1
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Shook AJ, Varga E, Boettcher I, Davis JP. Surface State Dissipation in Confined ^{3}He-A. PHYSICAL REVIEW LETTERS 2024; 132:156001. [PMID: 38682961 DOI: 10.1103/physrevlett.132.156001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/16/2023] [Accepted: 03/15/2024] [Indexed: 05/01/2024]
Abstract
We have studied the power dependence of superfluid Helmholtz resonators in flat (750 and 1800 nm) rectangular channels. In the A phase of superfluid ^{3}He, we observe a nonlinear response for velocities larger than a critical value. The small size of the channels stabilizes a static uniform texture that eliminates dissipative processes produced by changes in the texture. For such a static texture, the lowest velocity dissipative process is due to the pumping of surface bound states into the bulk liquid. We show that the temperature dependence of the critical velocity observed in our devices is consistent with this surface-state dissipation. Characterization of the force-velocity curves of our devices may provide a platform for studying the physics of exotic surface bound states in superfluid ^{3}He.
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Affiliation(s)
- Alexander J Shook
- Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E9, Canada
| | - Emil Varga
- Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E9, Canada
| | - Igor Boettcher
- Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E9, Canada
| | - John P Davis
- Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E9, Canada
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2
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Autti S, Haley RP, Jennings A, Pickett GR, Poole M, Schanen R, Soldatov AA, Tsepelin V, Vonka J, Zavjalov VV, Zmeev DE. Transport of bound quasiparticle states in a two-dimensional boundary superfluid. Nat Commun 2023; 14:6819. [PMID: 37919295 PMCID: PMC10622538 DOI: 10.1038/s41467-023-42520-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/17/2023] [Accepted: 10/13/2023] [Indexed: 11/04/2023] Open
Abstract
The B phase of superfluid 3He can be cooled into the pure superfluid regime, where the thermal quasiparticle density is negligible. The bulk superfluid is surrounded by a quantum well at the boundaries of the container, confining a sea of quasiparticles with energies below that of those in the bulk. We can create a non-equilibrium distribution of these states within the quantum well and observe the dynamics of their motion indirectly. Here we show that the induced quasiparticle currents flow diffusively in the two-dimensional system. Combining this with a direct measurement of energy conservation, we conclude that the bulk superfluid 3He is effectively surrounded by an independent two-dimensional superfluid, which is isolated from the bulk superfluid but which readily interacts with mechanical probes. Our work shows that this two-dimensional quantum condensate and the dynamics of the surface bound states are experimentally accessible, opening the possibility of engineering two-dimensional quantum condensates of arbitrary topology.
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Affiliation(s)
- Samuli Autti
- Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK.
| | - Richard P Haley
- Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK
| | - Asher Jennings
- Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK
- RIKEN Center for Quantum Computing, RIKEN, Wako, 351-0198, Japan
| | - George R Pickett
- Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK
| | - Malcolm Poole
- Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK
| | - Roch Schanen
- Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK
| | - Arkady A Soldatov
- P.L. Kapitza Institute for Physical Problems of RAS, 119334, Moscow, Russia
| | - Viktor Tsepelin
- Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK
| | - Jakub Vonka
- Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK
- Paul Scherrer Institute, Forschungsstrasse 111, 5232, Villigen PSI, Switzerland
| | | | - Dmitry E Zmeev
- Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK
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3
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Wu H, Sauls JA. Weyl Fermions and broken symmetry phases of laterally confined 3He films. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2023; 35:495402. [PMID: 37625425 DOI: 10.1088/1361-648x/acf42b] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/06/2023] [Accepted: 08/25/2023] [Indexed: 08/27/2023]
Abstract
Broken symmetries in topological condensed matter systems have implications for the spectrum of Fermionic excitations confined on surfaces or topological defects. The Fermionic spectrum of confined (quasi-2D)3He-A consists of branches of chiral edge states. The negative energy states are related to the ground-state angular momentum,Lz=(N/2)ℏ, forN/2Cooper pairs. The power law suppression of the angular momentum,Lz(T)≃(N/2)ℏ[1-23(πT/Δ)2]for0⩽T≪Tc, in the fully gapped 2D chiral A-phase reflects the thermal excitation of the chiral edge Fermions. We discuss the effects of wave function overlap, and hybridization between edge states confined near opposing edge boundaries on the edge currents, ground-state angular momentum and ground-state order parameter of superfluid3He thin films. Under strong lateral confinement, the chiral A phase undergoes a sequence of phase transitions, first to a pair density wave (PDW) phase with broken translational symmetry atDc2∼16ξ0. The PDW phase is described by a periodic array of chiral domains with alternating chirality, separated by domain walls. The period of PDW phase diverges as the confinement lengthD→Dc2. The PDW phase breaks time-reversal symmetry, translation invariance, but is invariant under the combination of time-reversal and translation by a one-half period of the PDW. The mass current distribution of the PDW phase reflects this combined symmetry, and originates from the spectra of edge Fermions and the chiral branches bound to the domain walls. Under sufficiently strong confinement a second-order transition occurs to the non-chiral 'polar phase' atDc1∼9ξ0, in which a single p-wave orbital state of Cooper pairs is aligned along the channel.
