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Register a new userNMR Frequency Shifts and Phase Identification in Superfluid $^3$He
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The pressure dependence of the order parameter in superfluid $^3$He is amazingly simple. In the Ginzburg-Landau regime, i.e. close to $T_c$, the square of the order parameter can be accurately measured by its proportionality to NMR frequency shifts and is strictly linear in pressure. This behavior is replicated for superfluid $^3$He imbibed in isotropic and anisotropic silica aerogels. The proportionality factor is constrained by the symmetry of the superfluid state and is an important signature of the corresponding superfluid phase. For the purpose of identifying various new superfluid states in the $p$-wave manifold, the order parameter amplitude of $^3$He-A is a useful reference, and this simple pressure dependence greatly facilitates identification.
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The superfluid $^3$He formed by spin-triplet $p$-wave Cooper pairs is a typical topological superfluid. In the superfluid $^3$He B-phase, several kinds of vortices classified by spatial symmetries $P_1$, $P_2$, and $P_3$ are produced, where $P_1$ is inversion symmetry, $P_2$ is magnetic reflection symmetry, and $P_3$ is magnetic $pi$-rotation symmetry. We have calculated the vortex bound states by the Bogoliubov-de Gennes theory and the quasiclassical Eilenberger theory, and also clarified symmetry protection of the low energy excitations by the spatial symmetries. On the symmetry protection, $P_3$ symmetry plays a key role which gives two-fold degenerate Majorana zero modes. Then, the bound states in the most symmetric $o$ vortex with $P_1$, $P_2$, and $P_3$ symmetries and in $w$ vortex with $P_3$ symmetry have the symmetry protected degenerate Majorana zero modes. On the other hand, zero energy modes in $v$ vortex, which is believed to be realized in the actual B-phase, are not protected, and in consequence become gapped by breaking axial symmetry. The excitation gap may have been observed as the variation of critical velocity. We have also suggested an experimental setup to create $o$ vortex with Majorana zero modes by a confinement and a magnetic field.
We consider fermionic states bound on domain walls in a Weyl superfluid $^3$He-A and on interfaces between $^3$He-A and a fully gapped topological superfluid $^3$He-B. We demonstrate that in both cases fermionic spectrum contains Fermi arcs which are continuous nodal lines of energy spectrum terminating at the projections of two Weyl points to the plane of surface states in momentum space. The number of Fermi arcs is determined by the index theorem which relates bulk values of topological invariant to the number of zero energy surface states. The index theorem is consistent with an exact spectrum of Bogolubov- de Gennes equation obtained numerically meanwhile the quasiclassical approximation fails to reproduce the correct number of zero modes. Thus we demonstrate that topology describes the properties of exact spectrum beyond quasiclassical approximation.
The discovery of superfluid $^{3}$He in high porosity silica aerogels, and subsequent experimental and theoretical work, have led to a better general understanding of quasiparticle scattering from impurities in unconventional pairing systems. It is immensely helpful for understanding impurity effects in the case of superfluid $^{3}$He that the structure of its order parameter is well-established. An overview of impurity effects is presented with emphasis on those experiments which have a quantitative interpretation in terms of theoretical models for homogeneous and inhomogeneous scattering. The latter can account successfully for most experimental results.
We have performed longitudinal ultrasound (9.5 MHz) attenuation measurements in the B-phase of superfluid $^3$He in 98% porosity aerogel down to the zero temperature limit for a wide range of pressures at zero magnetic field. The absolute attenuation was determined by direct transmission of sound pulses. Compared to the bulk fluid, our results revealed a drastically different behavior in attenuation, which is consistent with theoretical accounts with gapless excitations and a collision drag effect.
The specific heat of superfluid $^{3}$He, disordered by a silica aerogel, is found to have a sharp discontinuity marking the thermodynamic transition to superfluidity at a temperature reduced from that of bulk $^{3}$He. The magnitude of the discontinuity is also suppressed. This disorder effect can be understood from the Ginzburg-Landau theory which takes into account elastic quasiparticle scattering suppressing both the transition temperature and the amplitude of the order parameter. We infer that the limiting temperature dependence of the specific heat is linear at low temperatures in the disordered superfluid state, consistent with predictions of gapless excitations everywhere on the Fermi surface.
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