Cooper pairs in chiral superfluids carry quantized units of relative orbital angular momentum (OAM). Various predictions of the intrinsic OAM density or the macroscopic OAM of a two-dimensional chiral superfluid differ by several orders of magnitude, which constitute the so-called angular momentum paradox. Following several previous studies, we substantiate the semiclassical Bogoliubov--de Gennes theory of the single-particle edge current and OAM in two-dimensional chiral superfluids in the BCS limit. The analysis provides a simple intuitive understanding for the vanishing of OAM for a non-
p-wave chiral superfluid (such as
d+id) confined in a rigid potential. When generalized to anisotropic chiral superconductors and three-dimensional chiral superfluids, the theory similarly returns an accurate description. We also present a detailed numerical study of the chiral phases in the BEC limit. Our study suggests that, in both BCS and BEC phases, the relative OAM of the individual Cooper pairs contribute to the total OAM additively, and that in both phases the corresponding macroscopic OAM density distribution is localized at the boundary.
As an example of the skyrmion lattice, the structure of Mermin-Ho vortex lattice in superfluids
3He is studied by self-consistent Eilenberger theory and by Bogoliubov-de Gennes theory. We identify how the intrinsic orbital angular momentum
l of
p-wave Cooper pairs contributes to spatial structures of the pair potential, current flow, and quasiparticle states. There are two types of vortices depending on the
l direction relative to rotation. Only one of them has zero-energy bound states appearing even in coreless vortices due to intrinsic topological reasons.
We present a new calculation of the current $\stackrel{\ensuremath{\rightarrow}}{\mathrm{g}}$ flowing in a ground state of the Bardeen-Cooper-Schrieffer (BCS) form for a weakly inhomogeneous superfluid with the symmetry of
3He-
A. When the structure of the order parameter not determined by symmetry is appropriate to
3He-
A and when the mass density $\ensuremath{\rho}$ of the helium is essentially uniform, our current reduces to that calculated by Cross. If the mass density is allowed to vary, we find a generalization of the Cross current which shows that when ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}}_{s}=0$ and the anisotropy axis $\stackrel{\ensuremath{\rightarrow}}{\mathrm{l}}$ is uniform, then the current is simply $(\frac{\ensuremath{\hbar}}{4M}){\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\nabla}}}_{\ensuremath{\rho}}\ifmmode\times\else\texttimes\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{l}}$. We show that this property of the BCS ground state, which taken with the Cross definition leads to an "intrinsic angular momentum density" of $\frac{\ensuremath{\rho}\ensuremath{\hbar}}{2M}$ at zero temperature, also follows directly from the Gor'kov equations. If the range of the order parameter is taken to be small compared with the interatomic separation, then the ground state does not describe
3He-
A, but a Bose-Einstein condensate of tightly bound diatomic molecules. In this limit our current reduces to the form calculated by Ishikawa et al. We indicate why their analysis is only valid in this limit, and offer some rather more general remarks on the differences between Cooper pairing and the Bose-Einstein condensation of diatomic molecules.
The spectra of fermionic excitations, pairing correlations, and edge currents confined near the boundary of a chiral
p-wave superfluid are calculated to leading order in $\ensuremath{\hbar}/{p}_{f}\ensuremath{\xi}$. Results for the energy- and momentum-resolved spectral functions, including the spectral current density, of a chiral
p-wave superfluid near a confining boundary are reported. The spectral functions reveal the subtle role of the chiral edge states in relation to the edge current and the angular momentum of a chiral
p-wave superfluid, including the rapid suppression of
Lz(T) for $0\ensuremath{\lesssim}T\ensuremath{\ll}{T}_{c}$ in the fully gapped two-dimensional chiral superfluid. The edge current and ground-state angular momentum are shown to be sensitive to boundary conditions, and as a consequence the topology and geometry of the confining boundaries. For perfect specular boundaries, the edge current accounts for the ground-state angular momentum, ${L}_{z}=(N/2)\ensuremath{\hbar}$, of a cylindrical disk of a chiral superfluid with
N/2 fermion pairs. Nonspecular scattering can dramatically suppress the edge current. In the limit of perfect retroreflection, the edge states form a flat band of zero modes that are nonchiral and generate no edge current. For a chiral superfluid film confined in a cylindrical toroidal geometry, the ground-state angular momentum is, in general, nonextensive, and can have a value ranging from ${L}_{z}>(N/2)\ensuremath{\hbar}$ to ${L}_{z}<\ensuremath{-}(N/2)\ensuremath{\hbar}$ depending on the ratio of the inner and outer radii and the degree of backscattering on the inner and outer surfaces. Nonextensive scaling of
Lz, and the reversal of the ground-state angular momentum for a toroidal geometry, would provide a signature of broken time-reversal symmetry of the ground state of superfluid
3He-
A, as well as direct observation of chiral edge currents.
The kinetic energy of a charged system is calculated for Mermin's one-particle density matrix of Fermi superfluids. The final expression consists of two separate contributions from the center-of-mass and the relative motions of an effective bound-state wave function. The relative part naturally includes coupling of the intrinsic orbital angular momentum density to the magnetic field. For 3 He- A at T =0, its magnitude is
(ℏ/2)×(number density), in agreement with Ishikawa's result.
With proper care on the boundary conditions, the total orbital angular momentum L z at finite temperatures is microscopically calculated for the Anderson-Brinkman-Morel state in a cylinder with axial symmetry about the z axis, i.e. the textures considered by McClure and Takagi at T =0. The pairing in this geometry occurs between m and 1- m one-particle states, where m denotes the axial quantum number. It is found that L z decreases from its zero temperature value
2ℏN, with N the total particle number, in the same way as the components of the superfluid density tensor. Responsible for this reduction is identified to be the phase shift between m and 1- m quasiparticle states caused by the difference in the centrifugal potentials. This fact implies that the plane-wave representation is not suitable for the problem.