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Affiliation(s)
- Hao Wu
- Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, United States of America
| | - J A Sauls
- Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, United States of America
- Hearne Institute of Theoretical Physics, Louisiana State University, Baton Rouge, LA 70803, United States of America
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4
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Scott JW, Nguyen MD, Park D, Halperin WP. Magnetic Susceptibility of Andreev Bound States in Superfluid ^{3}He-B. PHYSICAL REVIEW LETTERS 2023; 131:046001. [PMID: 37566829 DOI: 10.1103/physrevlett.131.046001] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/03/2023] [Revised: 05/23/2023] [Accepted: 06/12/2023] [Indexed: 08/13/2023]
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.
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Affiliation(s)
- J W Scott
- Northwestern University, Evanston, Illinois 60208, USA
| | - M D Nguyen
- Northwestern University, Evanston, Illinois 60208, USA
| | - D Park
- Northwestern University, Evanston, Illinois 60208, USA
| | - W P Halperin
- Northwestern University, Evanston, Illinois 60208, USA
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5
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Abstract
A time crystal is a macroscopic quantum system in periodic motion in its ground state. In our experiments, two coupled time crystals consisting of spin-wave quasiparticles (magnons) form a macroscopic two-level system. The two levels evolve in time as determined intrinsically by a nonlinear feedback, allowing us to construct spontaneous two-level dynamics. In the course of a level crossing, magnons move from the ground level to the excited level driven by the Landau-Zener effect, combined with Rabi population oscillations. We demonstrate that magnon time crystals allow access to every aspect and detail of quantum-coherent interactions in a single run of the experiment. Our work opens an outlook for the detection of surface-bound Majorana fermions in the underlying superfluid system, and invites technological exploitation of coherent magnon phenomena – potentially even at room temperature. Recent work has reported a realization of a time crystal in the form of the Bose-Einstein condensate of magnons in superfluid 3He. Here, the authors study the dynamics of a pair of such quantum time crystals and show that it closely resembles the evolution of a two-level system, modified by nonlinear feedback.
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6
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Yapa PS, Boyack R, Maciejko J. Triangular Pair Density Wave in Confined Superfluid ^{3}He. PHYSICAL REVIEW LETTERS 2022; 128:015301. [PMID: 35061481 DOI: 10.1103/physrevlett.128.015301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/10/2021] [Revised: 10/21/2021] [Accepted: 12/15/2021] [Indexed: 06/14/2023]
Abstract
Recent advances in experiment and theory suggest that superfluid ^{3}He under planar confinement may form a pair density wave (PDW) whereby superfluid and crystalline orders coexist. While a natural candidate for this phase is a unidirectional stripe phase predicted by Vorontsov and Sauls in 2007, recent nuclear magnetic resonance measurements of the superfluid order parameter rather suggest a two-dimensional PDW with noncollinear wave vectors, of possibly square or hexagonal symmetry. In this Letter, we present a general mechanism by which a PDW with the symmetry of a triangular lattice can be stabilized, based on a superfluid generalization of Landau's theory of the liquid-solid transition. A soft-mode instability at a finite wave vector within the translationally invariant planar-distorted B phase triggers a transition from uniform superfluid to PDW that is first order due to a cubic term generally present in the PDW free-energy functional. This cubic term also lifts the degeneracy of possible PDW states in favor of those for which wave vectors add to zero in triangles, which in two dimensions uniquely selects the triangular lattice.
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Affiliation(s)
| | - Rufus Boyack
- Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada
- Theoretical Physics Institute, University of Alberta, Edmonton, Alberta T6G 2E1, Canada
| | - Joseph Maciejko
- Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada
- Theoretical Physics Institute, University of Alberta, Edmonton, Alberta T6G 2E1, Canada
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7
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Olde Olthof LAB, Johnsen LG, Robinson JWA, Linder J. Controllable Enhancement of p-Wave Superconductivity via Magnetic Coupling to a Conventional Superconductor. PHYSICAL REVIEW LETTERS 2021; 127:267001. [PMID: 35029472 DOI: 10.1103/physrevlett.127.267001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/21/2021] [Accepted: 11/16/2021] [Indexed: 06/14/2023]
Abstract
Unconventional superconductors are of high interest due to their rich physics, a topical example being topological edge states associated with p-wave superconductivity. A practical obstacle in studying such systems is the very low critical temperature T_{c} that is required to realize a p-wave superconducting phase in a material. We predict that the T_{c} of an intrinsic p-wave superconductor can be significantly enhanced by coupling to a conventional s-wave or d-wave superconductor with a higher critical temperature via an atomically thin ferromagnetic (F) layer. We show that this T_{c} boost is tunable via the direction of the magnetization in F. Moreover, we show that the enhancement in T_{c} can also be achieved using the Zeeman effect of an external magnetic field. Our findings provide a way to increase T_{c} in p-wave superconductors in a controllable way and make the exotic physics associated with such materials more easily accessible experimentally.
