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Physical Review B

Skyrmion lattice and intrinsic angular momentum effect in the A phase of superfluid H3e under rotation

Masanori Ichioka, Takeshi Mizushima, Kazushige Machida

Published Sep 23, 2010• 6 citations
Publisher

Abstract

As an example of the skyrmion lattice, the structure of Mermin-Ho vortex lattice in superfluids $^{3}\text{H}\text{e}$ is studied by self-consistent Eilenberger theory and by Bogoliubov-de Gennes theory. We identify how the intrinsic orbital angular momentum $\mathbf{l}$ 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 $\mathbf{l}$ direction relative to rotation. Only one of them has zero-energy bound states appearing even in coreless vortices due to intrinsic topological reasons.

DOI
10.1103/physrevb.82.094516

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As an example of the skyrmion lattice, the structure of Mermin-Ho vortex lattice in superfluids 3He^{3}\text{H}\text{e}3He is studied by self-consistent Eilenberger theory and by Bogoliubov-de Gennes theory. We identify how the intrinsic orbital angular momentum l\mathbf{l}l of ppp-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\mathbf{l}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 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.
In the superfluid 3 He A-phase, the edge mass current which accompanies surface Andreev bound states flows due to topological features. We discuss the relation between the angular momentum by the edge mass current and the macroscopic intrinsic angular momentum L = N ℏ/2 by the Cooper pairs with aligned angular momenta, based on the quasi-classical theory. At the low temperature limit, half of the edge mass current from the bound state is canceled by that from the continuum state. Then, the magnitude of the total angular momentum is L = N ℏ/2 which corresponds to the macroscopic intrinsic angular momentum of N particles. The net angular momentum decreases monotonously toward zero at the transition temperature. However, that is larger than the macroscopic intrinsic angular momentum expected from the same behavior of the superfluid density in middle temperatures.
We theoretically investigate vortex-lattice phases of rotating spinor Bose-Einstein condensates (BEC's) with a ferromagnetic spin interaction by numerically solving the Gross-Pitaevskii equation. The spinor BEC under slow rotation can sustain a rich variety of exotic vortices due to the multicomponent order parameters, such as Mermin-Ho and Anderson-Toulouse coreless vortices (the two-dimensional Skyrmion and meron) and nonaxisymmetric vortices with shifting vortex cores. Here, we present the spin texture of various vortex-lattice states at higher rotation rates and in the presence of an external magnetic field. In addition, the vortex phase diagram is constructed in the plane of the total magnetization MMM and the external rotation frequency $\ensuremath{\Omega}$ by comparing the free energies of possible vortices. It is shown that the vortex phase diagrams in the $M\text{\ensuremath{-}}\ensuremath{\Omega}$ plane may be divided into two categories: (i) the coreless vortex lattice formed by the several types of Mermin-Ho vortices and (ii) the vortex lattice filling in the cores with the pure polar (antiferromagnetic) state. In particular, it is found that the type-(ii) state forms composite lattices of coreless and polar-core vortices. The difference between type (i) and type (ii) results from the existence of the singularity of the spin textures, which may be experimentally confirmed by the spin imaging within polarized light recently proposed by Carusotto and Mueller. We also discuss the stability of triangular and square lattice states for rapidly rotating condensates.
Based on the microscopic Bogoliubov-de Gennes theory, we study vortical configurations in a mesoscopic superconducting square loop with mixed pairing orders. By choosing appropriate interaction strengths and chemical potential, the spin-triplet p -wave and subdominant spin-singlet d -wave pairing symmetries can be stabilized. In a small external magnetic field, unusual elliptical-like vortices in chiral p -wave orders as well as elongated vortices in the d -wave order tend to form in sample diagonals. For an enlarged flux, we find a two-quanta skyrmionic mode for the p -wave components, while the closed domain wall does not appear compared to the previous studies in superconducting systems with a pure p -wave state. Particularly, a novel multi-skyrmionic pattern containing four single-quanta skyrmions takes place with increasing flux. Such a single-quanta vortex structure consisting of two spatially separated half-quantum vortices is analogous to the nematic skyrmion with unit topological charge. In addition, several complex hybrid or multiple skyrmionic configurations are revealed in the strong flux range.
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.