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 evaluat
e 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 investigate how the vortex-vortex separation changes Majorana zero modes in the vicinity of the BCS-BEC (Bose-Einstein condensation) topological phase transition of p-wave resonant Fermi gases. By analytically and numerically solving the Bogoliubo
v-de Gennes equation for spinless p-wave superfluids with plural vortices, it is demonstrated that the quasiparticle tunneling between neighboring vortices gives rise to the quantum oscillation of the low-lying spectra on the scale of the Fermi wavelength in addition to the exponential splitting. This rapid oscillation, which appears in the weak coupling regime as a consequence of quantum oscillations of quasiparticle wave functions, disappears in the vicinity of the BCS-BEC topological phase transition. This is understandable from that the wave function of the Majorana zero modes is described by the modified Bessel function in the strong coupling regime and thus it becomes spread over the vortex core region. Due to the exponential divergence of the modified Bessel function, the concrete realization of the Majorana zero modes near the topological phase transition requires the neighboring vortices to be separated beyond the length scale defined by the coherence length and the dimensionless coupling constant. All these behaviors are also confirmed by carrying out the full numerical diagonalization of the non-local Bogoliubov-de Gennes equation in a two dimensional geometry. Furthermore, this argument is expanded into the case of three-vortex systems, where a pair of core-bound and edge-bound Majorana states survive at zero energy state regardless of the vortex separation.
Multiply quantized vortices in the BCS-to-BEC evolution of p-wave resonant Fermi gases are investigated theoretically. The vortex structure and the low-energy quasiparticle states are discussed, based on the self-consistent calculations of the Bogoli
ubov-de Gennes and gap equations. We reveal the direct relation between the macroscopic structure of vortices, such as particle densities, and the low-lying quasiparticle state. In addition, the net angular momentum for multiply quantized vortices with a vorticity $kappa$ is found to be expressed by a simple equation, which reflects the chirality of the Cooper pairing. Hence, the observation of the particle density depletion and the measurement of the angular momentum will provide the information on the core-bound state and $p$-wave superfluidity. Moreover, the details on the zero energy Majorana state are discussed in the vicinity of the BCS-to-BEC evolution. It is demonstrated numerically that the zero energy Majorana state appears in the weak coupling BCS limit only when the vortex winding number is odd. There exist the $kappa$ branches of the core bound states for a vortex state with vorticity $kappa$, whereas only one of them can be the zero energy. This zero energy state vanishes at the BCS-BEC topological phase transition, because of interference between the core-bound and edge-bound states.
Motivated by the on-going rotating cryostat experiments in ISSP, Univ. of Tokyo, we explore the textures and vortices in superfluid 3He-A phase confined in narrow cylinders, whose radii are R=50mum and 115mum. The calculations are based on the Ginzbu
rg-Landau (GL) framework, which fully takes into account the orbital (l-vector) and spin (d-vector) degrees of freedom for chiral p-wave pairing superfluid. The GL free energy functional is solved numerically by using best known GL parameters appropriate for the actual experimental situations at P=3.2MPa and H=21.6mT. We identify the ground state l-vector configuration as radial disgyration (RD) texture with the polar core both at rest and low rotations and associated d-vector textures for both narrow cylinder systems under high magnetic fields. The RD which has a singularity at center, changes into Mermin-Ho texture above the critical rotation speed which is determined precisely, providing an experimental check for own proposal.
It is found theoretically based on the Ginzburg-Landau framework that p-wave superfluids of neutral atom gases in three dimension harmonic traps exhibit spontaneous mass current at rest, whose direction depends on trap geometry. Under rotation variou
s types of the order parameter textures are stabilized, including Mermin-Ho and Anderson-Toulouse-Chechetkin vortices. In a cigar shape trap spontaneous current flows longitudial to the rotation axis and thus perpendicular to the ordinary rotational current. These features, spontaneous mass current at rest and texture formation, can be used as diagnoses for p-wave superfluidity.
We have found the precise stability region of the half quantum vortex (HQV) for superfluid $^3$He A phase confined in parallel plates with a narrow gap under rotation. Standard Ginzburg-Landau free energy, which is well established, is solved to loca
te the stability region spanned by temperature $T$ and rotation speed ($/Omega$). This $/Omega$-$T$ stability region is wide enough to check it experimentally in available experimental setup. The detailed order parameter structure of HQV characterized by A$_1$ core is given to facilitate the physical reasons of its stability over other vortices or textures.
The visualization of chiral p-wave superfluidity in Fermi gases near p-wave Feshbach resonances is theoretically examined. It is proposed that the superfluidity becomes detectable in the entire BCS-BEC regimes through (i) vortex visualization by the
density depletion inside the vortex core and (ii) intrinsic angular momentum in vortex free states. It is revealed that both (i) and (ii) are closely connected with the Majorana zero energy mode of the vortex core and the edge mode, which survive until the strong coupling BCS regime is approached from the weak coupling limit and vanish in the BEC regime.
We theoretically investigate the ground state of trapped neutral fermions with population imbalance in the BCS-BEC crossover regime. On the basis of the single-channel Hamiltonian, we perform full numerical calculations of the Bogoliubov-de Gennes eq
uation coupled with the regularized gap and number equations. The zero-temperature phase diagram in the crossover regime is presented, where the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing state governs the weak-coupling BCS region of a resonance. It is found that the FFLO oscillation vanishes in the BEC side, in which the system under population imbalance turns into a phase separation (PS) between locally binding superfluid and fully polarized spin domains. We also demonstrate numerical calculations with a large particle number O(10^5), comparable to that observed in recent experiments. The resulting density profile on a resonance yields the PS, which is in good agreement with the recent experiments, while the FFLO modulation exists in the pairing field. It is also proposed that the most favorable location for the detection of the FFLO oscillation is in the vicinity of the critical population imbalance in the weak coupling BCS regime, where the oscillation periodicity becomes much larger than the interparticle spacing. Finally, we analyze the radio-frequency (RF) spectroscopy in the imbalanced system. The clear difference in the RF spectroscopy between BCS and BEC sides reveals the structure of the pairing field and local ``magnetization.
The field dependence of the specific heat gamma(H) at lower temperatures in Sr2RuO4 is analyzed by solving microscopic Eilenberger equation numerically. We find that systematic gamma(H) behaviors from a concaved sqrt H to a convex H^{alpha} (alpha>1)
under H orientation change are understood by taking account of the Pauli paramagnetic effect. The magnetizations are shown to be consistent with it. This implies either a singlet pairing or a triplet one with d-vector locked in the basal plane, which allows us to explain other mysteries of this compound in a consistent way.