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Affiliation(s)
- Linde A B Olde Olthof
- Department of Materials Science & Metallurgy, University of Cambridge, CB3 0FS Cambridge, United Kingdom
| | - Lina G Johnsen
- Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
| | - Jason W A Robinson
- Department of Materials Science & Metallurgy, University of Cambridge, CB3 0FS Cambridge, United Kingdom
| | - Jacob Linder
- Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
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8
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Shook AJ, Vadakkumbatt V, Senarath Yapa P, Doolin C, Boyack R, Kim PH, Popowich GG, Souris F, Christani H, Maciejko J, Davis JP. Stabilized Pair Density Wave via Nanoscale Confinement of Superfluid ^{3}He. PHYSICAL REVIEW LETTERS 2020; 124:015301. [PMID: 31976679 DOI: 10.1103/physrevlett.124.015301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/05/2019] [Indexed: 06/10/2023]
Abstract
Superfluid ^{3}He under nanoscale confinement has generated significant interest due to the rich spectrum of phases with complex order parameters that may be stabilized. Experiments have uncovered a variety of interesting phenomena, but a complete picture of superfluid ^{3}He under confinement has remained elusive. Here, we present phase diagrams of superfluid ^{3}He under varying degrees of uniaxial confinement, over a wide range of pressures, which elucidate the progressive stability of both the A phase, as well as a growing region of stable pair density wave state.
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Affiliation(s)
- A J Shook
- Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada
| | - V Vadakkumbatt
- Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada
| | - P Senarath Yapa
- Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada
| | - C Doolin
- Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada
| | - R Boyack
- Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada
- Theoretical Physics Institute, University of Alberta, Edmonton, Alberta T6G 2E1, Canada
| | - P H Kim
- Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada
| | - G G Popowich
- Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada
| | - F Souris
- Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada
| | - H Christani
- Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada
| | - J Maciejko
- Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada
- Theoretical Physics Institute, University of Alberta, Edmonton, Alberta T6G 2E1, Canada
| | - J P Davis
- Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada
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9
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Kobayashi S, Yamakage A, Tanaka Y, Sato M. Majorana Multipole Response of Topological Superconductors. PHYSICAL REVIEW LETTERS 2019; 123:097002. [PMID: 31524460 DOI: 10.1103/physrevlett.123.097002] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/05/2018] [Indexed: 06/10/2023]
Abstract
In contrast to elementary Majorana particles, emergent Majorana fermions (MFs) in condensed-matter systems may have electromagnetic multipoles. We developed a general theory of magnetic multipoles for helical MFs on time-reversal-invariant superconductors. The results show that the multipole response is governed by crystal symmetry, and that a one-to-one correspondence exists between the symmetry of Cooper pairs and the representation of magnetic multipoles under crystal symmetry. The latter property provides a way to identify unconventional pairing symmetry via the magnetic response of helical MFs. We also find that most helical MFs exhibit a magnetic-dipole response, but those on superconductors with spin-3/2 electrons may display a magnetic-octupole response in leading order, which uniquely characterizes high-spin superconductors. Detection of such an octupole response provides direct evidence of high-spin superconductivity, such as in half-Heusler superconductors.
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Affiliation(s)
- Shingo Kobayashi
- Institute for Advanced Research, Nagoya University, Nagoya 464-8601, Japan
- Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan
| | - Ai Yamakage
- Department of Physics, Nagoya University, Nagoya 464-8602, Japan
| | - Yukio Tanaka
- Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan
| | - Masatoshi Sato
- Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
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10
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Mizushima T, Machida K. Multifaceted properties of Andreev bound states: interplay of symmetry and topology. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2018; 376:20150355. [PMID: 29941630 PMCID: PMC6030149 DOI: 10.1098/rsta.2015.0355] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 05/11/2018] [Indexed: 06/08/2023]
Abstract
Andreev bound states (ABSs) ubiquitously emerge as a consequence of non-trivial topological structures of the order parameter of superfluids and superconductors and significantly contribute to thermodynamics and low-energy quantum transport phenomena. We here share the current status of our knowledge on their multifaceted properties such as Majorana fermions and odd-frequency pairing. A unified concept behind ABSs originates from a soliton state in the one-dimensional Dirac equation with mass domain wall and interplay of ABSs with symmetry and topology enrich their physical characteristics. We make an overview of ABSs with a special focus on superfluid 3He. The quantum liquid confined to restricted geometries serves as a rich repository of noteworthy quantum phenomena, such as the mass acquisition of Majorana fermions driven by spontaneous symmetry breaking, topological quantum criticality, Weyl superfluidity and the anomalous magnetic response. The marriage of the superfluid 3He and nano-fabrication techniques will take one to a new horizon of topological quantum phenomena associated with ABSs.This article is part of the theme issue 'Andreev bound states'.
