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arXiv:2506.08468 · Physical review. B./Physical review. B

Vortices in two-dimensional chiral superfluids

Yan He, Wenxing Nie

Published Sep 3, 2025• 0 citations
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Abstract

We study the orbital angular momentum (OAM) $L_z$ of two-dimensional chiral $(p_x+ip_y)^ν$-wave superfluids (SFs) in the presence of an axisymmetric multiply quantized vortex (MQV) with vorticity $k$ 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 $L_z=(k+ν)N/2$ for any integer $ν$ and $k$ in the Bose-Einstein Condensation (BEC) regime, where $N$ is the total number of fermions. While in the weak-pairing Bardeen-Cooper-Schrieffer (BCS) regime, only for chiral $p+ip$-wave SF with $k=\pm 1$, $L_z=(k+ν)N/2$ still holds. For chiral SFs with $ν\ge2$ or $|k|\ge2$ in the BCS regime, the OAM $L_z$ is remarkably reduced from its ``full" value in the BEC regime. However, the deviations differ in these two cases. For chiral SFs with $ν\ge2$, $L_z$ is sharply suppressed in this ideal setting with a specular wall, while the suppression caused by the $|k| \ge 2$ vortex is moderate, which is core-size dependent. Furthermore, for $p+ip$-wave SF with $k=-1$, the total OAM $L_z$ is zero, but the distribution $L_z(r)$ is nontrivial compared with that of vortex-free $s$-wave SF, in which the total OAM is zero as well. For chiral SFs with $ν\ge2$ and $|k|\ge2$, 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.

DOI
10.1103/mzjx-npdd

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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.
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 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.
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.
Multiply quantized vortices in the BCS-to-BEC (Bose-Einstein condensation) evolution of ppp-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 Bogoliubov-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 $\ensuremath{\kappa}$ 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 ppp-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. The $\ensuremath{\kappa}$ branches of the core-bound states for a vortex state with vorticity $\ensuremath{\kappa}$ exist; however, 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.
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.
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