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Register a new userHalf-Quantum Vortices in Nematic and Chiral Phases of $^3$He
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We report theoretical results for the stability of half-quantum vortices (HQVs) in the superfluid phases of $^3$He confined in highly anisotropic Nafen aerogel. Superfluidity of $^3$He confined in Nafen is the realization of a nematic superfluid with Cooper pairs condensed into a single p-wave orbital aligned along the anisotropy axis of the Nafen aerogel. In addition to the nematic phase, we predict a second chiral phase that onsets at a lower transition temperature. This chiral phase spontaneously breaks time-reversal symmetry and is a topological superfluid. Both superfluid phases are equal-spin pairing condensates that host arrays of HQVs as equilibrium states of rotating superfluid $^3$He. We present results for the structure of HQVs, including magnetic and topological signatures of HQVs in both the nematic and chiral phases of $^3$He-Nafen.
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The existence and stability of non-Abelian half-quantum vortices (HQVs) are established in ${}^{3}P_{2}$ superfluids in neutron stars with strong magnetic fields, the largest topological quantum matter in our Universe. Using a self-consistent microscopic framework, we find that one integer vortex is energetically destabilized into a pair of two non-Abelian HQVs due to the strong spin-orbit coupled gap functions. We find a topologically protected Majorana fermion on each HQV, thereby providing two-fold non-Abelian anyons characterized by both Majorana fermions and a non-Abelian first homotopy group.
We start by showing that the most generic spin-singlet pairing in a superconducting Weyl/Dirac semimetal is specified by a $U(1)$ phase $e^{iphi}$ and $two~real~numbers$ $(Delta_s,Delta_5)$ that form a representation of complex algebra. Such a complex superconducting state realizes a $Z_2times U(1)$ symmetry breaking in the matter sector where $Z_2$ is associated with the chirality. The resulting effective XY theory of the fluctuations of the $U(1)$ phase $phi$ will be now augmented by coupling to another dynamical variable, the $chiral~angle$ $chi$ that defines the polar angle of the complex number $(Delta_s,Delta_5)$. We compute this coupling by considering a Josephson set up. Our energy functional of two phase variables $phi$ and $chi$ allows for the realization of a half-vortex (or double Cooper pair) state and its BKT transition. The half-vortex state is sharply characterized by a flux quantum which is half of the ordinary superconductors. Such a $pi$-periodic Josephson effect can be easily detected as doubled ac Josephson frequency. We further show that the Josephson current $I$ is always accompanied by a $chiral~Josephson~current$ $I_5$. Strain pseudo gauge fields that couple to the $chi$, destabilize the half-vortex state. We argue that our complex superconductor realizes an extension of XY model that supports confinement transition from half-vortex to full vortex excitations.
Role of inhomogeneous perturbations of nematic aerogel on the form of the order parameter emerging at the transition of liquid $^3$He in the superfluid state is considered. It is shown that a region of stability of the polar distorted ABM phase can begin right from the transition temperature. The symmetry argument is given, which selects the most favorable aerogel for stabilization of pure polar phase.
Majorana zero-modes in a superconductor are midgap states localized in the core of a vortex or bound to the end of a nanowire. They are anyons with non-Abelian braiding statistics, but when they are immobile one cannot demonstrate this by exchanging them in real space and indirect methods are needed. As a real-space alternative, we propose to use the chiral motion along the boundary of the superconductor to braid a mobile vortex in the edge channel with an immobile vortex in the bulk. The measurement scheme is fully electrical and deterministic: edge vortices ($pi$-phase domain walls) are created on demand by a voltage pulse at a Josephson junction and the braiding with a Majorana zero-mode in the bulk is detected by the charge produced upon their fusion at a second Josephson junction.
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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