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Affiliation(s)
- T Mizushima
- Department of Materials Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan
| | - K Machida
- Department of Physics, Ritsumeikan University, Kusatsu 525-8577, Japan
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11
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Sato M, Ando Y. Topological superconductors: a review. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2017; 80:076501. [PMID: 28367833 DOI: 10.1088/1361-6633/aa6ac7] [Citation(s) in RCA: 253] [Impact Index Per Article: 36.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/12/2023]
Abstract
This review elaborates pedagogically on the fundamental concept, basic theory, expected properties, and materials realizations of topological superconductors. The relation between topological superconductivity and Majorana fermions are explained, and the difference between dispersive Majorana fermions and a localized Majorana zero mode is emphasized. A variety of routes to topological superconductivity are explained with an emphasis on the roles of spin-orbit coupling. Present experimental situations and possible signatures of topological superconductivity are summarized with an emphasis on intrinsic topological superconductors.
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Affiliation(s)
- Masatoshi Sato
- Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
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12
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Tsutsumi Y. Scattering Theory on Surface Majorana Fermions by an Impurity in ^{3}He-B. PHYSICAL REVIEW LETTERS 2017; 118:145301. [PMID: 28430515 DOI: 10.1103/physrevlett.118.145301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/02/2016] [Indexed: 06/07/2023]
Abstract
We have formulated the scattering theory on Majorana fermions emerging in the surface bound state of the superfluid ^{3}He B phase (^{3}He-B) by an impurity. By applying the theory to the electron bubble, which is regarded as the impurity, trapped below a free surface of ^{3}He-B, the observed mobility of the electron bubble [J. Phys. Soc. Jpn. 82, 124607 (2013)JUPSAU0031-901510.7566/JPSJ.82.124607] is quantitatively reproduced. The mobility is suppressed in low temperatures from the expected value in the bulk ^{3}He-B by the contribution from the surface Majorana fermions. By contrast, the mobility does not depend on the trapped depth of the electron bubble in spite of the spatial variation of the wave function of the surface Majorana fermions. Our formulated theory demonstrates the depth-independent mobility by considering intermediate states in the scattering process. Therefore, we conclude that the experiment has succeeded in observing Majorana fermions in the surface bound state.
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Affiliation(s)
- Yasumasa Tsutsumi
- Department of Basic Science, University of Tokyo, Meguro, Tokyo 153-8902, Japan
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13
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14
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Zheng P, Jiang WG, Barquist CS, Lee Y, Chan HB. Critical Velocity in the Presence of Surface Bound States in Superfluid ^{3}He-B. PHYSICAL REVIEW LETTERS 2017; 118:065301. [PMID: 28234542 DOI: 10.1103/physrevlett.118.065301] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/06/2016] [Indexed: 06/06/2023]
Abstract
A microelectromechanical oscillator with a gap of 1.25 μm was immersed in superfluid ^{3}He-B and cooled below 250 μK at various pressures. Mechanical resonances of its shear motion were measured at various levels of driving force. The oscillator enters into a nonlinear regime above a certain threshold velocity. The damping increases rapidly in the nonlinear region and eventually prevents the velocity of the oscillator from increasing beyond the critical velocity which is much lower than the Landau critical velocity. We propose that this peculiar nonlinear behavior stems from the escape of quasiparticles from the surface bound states into the bulk fluid.
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Affiliation(s)
- P Zheng
- Department of Physics, University of Florida, Gainesville, Florida 32611-8440, USA
| | - W G Jiang
- Department of Physics, University of Florida, Gainesville, Florida 32611-8440, USA
| | - C S Barquist
- Department of Physics, University of Florida, Gainesville, Florida 32611-8440, USA
| | - Y Lee
- Department of Physics, University of Florida, Gainesville, Florida 32611-8440, USA
| | - H B Chan
- Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
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15
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Kozii V, Venderbos JWF, Fu L. Three-dimensional Majorana fermions in chiral superconductors. SCIENCE ADVANCES 2016; 2:e1601835. [PMID: 27957543 PMCID: PMC5142806 DOI: 10.1126/sciadv.1601835] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/06/2016] [Accepted: 11/03/2016] [Indexed: 05/31/2023]
Abstract
Using a systematic symmetry and topology analysis, we establish that three-dimensional chiral superconductors with strong spin-orbit coupling and odd-parity pairing generically host low-energy nodal quasiparticles that are spin-nondegenerate and realize Majorana fermions in three dimensions. By examining all types of chiral Cooper pairs with total angular momentum J formed by Bloch electrons with angular momentum j in crystals, we obtain a comprehensive classification of gapless Majorana quasiparticles in terms of energy-momentum relation and location on the Fermi surface. We show that the existence of bulk Majorana fermions in the vicinity of spin-selective point nodes is rooted in the nonunitary nature of chiral pairing in spin-orbit-coupled superconductors. We address experimental signatures of Majorana fermions and find that the nuclear magnetic resonance spin relaxation rate is significantly suppressed for nuclear spins polarized along the nodal direction as a consequence of the spin-selective Majorana nature of nodal quasiparticles. Furthermore, Majorana nodes in the bulk have nontrivial topology and imply the presence of Majorana bound states on the surface, which form arcs in momentum space. We conclude by proposing the heavy fermion superconductor PrOs4Sb12 and related materials as promising candidates for nonunitary chiral superconductors hosting three-dimensional Majorana fermions.
