ are pseudo-Weyl points. Their topology is illustrated as the hedgehog in momentum space, with the topological invariant in Eq. (4). Arrows show direction of the $\hat{n}$ vector and the parameters in Eq. (2) are chosen as $m^{*} c / p_{F} = 1 / 12$ and $v_{s} / c = 1 / 2$ .
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Figure 2
Types of quantized vortices in the polar phase. The order parameter phase $ϕ$ (background color) winds by $2 π$ around a single-quantum vortex and by $π$ around a half-quantum vortex. To keep the order parameter single-valued, vector $\hat{d}$ (red arrows) also rotates around the HQV core, so that $\hat{d} \to - \hat{d}$ when $ϕ \to ϕ + π$ . In nonaxial magnetic field (green arrows), this leads to the formation of $\hat{d}$ solitons connecting HQVs pairwise (blue dashed line). Vortex cores, vector $\hat{m}$ , NAFEN strands, and the axis of rotation are perpendicular to the plane of the picture.
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Figure 3
Increase of the relaxation rate of the magnon BEC $τ_{M}^{- 1}$ due to vortices in the polar phase. (a) The relaxation rate grows as a function of rotation velocity $Ω$ applied at the transition to the superfluid state. For HQVs, the shown slope is extracted from the data in Ref. [32] with the contribution of vortices, created by the Kibble-Zurek mechanism (KZM), removed. For SQVs, KZM is suppressed by the symmetry-violating bias [33]. The measured points are shown by symbols and the line is a fit to $Ω^{1 / 2}$ dependence. (b) For HQVs, $τ_{M}^{- 1}$ (circles) is proportional to the total volume of the $\hat{d}$ solitons between HQV cores, measured by the area of the satellite peak $I_{sat}$ in the normalized NMR spectrum, like in Fig. 4. The line is a linear fit.
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Figure 4
Continuous-wave NMR spectra of the polar phase measured in the magnetic field transverse to NAFEN strands with HQVs and SQVs present in the sample. (a) The absorption normalized to the total spectrum area is plotted vs the frequency shift from the Larmor value $f_{0} = | γ | H / 2 π$ , where $γ$ is the gyromagnetic ratio of ${}^{3}{He}$ . The HQVs produce a satellite in the NMR spectrum, while for SQVs no clear distinguishing features are seen. (b) SQVs are produced with the slow sweep of the angular velocity $Ω (t)$ . (c) Rapid changes in rotation velocity produce HQVs in addition to SQVs. Last of the ten periods of such drive applied in the course of the measurement in zero magnetic field is shown.
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Figure 5
Change of the magnon BEC relaxation $τ_{M}^{- 1}$ when the rotation velocity $Ω$ is gradually increased from 0 to 2 rad/s and then decreased back to 0 at constant temperature, starting from the state with no vortices. Two traces for two independently prepared initial states are shown; the respective $Ω (t)$ dependences and final spectra are plotted in Figs. 4 and 4. Single-quantum vortices form first at $Ω \approx 1 rad$ /s. Pinning prevents vortices from disappearance when $Ω$ is decreased. Before vortex formation, stable superflow with velocity up to $v_{c} \approx 0.2 cm$ /s exists in the sample, while the Landau critical velocity $v_{cL} = 0$ in the polar phase.
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Figure 6
Simulation of the vortex formation with pinning: (a) Number of vortices (red dash-dotted line), antivortices (blue dashed line), and the sum of the two populations (solid black line), as a function rotation velocity $Ω$ which is changed from 0 to 2.5 rad/s and back. (b) The configuration of vortices (red circles) and antivortices (blue crosses) at $Ω = 2.5 rad$ /s. (c) The final vortex configuration at $Ω = 0$ .
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