Majorana surface states of superfluid<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mmultiscripts><mml:mi mathvariant="normal">He</mml:mi><mml:mprescripts /><mml:none /><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:mmultiscripts></mml:math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>B</mml:mi></mml:mrow>

Ichioka, Masanori; Machida, Kazushige

doi:10.1103/physrevb.83.094510

Cited by 67 publications

(106 citation statements)

References 49 publications

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“…The ABM and planar states can be stabilized in a thin film that the thick is comparable with the superfluid coherence length [71,75,76,105,106,107,108,143,144], while the one-dimensional polar phase may be stabilized in a narrow cylinder [138,373]. The topology of the ABM state that spontaneously breaks time-reversal symmetry is characterized by the mirror Chern number [51,56].…”

Section: Discussionmentioning

confidence: 99%

“…The linear dependence is also obtained by numerical calculation [71,75,144,152]. The linear behavior of the low-energy density of states is responsible for the power-law behavior of the temperature-dependence of the specific heat in 3 He-B confined in a slab geometry [69], in contrary to the exponential behavior .…”

Section: Exact Solutions For Superfluid 3 He-bmentioning

confidence: 99%

“…[130,131,132,133,134,135,136,137,138,139,140,141,142] with the Ginzburg-Landau theory and Refs. [71,75,76,105,106,107,108,143,144] with the quasiclassical theory. The latter underlines the role of the surface Andreev bound states on thermodynamics.…”

Section: 3mentioning

confidence: 99%

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Symmetry protected topological superfluid³He-B

Mizushima

Tsutsumi

Sato

et al. 2015

J. Phys.: Condens. Matter

158

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Abstract.Owing to the richness of symmetry and well-established knowledge on the 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 consideration, 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 off 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 3 He nucleus. It is also demonstrated that odd-frequency evenparity 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 UPt 3 and Rashba superconductors as examples.

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Section: Discussionmentioning

confidence: 99%

Section: Exact Solutions For Superfluid 3 He-bmentioning

confidence: 99%

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Symmetry protected topological superfluid³He-B

Mizushima

Tsutsumi

Sato

et al. 2015

J. Phys.: Condens. Matter

158

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“…This system is realized by confining the superfluid 3 He A-phase in a slab of sub-µm thickness [4] that is already realized experimentally [21]. Since the d-vector is fixed, we consider the orbital part of the order parameter ∆(x, k) = A x (x)k x + iA y (x)k y in the one-dimension toward the radial direction x, where k normalized by k F is the relative momentum of a Cooper pair.…”

mentioning

confidence: 99%

“…The 3 He A-phase is a chiral superfluid with the spontaneous edge mass current while B-phase is a helical superfluid with the edge spin current [4]. The Majorana nature for the A-and B-phases is also distinctive; Majorana Fermions have the surface Andreev bound state with linear dispersion relation centered at the zero energy forming a "Majorana valley" in the A-phase [4,5] and a "Majorana cone" in the B-phase [6][7][8]. This Majorana cone is emerging by recent experiments [9,10].…”