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Affiliation(s)
- Vladyslav Kozii
- Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
| | - Jörn W. F. Venderbos
- Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
| | - Liang Fu
- Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
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16
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Saitoh M, Ikegami H, Kono K. Onset of Superfluidity in ^{3}He Films. PHYSICAL REVIEW LETTERS 2016; 117:205302. [PMID: 27886487 DOI: 10.1103/physrevlett.117.205302] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/18/2016] [Indexed: 06/06/2023]
Abstract
We elucidate, for the first time, the overall behavior of the onset temperature of superfluidity in ^{3}He films for a wide range of film thicknesses d between 0.06 and 10 μm by taking advantage of the tunability of d implemented using microfabricated devices. We observe a suppression of the onset temperature of superfluidity T_{c}^{f} in a film from the bulk transition temperature as d decreases. In particular, T_{c}^{f} is strongly suppressed when d approaches the coherence length (∼77 nm). The observed T_{c}^{f} as a function of d is similar to that expected from the quasiclassical theory, but it unexpectedly deviates from the theoretical value by several percent when 0.5≲d≲5 μm.
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Affiliation(s)
- Masamichi Saitoh
- The Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan
| | - Hiroki Ikegami
- The Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan
| | - Kimitoshi Kono
- The Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan
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17
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Zheng P, Jiang WG, Barquist CS, Lee Y, Chan HB. Anomalous Damping of a Microelectromechanical Oscillator in Superfluid ^{3}He-B. PHYSICAL REVIEW LETTERS 2016; 117:195301. [PMID: 27858447 DOI: 10.1103/physrevlett.117.195301] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/13/2016] [Indexed: 06/06/2023]
Abstract
The mechanical resonance properties of a microelectromechanical oscillator with a gap of 1.25 μm was studied in superfluid ^{3}He-B at various pressures. The oscillator was driven in the linear damping regime where the damping coefficient is independent of the oscillator velocity. The quality factor of the oscillator remains low (Q≈80) down to 0.1T_{c}, 4 orders of magnitude less than the intrinsic quality factor measured in vacuum at 4 K. In addition to the Boltzmann temperature dependent contribution to the damping, a damping proportional to temperature was found to dominate at low temperatures. We propose a multiple scattering mechanism of the surface Andreev bound states to be a possible cause for the anomalous damping.
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Affiliation(s)
- P Zheng
- Department of Physics, University of Florida, Gainesville, Florida 32611-8440, USA
| | - W G Jiang
- Department of Physics, University of Florida, Gainesville, Florida 32611-8440, USA
| | - C S Barquist
- Department of Physics, University of Florida, Gainesville, Florida 32611-8440, USA
| | - Y Lee
- Department of Physics, University of Florida, Gainesville, Florida 32611-8440, USA
| | - H B Chan
- Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
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18
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Yang W, Li Y, Wu C. Topological Septet Pairing with Spin-3/2 Fermions: High-Partial-Wave Channel Counterpart of the ^{3}He-B Phase. PHYSICAL REVIEW LETTERS 2016; 117:075301. [PMID: 27563972 DOI: 10.1103/physrevlett.117.075301] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/01/2016] [Indexed: 06/06/2023]
Abstract
We systematically generalize the exotic ^{3}He-B phase, which not only exhibits unconventional symmetry but is also isotropic and topologically nontrivial, to arbitrary partial-wave channels with multicomponent fermions. The concrete example with four-component fermions is illustrated including the isotropic f-, p-, and d-wave pairings in the spin septet, triplet, and quintet channels, respectively. The odd partial-wave channel pairings are topologically nontrivial, while pairings in even partial-wave channels are topologically trivial. The topological index reaches the largest value of N^{2} in the p-wave channel (N is half of the fermion component number). The surface spectra exhibit multiple linear and even high order Dirac cones. Applications to multiorbital condensed matter systems and multicomponent ultracold large spin fermion systems are discussed.
