Surface Majorana Cone of the Superfluid ³He B Phase

Abstract

The superfluid3 He B phase, one of the oldest unconventional fermionic condensates experimentally realized, is recently predicted to support Majorana fermion surface states. Majorana fermion, which is characterized by the equivalence of particle and antiparticle, has a linear dispersion relation referred to as the Majorana cone. We measured the transverse acoustic impedance Z of the superfluid 3 He B phase changing its boundary condition and found a growth of peak in Z on a higher specularity wall. Our theoretical analysis indicates that the variation of Z is induced by the formation of the cone-like dispersion relation and thus confirms the important feature of the Majorana fermion in the specular limit. KEYWORDS: superfluid3 He, surface Andreev bound states, Majorana fermions, Majorana cone, acoustic impedanceSurface Andreev bound states (SABS) of the superfluid 3 He B phase are receiving renewed attention as ''edge states'' of a three-dimensional (3D) time reversal invariant topological superfluid.1-9) Topological superfluids and superconductors are characterized by a non-trivial topological number in the gapped bulk state and a gapless edge state on their edges or surfaces. SABS of the superfluid 3 He B phase can be regarded as Majorana fermions as they satisfy the Majorana condition, i.e., a particle and its antiparticle are equivalent, because the degrees of freedom of the bound states are halved. When a surface is specular, linear dispersions are predicted for Majorana fermions, 10) which have been recently referred to as a Majorana cone. 4,5) We measured the transverse acoustic impedance Z of the superfluid 3 He B phase changing its boundary condition up to practically specular scattering. The observed variation of Z is well reproduced by theoretical analysis and is shown to be induced by the formation of the cone-like dispersion relation of SABS at higher specularities. This provides an evidence of the existence of the Majorana cone in superfluid 3 He B in the specular limit. Superfluid 3 He is a well established spin-triplet p-wave superfluid where the B phase is the realization of the BalianWerthamer state, which breaks the relative spin-orbit symmetry.11,12) The surface Majorana states of the 3 He B phase should exhibit peculiar features such as a surface spin current and an anisotropic magnetic response.5,7-9) Under the specular scattering boundary condition of the surface, SABS with a finite energy are not degenerate, and have a dispersion relation linear to the momentum, which takes the shape of a 2D cone for massless Dirac fermions. In the spin-triplet p-wave pairing system, the annihilation operator of the negative energy state is equivalent to the creation operator of the positive energy state. Hence, only the positive energy cone, which is called a Majorana cone, is physical.4,5) To date, Majorana surface fermion states have been discussed mostly on smooth surfaces (specular scattering limit). However, intriguing problems are to elucidate how the Majorana cone transforms as the roughness of th...

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

“…The angular momenta L MJ and L cont can be suppressed by the different rates as a function of the surface roughness. Here we notice in the case of the superfluid 3 He B-phase that the surface Majorana bound state remains at a partially specular surface [9,10]. The concrete calculation to estimate quantitatively this change is a future problem.…”

“…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.…”

Symmetry protected topological superfluid3He-B

“…At the weak coupling limit, the planar is energetically degenerate to the A-phase that is the time-reversal symmetry breaking phase with point nodes, while the strong coupling effect that is the spin-fluctuation feedback effect can stabilize the A-phase relative to the planar phase and the A-B phase boundary becomes the first order transition. The pair breaking effect and the enhancement of the surface density of states due to the surface bound state have been observed by several experiments, by using the NMR techniques [112,113,114,115,116,117,118,155,156,157], the motion of a vibrating wire resonator [158], transverse acoustic impedance [162,163,164,165,166,167,168,169], surface contributions of the heat capacity [160], and anomalous attenuation of transverse sound [161]. Among them, Murakawa et al [165,168] has observed the specularity dependence of the surface density of states by systematically controlling the surface specularity by coating the surface with 4 He layers.…”

“…The surface density of states has a very sharp edge at E = ∆ * that is smaller than the bulk energy gap ∆ 0 . The temperature dependence of the complex impedance observed two different energy scales ∆ * and ∆ 0 [162,163,164,165,166,167,168,169] as a kink and peak in Z ′ (T ) and Z ′′ (T ). The remarkable point is that the surface specularity is controllable by coating the wall of the transducer with thin layers of 4 He.…”

Quantum thermal Hall effect of Majorana fermions on the surface of superconducting topological insulators

“…On the surface of 3 He B phase, a clear linear dispersion of in-gap states was measured experimentally 18 . In addition, the newly found superconducting phase in Cudoped Bi 2 Se 3 19 has been proposed to be a 3d-TSC 20 .…”