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confidence: 99%

Edge mass current and the role of Majorana fermions inA-phase superfluid3He

Machida

2012

Phys. Rev. B

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The total angular momentum associated with the edge mass current flowing at the boundary in the superfluid 3 He A-phase confined in a disk is proved to be L = N /2, consisting of L MJ = N from the Majorana quasi-particles (QPs) and L cont = −N /2 from the continuum state. We show it based on an analytic solution of the chiral order parameter for quasi-classical Eilenberger equation. Important analytic expressions are obtained for mass current, angular momentum, and density of states (DOS). Notably the DOS of the Majorana QPs is exactly N0/2 (N0: normal state DOS) responsible for the factor 2 difference between L MJ and L cont . The current decreases as E −3 against the energy E, and L(T ) ∝ −T 2 . This analytic solution is fully backed up by numerically solving the Eilenberger equation. We touch on the so-called intrinsic angular momentum problem. The superfluid is a paradigmatic and fertile testing ground to investigate the central questions concerning the origin of anisotropic superfluidity since the p-wave pairing symmetry for the 3 He A-phase and B-phase is firmly established [1]. Recently the topological aspect of the 3 He A-and B-phases has become a focus because there has been no other concrete topological superfluids or superconductors established so far. The topological superfluid defined by a topological number [2]. At an edge of the superfluid 3 He faced to a topologically trivial vacuum, a superfluid gap is closed by a topological phase transition at the interface. The surface Andreev bound state of Majorana Fermions must appear [3]. The topological features are quite different in the 3 He A-and B-phases. The 3 He A-phase is a chiral superfluid with the spontaneous edge mass current while B-phase is a helical superfluid with the edge spin current [4]. The Majorana nature for the A-and B-phases is also distinctive; Majorana Fermions have the surface Andreev bound state with linear dispersion relation centered at the zero energy forming a "Majorana valley" in the A-phase [4,5] and a "Majorana cone" in the B-phase [6][7][8]. This Majorana cone is emerging by recent experiments [9,10]. However, there has been no firm evidence for detecting the Majorana zero mode. We are still in the process to better characterize this elusive quasi-particle.The edge mass current spontaneously generated at the boundary in the superfluid 3 He A-phase is carried both by the Majorana quasi-particles (QPs) within the energy gap and by QPs in the continuum state outside the gap. Those two kinds of QPs play a different role for the edge mass current and associated total angular momentum L of the whole system, such as a disk. Stone and Roy [11] proved that in the A-phase the magnitude of the total angular momentum by the edge mass current is N /2 at the zero temperature (N is the total number of 3 He atoms in disk geometry).Here we pose problems on what amount of this mass current or the angular momentum L carried by the Majorana QPs and the remaining continuum state. It will turn out that exactly L MJ = N for Majorana QPs and L...

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The Topology of the Quantum Vacuum

Volovik

2013

Lecture Notes in Physics

111

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Topology in momentum space is the main characteristics of the ground states of a system at zero temperature, the quantum vacua. The gaplessness of fermions in bulk, on the surface or inside the vortex core is protected by topology, and is not sensitive to details of the microscopic physics (atomic or trans-Planckian). Irrespective of the deformation of the parameters of the microscopic theory, the energy spectrum of these fermions remains strictly gapless. This solves the main hierarchy problem in particle physics: for fermionic vacua with Fermi points the masses of elementary particles are naturally small. The quantum vacuum of Standard Model is one of the representatives of topological matter alongside with topological superfluids and superconductors, topological insulators and semi-metals, etc. There is a number of of topological invariants in momentum space of different dimensions. They determine universality classes of the topological matter and the type of the effective theory which emerges at low energy. In many cases they also give rise to emergent symmetries, including the effective Lorentz invariance, and emergent phenomena such as effective gauge and gravitational fields. The topological invariants in extended momentum and coordinate space determine the bulk-surface and bulkvortex correspondence. They connect the momentum space topology in bulk with the real space. These invariants determine the gapless fermions living on the surface of a system or in the core of topological defects (vortices, strings, domain walls, solitons, monopoles, etc.). The momentum space topology gives some lessons for quantum gravity. In effective gravity emerging at low energy, the collective variables are the tetrad field and spin connections, while the metric is the composite object of tetrad field. This suggests that the Einstein-Cartan-Sciama-Kibble theory with torsion field is more relevant. There are also several scenarios of Lorentz invariance violation governed by topology, including splitting of Fermi point and development of the Dirac points with quadratic and cubic spectrum. The latter leads to the natural emergence of the Hořava-Lifshitz gravity.

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Majorana surface states of superfluidHe3AandB

Cited by 67 publications

References 49 publications

Symmetry protected topological superfluid³He-B

Symmetry protected topological superfluid³He-B

Edge mass current and the role of Majorana fermions inA-phase superfluid3He

The Topology of the Quantum Vacuum

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Majorana surface states of superfluidHe3AandB

Cited by 67 publications

References 49 publications

Symmetry protected topological superfluid3He-B

Symmetry protected topological superfluid3He-B

Edge mass current and the role of Majorana fermions inA-phase superfluid3He

The Topology of the Quantum Vacuum

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Symmetry protected topological superfluid³He-B

Symmetry protected topological superfluid³He-B