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Affiliation(s)
- Wang Yang
- Department of Physics, University of California, San Diego, California 92093, USA
| | - Yi Li
- Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, Maryland 21218, USA
| | - Congjun Wu
- Department of Physics, University of California, San Diego, California 92093, USA
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Mizushima T, Tsutsumi Y, Sato M, Machida K. Symmetry protected topological superfluid (3)He-B. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2015; 27:113203. [PMID: 25730099 DOI: 10.1088/0953-8984/27/11/113203] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
Owing to the richness of symmetry and well-established knowledge of bulk superfluidity, the superfluid (3)He has offered a prototypical system to study intertwining of topology and symmetry. This article reviews recent progress in understanding the topological superfluidity of (3)He in a multifaceted manner, including symmetry considerations, the Jackiw-Rebbi's index theorem, and the quasiclassical theory. Special focus is placed on the symmetry protected topological superfuidity of the (3)He-B confined in a slab geometry. The (3)He-B under a magnetic field is separated to two different sub-phases: the symmetry protected topological phase and non-topological phase. The former phase is characterized by the existence of symmetry protected Majorana fermions. The topological phase transition between them is triggered by the spontaneous breaking of a hidden discrete symmetry. The critical field is quantitatively determined from the microscopic calculation that takes account of magnetic dipole interaction of the (3)He nucleus. It is also demonstrated that odd-frequency even-parity Cooper pair amplitudes are emergent in low-lying quasiparticles. The key ingredients, symmetry protected Majorana fermions and odd-frequency pairing, bring an important consequence that the coupling of the surface states to an applied field is prohibited by the hidden discrete symmetry, while the topological phase transition with the spontaneous symmetry breaking is accompanied by anomalous enhancement and anisotropic quantum criticality of surface spin susceptibility. We also illustrate common topological features between topological crystalline superconductors and symmetry protected topological superfluids, taking UPt3 and Rashba superconductors as examples.
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Grover T, Sheng DN, Vishwanath A. Emergent Space-Time Supersymmetry at the Boundary of a Topological Phase. Science 2014; 344:280-3. [DOI: 10.1126/science.1248253] [Citation(s) in RCA: 192] [Impact Index Per Article: 19.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/02/2022]
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21
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He JJ, Ng TK, Lee PA, Law KT. Selective equal-spin Andreev reflections induced by Majorana fermions. PHYSICAL REVIEW LETTERS 2014; 112:037001. [PMID: 24484161 DOI: 10.1103/physrevlett.112.037001] [Citation(s) in RCA: 31] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/10/2013] [Indexed: 06/03/2023]
Abstract
In this work, we find that Majorana fermions induce selective equal spin Andreev reflections (SESARs), in which incoming electrons with certain spin polarization in the lead are reflected as counterpropagating holes with the same spin. The spin polarization direction of the electrons of this Andreev reflected channel is selected by the Majorana fermions. Moreover, electrons with opposite spin polarization are always reflected as electrons with unchanged spin. As a result, the charge current in the lead is spin polarized. Therefore, a topological superconductor which supports Majorana fermions can be used as a novel device to create fully spin-polarized currents in paramagnetic leads. We point out that SESARs can also be used to detect Majorana fermions in topological superconductors.
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Affiliation(s)
- James J He
- Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China
| | - T K Ng
- Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China
| | - Patrick A Lee
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - K T Law
- Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China
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22
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Levitin LV, Bennett RG, Surovtsev EV, Parpia JM, Cowan B, Casey AJ, Saunders J. Surface-induced order parameter distortion in superfluid ³He-B measured by nonlinear NMR. PHYSICAL REVIEW LETTERS 2013; 111:235304. [PMID: 24476290 DOI: 10.1103/physrevlett.111.235304] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/30/2013] [Indexed: 06/03/2023]
Abstract
The B phase of superfluid 3He is a three-dimensional time-reversal invariant topological superfluid, predicted to support gapless Majorana surface states. We confine superfluid 3He into a thin nanofluidic slab geometry. In the presence of a weak symmetry-breaking magnetic field, we have observed two possible states of the confined 3He-B phase manifold, through the small tipping angle NMR response. Large tipping angle nonlinear NMR has allowed the identification of the order parameter of these states and enabled a measurement of the surface-induced gap distortion. The results for two different quasiparticle surface scattering boundary conditions are compared with the predictions of weak-coupling quasiclassical theory. We identify a textural domain wall between the two B phase states, the edge of which at the cavity surface is predicted to host gapless states, protected in the magnetic field.
