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arXiv:1511.03502 · Physical review. B./Physical review. B

Chiral p±ip superfluid on a sphere

Sergej Moroz, Carlos Hoyos, Leo Radzihovsky

Published Jan 26, 2016• 15 citations
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Abstract

We consider a spinless fermionic $p\ifmmode\pm\else\textpm\fi{}ip$ superfluid living on a two-dimensional sphere. Using superfluid hydrodynamics, we show that the ground state necessarily exhibits topological defects: either a pair of elementary vortices or a domain wall between $p\ifmmode\pm\else\textpm\fi{}ip$ phases. In the topologically nontrivial BCS phase we identify the chiral fermion modes localized on the topological defects and compute their low-energy spectrum.

DOI
10.1103/physrevb.93.024521

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We consider a spinless fermionic p\ifmmode±\else\textpm\fiipp\ifmmode\pm\else\textpm\fi{}ipp\ifmmode±\else\textpm\fiip superfluid living on a two-dimensional sphere. Using superfluid hydrodynamics, we show that the ground state necessarily exhibits topological defects: either a pair of elementary vortices or a domain wall between p\ifmmode±\else\textpm\fiipp\ifmmode\pm\else\textpm\fi{}ipp\ifmmode±\else\textpm\fiip phases. In the topologically nontrivial BCS phase we identify the chiral fermion modes localized on the topological defects and compute their low-energy spectrum.
Curvature Induced Topological Defects of ppp-wave Superfluid on a Sphere
We study the ground state of spinless fermions living on a sphere across ppp-wave Feschbach resonances. By construsting a microscopic model of fermions on a general curved surface, we show that the Guassian curvature induces an emergent magnetic field coupled to the p±ipp\pm ipp±ip order parameters. In the case of a sphere, the magnetic field corresponds to a Dirac monopole field, which causes topological defects in the superfluid ground state. Using the BCS mean field theory, we calculate its many-body ground state self consistently and give the phase diagram. The ground state may exhibit two types of topological defects, two voritces on the south and north pole or a domain wall which separates pθ+ipϕp_θ+ ip_ϕpθ​+ipϕ​ and pθ−ipϕp_θ-ip_ϕpθ​−ipϕ​ superfluids.
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.
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.
Two-fold degeneracy of the ground state in chiral p -wave superfluids or superconductors makes it possible for domain walls separating regions of opposite chirality to exist. In addition to affecting the quasiparticle spectrum in the bulk, the domain walls also carry localized low-energy fermionic excitations, whose spectrum is determined by the phase difference between the domains. In this letter we show how a moving domain wall acquires inertia and friction due to transitions between different quasiparticle states.
We study Majorana modes in the vortex core of a two-dimensional p+ipp+ipp+ip Fermi superfluid interacting with a Bose-Einstein condensate. Under a repulsive sss-wave contact interaction between fermions and bosons, fermions are depleted from the vortex core when the bosonic density becomes sufficiently high. This gives rise to a dynamically driven local interface that emerges between fermions and bosons, along which chiral Majorana edge states should appear. We examine in detail the variation of vortex-core structures as well as the formation of chiral Majorana edge states with increasing bosonic density, where the circulation of the vortex plays an important role. Whereas both the Majorana modes and the vortex-core structures can be controlled and manipulated by tuning the bosonic density and the Bose-Fermi interaction strength, our study presents an illuminating example of how topological defects can be dynamically controlled in the context of cold-atomic gases.
In this work, we utilize semiclassical kinetic equations to investigate the order parameter collective modes of a class of two-dimensional superfluids. Extending the known results for ppp-wave superfluids, we show that for any chiral ground state of angular momentum $L\ensuremath{\ge}1$, there exists a subgap mode with a mass $\sqrt{2}\phantom{\rule{0.16em}{0ex}}\mathrm{\ensuremath{\Delta}}$ in the BCS limit, where $\mathrm{\ensuremath{\Delta}}$ is the magnitude of the ground-state gap. We determine the most significant Landau parameter that contributes to the mass renormalization and show explicitly that the renormalized modes become massless at the Pomeranchuk instability of the fermion vacuum. Particularly for L=1L=1L=1, we propose a continuous field theory to include the Fermi liquid effect in a quadrupolar channel and produce the same result under consistent approximations. They provide potential diagnostics for distinguishing two-dimensional chiral ground states of different angular momenta with the order parameter collective modes and reveal another low-energy degree of freedom near the nematic transition.