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Register a new userEdge Current due to Majorana Fermions in Superfluid $^3$He A- and B-Phases
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We propose a method utilizing edge current to observe Majorana fermions in the surface Andreev bound state for the superfluid $^3$He A- and B-phases. The proposal is based on self-consistent analytic solutions of quasi-classical Greens function with an edge. The local density of states and edge mass current in the A-phase or edge spin current in the B-phase can be obtained from these solutions. The edge current carried by the Majorana fermions is partially cancelled by quasiparticles (QPs) in the continuum state outside the superfluid gap. QPs contributing to the edge current in the continuum state are distributed in energy even away from the superfluid gap. The effect of Majorana fermions emerges in the depletion of the edge current by temperature within a low-temperature range. The observations that the reduction in the mass current is changed by $T^2$-power in the A-phase and the reduction in the spin current is changed by $T^3$-power in the B-phase establish the existence of Majorana fermions. We also point out another possibility for observing Majorana fermions by controlling surface roughness.
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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 $L=Nhbar/2$, consisting of $L^{rm MJ}=Nhbar$ from the Majorana quasi-particles (QPs) and $L^{rm cont}=-Nhbar/2$ 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 $N_0/2$ ($N_0$: normal state DOS) responsible for the factor 2 difference between $L^{rm MJ}$ and $L^{rm cont}$. The current decreases as $E^{-3}$ against the energy $E$, and $L(T) propto -T^2$. This analytic solution is fully backed up by numerically solving the Eilenberger equation. We touch on the so-called intrinsic angular momentum problem.
Motivated by experiments on the superfluid 3He confined in a thin slab, we design a concrete experimental setup for observing the Majorana surface states. We solve the quasi-classical Eilenberger equation, which is quantitatively reliable, to evaluate several quantities, such as local density of states (LDOS), mass current for the A-phase, and spin current for the B-phase. In connection with realistic slab samples, we consider the upper and lower surfaces and the side edges including the corners with several thicknesses. Consequently the influence on the Majorana zero modes from the spatial variation of l-vector for the A-phase in thick slabs and the energy splitting of the zero-energy quasi-particles for the B-phase confined in thin slabs are demonstrated. The corner of slabs in the B-phase is accompanied by the unique zero-energy LDOS of corner modes. On the basis of the quantitative calculation, we propose several feasible and verifiable experiments to check the existence of the Majorana surface states, such as the measurement of specific heat, edge current, and anisotropic spin susceptibility.
The theoretical study of topological superfluids and superconductors has so far been carried out largely as a translation of the theory of noninteracting topological insulators into the superfluid language, whereby one replaces electrons by Bogoliubov quasiparticles and single-particle band Hamiltonians by Bogoliubov-de Gennes Hamiltonians. Band insulators and superfluids are, however, fundamentally different: while the former exist in the absence of inter-particle interactions, the latter are broken symmetry states that owe their very existence to such interactions. In particular, unlike the static energy gap of a band insulator, the gap in a superfluid is due to a dynamical order parameter that is subject to both thermal and quantum fluctuations. In this work, we explore the consequences of bulk quantum fluctuations of the order parameter in the $B$ phase of superfluid $^3$He on the topologically protected Majorana surface states. Neglecting the high-energy amplitude modes, we find that one of the three spin-orbit Goldstone modes in $^3$He-$B$ couples to the surface Majorana fermions. This coupling in turn induces an effective short-range two-body interaction between the Majorana fermions, with coupling constant inversely proportional to the strength of the nuclear dipole-dipole interaction in bulk $^3$He. A mean-field theory estimate of the value of this coupling suggests that the surface Majorana fermions in $^3$He-$B$ are in the vicinity of a quantum phase transition to a gapped time-reversal symmetry breaking phase.
We investigate torsional chiral magnetic effect (TCME) induced by skyrmion-vortex textures in the A phase of the superfluid $^3$He. In $^3$He-A, Bogoliubov quasiparticles around point nodes behave as Weyl fermions, and the nodal direction represented by the $ell$-vector may form a spatially modulated texture. $ell$-textures generate a chiral gauge field and a torsion field directly acting on the chirality of Weyl-Bogoliubov quasiparticles. It has been clarified by G. E. Volovik [Pisma Zh. Eksp. Teor. Fiz. {bf 43}, 428 (1986)] that, if the $ell$-vector is twisted, the chiral gauge field is responsible for the chiral anomaly, leading to an anomalous current along ${ell}$. Here we show that, even for non-twisted $ell$-vector fields, a torsion arising from $ell$-textures brings about contributions to the equilibrium currents of Weyl-Bogoliubov quasiparticles along ${rm curl}{ell}$. This implies that while the anomalous current appears only for the twisted (Bloch-type) skyrmion of the $ell$-vector, the extra mass current due to TCME always exists regardless of the skyrmion type. Solving the Bogoliubov-de Gennes equation, we demonstrate that both Bloch-type and N{e}el-type skyrmions induce chiral fermion states with spectral asymmetry, and possess spatially inhomogeneous structures of Weyl bands in the real coordinate space. Furthermore, we discuss the contributions of Weyl-Bogoliubov quasiparticles and continuum states to the mass current density in the vicinity of the topological phase transition. In the weak coupling limit, continuum states give rise to backflow to the mass current generated by Weyl-Bogoliubov quasiparticles, which makes a non-negligible contribution to the orbital angular momentum. As the topological transition is approached, the mass current density is governed by the contribution of continuum states.
The superfluid $^3$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.
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