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Affiliation(s)
- Lev V Levitin
- Department of Physics, Royal Holloway University of London, Egham, TW20 0EX Surrey, United Kingdom
| | - Robert G Bennett
- Department of Physics, Royal Holloway University of London, Egham, TW20 0EX Surrey, United Kingdom and Department of Physics, Cornell University, Ithaca, New York 14853, USA
| | - Evgeny V Surovtsev
- Kapitza Institute for Physical Problems, ul. Kosygina 2, Moscow 119334, Russia
| | - Jeevak M Parpia
- Department of Physics, Cornell University, Ithaca, New York 14853, USA
| | - Brian Cowan
- Department of Physics, Royal Holloway University of London, Egham, TW20 0EX Surrey, United Kingdom
| | - Andrew J Casey
- Department of Physics, Royal Holloway University of London, Egham, TW20 0EX Surrey, United Kingdom
| | - John Saunders
- Department of Physics, Royal Holloway University of London, Egham, TW20 0EX Surrey, United Kingdom
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23
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Ueno Y, Yamakage A, Tanaka Y, Sato M. Symmetry-protected Majorana fermions in topological crystalline superconductors: theory and application to Sr2RuO4. PHYSICAL REVIEW LETTERS 2013; 111:087002. [PMID: 24010466 DOI: 10.1103/physrevlett.111.087002] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/01/2013] [Indexed: 06/02/2023]
Abstract
Crystal point group symmetry is shown to protect Majorana fermions (MFs) in spinfull superconductors (SCs). We elucidate the condition necessary to obtain MFs protected by the point group symmetry. We argue that superconductivity in Sr2RuO4 hosts a topological phase transition to a topological crystalline SC, which accompanies a d-vector rotation under a magnetic field along the c axis. Taking all three bands and spin-orbit interactions into account, symmetry-protected MFs in the topological crystalline SC are identified. Detection of such MFs provides evidence of the d-vector rotation in Sr2RuO4 expected from Knight shift measurements but not yet verified.
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Affiliation(s)
- Yuji Ueno
- Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan
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24
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Zyuzin AA, Rainis D, Klinovaja J, Loss D. Correlations between Majorana fermions through a superconductor. PHYSICAL REVIEW LETTERS 2013; 111:056802. [PMID: 23952428 DOI: 10.1103/physrevlett.111.056802] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/21/2013] [Indexed: 06/02/2023]
Abstract
We consider a model of ballistic quasi-one-dimensional semiconducting wire with intrinsic spin-orbit interaction placed on the surface of a bulk s-wave superconductor (SC), in the presence of an external magnetic field. This setup has been shown to give rise to a topological superconducting state in the wire, characterized by a pair of Majorana-fermion (MF) bound states formed at the two ends of the wire. Here, we demonstrate that besides the well-known direct-overlap-induced energy splitting, the two MF bound states may hybridize via elastic tunneling processes through virtual quasiparticle states in the SC, giving rise to an additional energy splitting between MF states from the same as well as from different wires.
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Affiliation(s)
- A A Zyuzin
- Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
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Levitin LV, Bennett RG, Casey A, Cowan B, Saunders J, Drung D, Schurig T, Parpia JM. Phase Diagram of the Topological Superfluid
3
He Confined in a Nanoscale Slab Geometry. Science 2013; 340:841-4. [DOI: 10.1126/science.1233621] [Citation(s) in RCA: 64] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/02/2022]
Affiliation(s)
- L. V. Levitin
- Department of Physics, Royal Holloway University of London, Egham, Surrey, TW20 0EX, UK
| | - R. G. Bennett
- Department of Physics, Royal Holloway University of London, Egham, Surrey, TW20 0EX, UK
| | - A. Casey
- Department of Physics, Royal Holloway University of London, Egham, Surrey, TW20 0EX, UK
| | - B. Cowan
- Department of Physics, Royal Holloway University of London, Egham, Surrey, TW20 0EX, UK
| | - J. Saunders
- Department of Physics, Royal Holloway University of London, Egham, Surrey, TW20 0EX, UK
| | - D. Drung
- Physikalisch-Technische Bundesanstalt, Abbestrasse 2-12, D-10587 Berlin, Germany
| | - Th. Schurig
- Physikalisch-Technische Bundesanstalt, Abbestrasse 2-12, D-10587 Berlin, Germany
| | - J. M. Parpia
- Department of Physics, Cornell University, Ithaca, NY 14853, USA
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26
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Mizushima T, Sato M, Machida K. Symmetry protected topological order and spin susceptibility in superfluid 3He-B. PHYSICAL REVIEW LETTERS 2012; 109:165301. [PMID: 23215090 DOI: 10.1103/physrevlett.109.165301] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/19/2012] [Revised: 08/01/2012] [Indexed: 06/01/2023]
Abstract
We here demonstrate that the superfluid (3)He-B under a magnetic field in a particular direction stays topological due to a discrete symmetry, that is, in a symmetry protected topological order. Because of the symmetry protected topological order, helical surface Majorana fermions in the B phase remain gapless and their Ising spin character persists. We unveil that the competition between the Zeeman magnetic field and dipole interaction involves an anomalous quantum phase transition in which a topological phase transition takes place together with spontaneous symmetry breaking. Based on the quasiclassical theory, we illustrate that the phase transition is accompanied by anisotropic quantum criticality of spin susceptibilities on the surface, which is detectable in NMR experiments.
