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Register a new userOdd-frequency Cooper pairs and zero-energy surface bound states in superfluid 3He
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We study the odd-frequency Cooper pairs formed near the surface of superfluid 3He. The odd-frequency pair amplitude is closely related to the local density of states in the low energy limit. We derive a formula relating explicitly the two quantities. This formula holds for arbitrary boundary condition at the surface. We also present some numerical results on the surface odd-frequency pair amplitude in superfluid 3He-B. Those analytical and numerical results allow one to interpret the midgap surface density of states, observed recently by transverse acoustic impedance measurements on superfluid 3He-B, as the manifestation of the surface odd-frequency state.
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We propose that the odd-frequency $s$ wave ($s^{rm{odd}}$ wave) superconducting gap function, which is usually unstable in the bulk, naturally emerges at the edge of $d$ wave superconductors. This prediction is based on the surface spin fluctuation pairing mechanism owing to the zero-energy surface Andreev bound state. The interference between bulk and edge gap functions triggers the $d+s^{rm{odd}}$ state, and the generated spin current is a useful signal uncovering the ``hidden odd-frequency gap. In addition, the edge $s^{rm{odd}}$ gap can be determined via the proximity effect on the diffusive normal metal. Furthermore, this study provides a decisive validation of the ``Hermite odd-frequency gap function, which has been an open fundamental challenge to this field.
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.
We theoretically study the magnetization inside a normal metal induced in an s-wave superconductor/ferromagnetic metal/normal metal/ferromagnetic metal/s-wave superconductor (S/F1/N/F2/S) Josephson junction. Using quasiclassical Greens function method, we show that the magnetization becomes finite inside N. The origin of this magnetization is due to odd-frequency spin-triplet Cooper pairs formed by electrons of equal and opposite spins, which are induced by proximity effect in the S/F1/N/F2/S junction. We find that the magnetization M(d,q) in N can be decomposed into two parts, M(d,q)=MI(d)+MII(d,q), where q is the superconducting phase difference between two Ss and d is the thickness of N. MI(d) exists generally in S/F junctions, while MII(d,q) carries all q dependence and represents the fingerprint of phase coherence between two Ss in Josephson junctions. The q dependence thus allows us to control the magnetization in N by tuning q for a fixed d. We show that MI(d) weakly decreases with increasing d, while the q dependent magnetization MII(d,q) rapidly decays with d. Moreover, we find that the time-averaged magnetization <MII(d,q)> exhibits discontinuous peak at each resonance DC voltage Vn=nhw_S/2e(n: integer) when DC voltage V as well as AC voltage v_ac(t) with frequency w_S are both applied to the S/F1/N/F2/S junction. This is because MII(d,q) oscillates generally in time t (AC magnetization) with dq/dt=2e[V+v_ac(t)]/h and thus <MII(d,q)>=0, but can be converted into the time-independent DC magnetization for DC voltage at Vn. We also discuss that the magnetization induced in N can be measurably large in realistic systems. Therefore, the measurement of the induced magnetization serves as an alternative way to detect the phase coherence between two Ss in Josephson junctions. Our results also provide a basic concept for tunable magnetization in superconducting spintronics devices.
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.
A concrete and experimentally feasible example for testing the putative Majorana zero energy state bound in a vortex is theoretically proposed for a parallel plate geometry of superfluid $^3$He-A phase. We examine the experimental setup in connection with ongoing rotating cryostat experiments. The theoretical analysis is based on the well-established Ginzburg--Landau functional, supplemented by microscopic calculations of the Bogoliubov--de Gennes equation, both of which allow the precise location of the parameter regions of the Majorana state to be found in realistic situations.
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