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arXiv:1209.5501 · Physical Review B

Surface states, edge currents, and the angular momentum of chiral p -wave superfluids

James A. Sauls

Published Dec 7, 2011• 124 citations
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Abstract

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 ${L}_{z}(T)$ 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 ${L}_{z}$, 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 ${}^{3}$He-$A$, as well as direct observation of chiral edge currents.

DOI
10.1103/physrevb.84.214509

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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 \ensuremathℏ/pf\ensuremathξ\ensuremath{\hbar}/{p}_{f}\ensuremath{\xi}\ensuremathℏ/pf​\ensuremathξ. 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≲T\ensuremath≪Tc0\ensuremath{\lesssim}T\ensuremath{\ll}{T}_{c}0\ensuremath≲T\ensuremath≪Tc​ 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, Lz=(N/2)\ensuremathℏ{L}_{z}=(N/2)\ensuremath{\hbar}Lz​=(N/2)\ensuremathℏ, 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 Lz>(N/2)\ensuremathℏ{L}_{z}>(N/2)\ensuremath{\hbar}Lz​>(N/2)\ensuremathℏ to Lz<\ensuremath−(N/2)\ensuremathℏ{L}_{z}<\ensuremath{-}(N/2)\ensuremath{\hbar}Lz​<\ensuremath−(N/2)\ensuremathℏ 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 study the orbital angular momentum (OAM) Lz{L}_{z}Lz​ in two-dimensional chiral $({p}_{x}+i{p}_{y}{)}^{\ensuremath{\nu}}$-wave superfluids (SFs) of NNN 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 Lz=N/2{L}_{z}=N/2Lz​=N/2 for the p+ipp+ipp+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 ϵF{ϵ}_{F}ϵF​ is the Fermi energy. We demonstrate that the difference between the p+ipp+ipp+ip-wave SF and the other chiral SFs in the BCS regimes originates from the nature of edge modes and related depairing effects.
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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.
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