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27
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Okuda Y, Nomura R. Surface Andreev bound states of superfluid 3He and Majorana fermions. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2012; 24:343201. [PMID: 22871689 DOI: 10.1088/0953-8984/24/34/343201] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
Superfluid (3)He is an intensively investigated and well characterized p-wave superfluid. In the bulk Balian-Werthamer state, which is commonly called the (3)He B phase, the superfluid gap is opened isotropically but near a flat boundary such as a wall of a container it can harbor interesting quasi-particle states inside the gap. These states are called surface Andreev bound states, and have not been experimentally explored in detail. Transverse acoustic impedance measurement has revealed their existence and provided spectroscopic details of the dispersion of the bound states. Recent theoretical arguments claim that the surface Andreev bound states of the superfluid (3)He B phase can be recognized as the edge states of the topological superfluid and be regarded as a Majorana fermion, a fancy particle which has not been confirmed in elementary particle physics. In this review, we present up-to-date knowledge on the surface Andreev bound states of the (3)He B phase revealed by acoustic spectroscopy and the possible realization of a Majorana fermion, along with related studies on this topic.
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Affiliation(s)
- Y Okuda
- Department of Physics, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguro-ku, Tokyo 152-8551, Japan.
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Machida Y, Itoh A, So Y, Izawa K, Haga Y, Yamamoto E, Kimura N, Onuki Y, Tsutsumi Y, Machida K. Twofold spontaneous symmetry breaking in the heavy-fermion superconductor UPt3. PHYSICAL REVIEW LETTERS 2012; 108:157002. [PMID: 22587277 DOI: 10.1103/physrevlett.108.157002] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/15/2011] [Indexed: 05/31/2023]
Abstract
The field-orientation dependent thermal conductivity of the heavy-fermion superconductor UPt3 was measured down to very low temperatures and under magnetic fields throughout the distinct superconducting phases: B and C phases. In the C phase, a striking twofold oscillation of the thermal conductivity within the basal plane is resolved reflecting the superconducting gap structure with a line of node along the a axis. Moreover, we find an abrupt vanishing of the oscillation across a transition to the B phase, as a clear indication of a change of gap symmetries. We also identify extra two line nodes below and above the equator in both B and C phases. From these results together with the symmetry consideration, the gap function of UPt3 is determined as a E(1u) representation characterized by a combination of two line nodes at the tropics and point nodes at the poles.
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Affiliation(s)
- Y Machida
- Department of Physics, Tokyo Institute of Technology, Meguro 152-8551, Japan
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29
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Autti S, Bunkov YM, Eltsov VB, Heikkinen PJ, Hosio JJ, Hunger P, Krusius M, Volovik GE. Self-trapping of magnon Bose-Einstein condensates in the ground state and on excited levels: from harmonic to box confinement. PHYSICAL REVIEW LETTERS 2012; 108:145303. [PMID: 22540804 DOI: 10.1103/physrevlett.108.145303] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/14/2011] [Indexed: 05/31/2023]
Abstract
Long-lived coherent spin precession of (3)He-B at low temperatures around 0.2T(c) is a manifestation of Bose-Einstein condensation of spin-wave excitations or magnons in a magnetic trap which is formed by the order-parameter texture and can be manipulated experimentally. When the number of magnons increases, the orbital texture reorients under the influence of the spin-orbit interaction and the profile of the trap gradually changes from harmonic to a square well, with walls almost impenetrable to magnons. This is the first experimental example of Bose condensation in a box. By selective rf pumping the trap can be populated with a ground-state condensate or one at any of the excited energy levels. In the latter case the ground state is simultaneously populated by relaxation from the exited level, forming a system of two coexisting condensates.
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Affiliation(s)
- S Autti
- Low Temperature Laboratory, School of Science, Aalto University, Finland
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30
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Asano Y, Tanaka Y, Nagaosa N. Tunneling between two helical superconductors via Majorana edge channels. PHYSICAL REVIEW LETTERS 2010; 105:056402. [PMID: 20867939 DOI: 10.1103/physrevlett.105.056402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/22/2010] [Indexed: 05/29/2023]
Abstract
We discuss electric transport through a point contact which bridges Majorana fermion modes appearing at edges of two helical superconductors. The contents focus on the effects of interference and interaction unique to the Majorana fermions and the role of spin-orbit interaction (SOI). Besides the Josephson current, the quasiparticle conductance depends sensitively on the phase difference and relative helicity between the two superconductors. The interaction among the Majorana fermions causes the power-law temperature dependences of conductance for various tunneling channels. Especially, in the presence of SOI, the conductance always increases as the temperature is lowered.
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Affiliation(s)
- Yasuhiro Asano
- Department of Applied Physics & Center for Topological Science and Technology, Hokkaido University, Sapporo, 060-8628, Japan
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