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arXiv:1409.7459 · Physical Review Letters

Orbital Angular Momentum and Spectral Flow in Two-Dimensional Chiral Superfluids

Yasuhiro Tada, Wenxing Nie, Masaki Oshikawa

Published May 13, 2015• 68 citations
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Abstract

We study the orbital angular momentum (OAM) ${L}_{z}$ in two-dimensional chiral $({p}_{x}+i{p}_{y}{)}^{\ensuremath{\nu}}$-wave superfluids (SFs) of $N$ fermions on a disk at zero temperature, in terms of spectral asymmetry and spectral flow. It is shown that ${L}_{z}=\ensuremath{\nu}N/2$ for any integer $\ensuremath{\nu}$, in the Bose-Einstein condensation regime. In contrast, in the BCS limit, while the OAM is ${L}_{z}=N/2$ for the $p+ip$-wave SF, for chiral SFs with $\ensuremath{\nu}\ensuremath{\ge}2$, the OAM is remarkably suppressed as ${L}_{z}=N\ifmmode\times\else\texttimes\fi{}O({\mathrm{\ensuremath{\Delta}}}_{0}/{ϵ}_{F})\ensuremath{\ll}N$, where ${\mathrm{\ensuremath{\Delta}}}_{0}$ is the gap amplitude and ${ϵ}_{F}$ is the Fermi energy. We demonstrate that the difference between the $p+ip$-wave SF and the other chiral SFs in the BCS regimes originates from the nature of edge modes and related depairing effects.

DOI
10.1103/physrevlett.114.195301

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We study the orbital angular momentum (OAM) LzL_zLz​ of two-dimensional chiral (px+ipy)ν(p_x+ip_y)^ν(px​+ipy​)ν-wave superfluids (SFs) in the presence of an axisymmetric multiply quantized vortex (MQV) with vorticity kkk on a disk at zero temperature, in the framework of Bogoliubov-de Gennes (BdG) theory. Focusing on spectral asymmetry (or spectral flow), we find that Lz=(k+ν)N/2L_z=(k+ν)N/2Lz​=(k+ν)N/2 for any integer ννν and kkk in the Bose-Einstein Condensation (BEC) regime, where NNN is the total number of fermions. While in the weak-pairing Bardeen-Cooper-Schrieffer (BCS) regime, only for chiral p+ipp+ipp+ip-wave SF with k=±1k=\pm 1k=±1, Lz=(k+ν)N/2L_z=(k+ν)N/2Lz​=(k+ν)N/2 still holds. For chiral SFs with ν≥2ν\ge2ν≥2 or ∣k∣≥2|k|\ge2∣k∣≥2 in the BCS regime, the OAM LzL_zLz​ is remarkably reduced from its ``full" value in the BEC regime. However, the deviations differ in these two cases. For chiral SFs with ν≥2ν\ge2ν≥2, LzL_zLz​ is sharply suppressed in this ideal setting with a specular wall, while the suppression caused by the ∣k∣≥2|k| \ge 2∣k∣≥2 vortex is moderate, which is core-size dependent. Furthermore, for p+ipp+ipp+ip-wave SF with k=−1k=-1k=−1, the total OAM LzL_zLz​ is zero, but the distribution Lz(r)L_z(r)Lz​(r) is nontrivial compared with that of vortex-free sss-wave SF, in which the total OAM is zero as well. For chiral SFs with ν≥2ν\ge2ν≥2 and ∣k∣≥2|k|\ge2∣k∣≥2, the effects of circulation due to vortex and chiral pairing can coexist, and hence depress the OAM simultaneously. These observations can be explained by spectral asymmetry and unpaired fermions in the ground state of the BdG Hamiltonian. We also investigate the spatial distribution of particle density, OAM, by solving the BdG equation.
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-ppp-wave chiral superfluid (such as d+idd+idd+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.
The spectra of fermionic excitations, pairing correlations, and edge currents confined near the boundary of a chiral ppp-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 ppp-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 ppp-wave superfluid, including the rapid suppression of Lz(T){L}_{z}(T)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/2N/2N/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{L}_{z}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 3{}^{3}3He-AAA, as well as direct observation of chiral edge currents.
We compute the orbital angular momentum L_{z} of an s-wave paired superfluid in the presence of an axisymmetric multiply quantized vortex. For vortices with a winding number |k|>1, we find that in the weak-pairing BCS regime, L_{z} is significantly reduced from its value ℏNk/2 in the Bose-Einstein condensation (BEC) regime, where N is the total number of fermions. This deviation results from the presence of unpaired fermions in the BCS ground state, which arise as a consequence of spectral flow along the vortex subgap states. We support our results analytically and numerically by solving the Bogoliubov-de Gennes equations within the weak-pairing BCS regime.
We analyze the evolution of superfluidity for nonzero orbital angular momentum channels from the Bardeen-Cooper-Schrieffer (BCS) to the Bose-Einstein condensation (BEC) limit in three dimensions. First, we analyze the low-energy scattering properties of finite range interactions for all possible angular momentum channels. Second, we discuss ground-state (T=0)(T=0)(T=0) superfluid properties including the order parameter, chemical potential, quasiparticle excitation spectrum, momentum distribution, atomic compressibility, ground-state energy, and low-energy collective excitations. We show that a quantum phase transition occurs for nonzero angular momentum pairing, unlike the sss-wave case where the BCS to BEC evolution is just a crossover. Third, we present a Gaussian fluctuation theory near the critical temperature (T=Tc)(T={T}_{\mathrm{c}})(T=Tc​), and we analyze the number of bound, scattering, and unbound fermions as well as the chemical potential. Finally, we derive the time-dependent Ginzburg-Landau functional near Tc{T}_{\mathrm{c}}Tc​, and compare the Ginzburg-Landau coherence length with the zero-temperature average Cooper pair size.
Two-dimensional spin-orbit-coupled Fermi gases subject to sss-wave pairing can be driven into a topological phase by increasing the Zeeman spin splitting beyond a critical value. In the topological phase, the system exhibits the hallmarks of chiral ppp-wave superfluidity, including exotic Majorana excitations. Previous theoretical studies of this realization of a two-dimensional topological Fermi superfluid have focused on the BCS regime where the sss-wave Cooper pairs are only weakly bound and, hence, the induced chiral ppp-wave order parameter has a small magnitude. Motivated by the goal to identify potential new ways for the experimental realization of robust topological superfluids in ultracold-atom gases, we study the BCS-to-BEC crossover driven by increasing the Cooper-pair binding energy for this system. In particular, we obtain phase diagrams in the parameter space of two-particle bound-state energy and Zeeman spin-splitting energy. Ordinary characteristics of the BCS-to-BEC crossover, in particular the shrinking and eventual disappearance of the Fermi surface, are observed in the nontopological phase. In contrast, the topological phase retains all features of chiral ppp-wave superfluidity, including a well-defined underlying Fermi surface, even for large sss-wave pair-binding energies. Compared to the BCS limit, the topological superfluid in the BEC regime turns out to be better realizable even for only moderate magnitude of spin-orbit coupling because the chiral ppp-wave order parameter is generally larger and remnants of sss-wave pairing are suppressed. We identify optimal parameter ranges that can aid further experimental investigations and elucidate the underlying physical reason for the persistence of the chiral ppp-wave superfluid.