Multiply Quantized Vortices in Fermionic Superfluids: Angular Momentum, Unpaired Fermions, and Spectral Asymmetry

Abstract

We compute the orbital angular momentum Lz of an s-wave paired superfluid in the presence of an axisymmetric multiply quantized vortex. For vortices with winding number |k| > 1, we find that in the weak-pairing BCS regime Lz is significantly reduced from its value N k/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 sub-gap states. We support our results analytically and numerically by solving the Bogoliubov-de-Gennes equations within the weak-pairing BCS regime.Quantized vortices are a hallmark of superfluids (SFs) and superconductors. These topological defects form in response to external rotation or magnetic field and play a key role in understanding a broad spectrum of phenomena, such as the Berezinskii-Kosterlitz-Thouless transition in two-dimensional (2D) SFs [1,2], superconductor/insulator transitions [3][4][5], turbulence [6], and dissipation [7,8]. In fermionic s-wave paired states, the structure of the ground state and low lying excitations of an axisymmetric singly quantized vortex has been established through analytical and numerical studies in both the strong-pairing regime (where the SF phase is understood as a Bose-Einstein condensate (BEC) of bosonic molecules) and in the weak-pairing Bardeen Cooper Schrieffer (BCS) regime. In the BEC regime, the microscopic Gross-Pitaevskii equation provides a reliable framework [9,10], while in the BCS regime the (selfconsistent) Bogoliubov-deGennes (BdG) theory is key in identifying the structure of the ground state [11,12] and the spectrum of sub-gap fermionic excitations [13].Multiply quantized vortices (MQVs) have however not received much attention. Generically in a homogeneous bulk system, the logarithmic repulsion between vortices, which scales as the square of the vortex winding number k, energetically favors an instability of a multiply quantized vortex into separated elementary unit vortices [14]. However, MQVs are of interest since under certain circumstances, the interaction between vortices is not purely repulsive and can support multi-vortex bound states, at least as metastable defects. This can happen, for instance, in type-II mesoscopic superconductors, where MQVs have been predicted [15] and experimentally observed [16][17][18][19]. In addition, it has been argued that MQVs are expected to be energetically stable in multicomponent superconductors [20,21] and in chiral p-wave superconductors [22,23]. In fermionic SFs, a doubly quantized vortex was predicted [24] and observed in 3 He-A [25]. It has further been argued that fast rotating Fermi gases trapped in an anharmonic potential will support an MQV state [26][27][28]. Similar vortex states have been created in rotating BEC experiments [29][30][31][32].Surprisingly, as we demonstrate in this Letter, there is a fundamental difference between a singly quantized vortex (|k| = 1) and an MQV (|k| > 1) in a wea...

Fulde-Ferrell state in a ferromagnetic chiral superconductor with magnetic domain walls

“…7. Some eigenvalues of the DW modes change signs by δH, which is called spectral flow of BdG eigenvalues 14,16,19,20,[36][37][38][39][40] . The spectral flow is related to physical quantities in some models, and also to ground state wavefunctions of mean field Hamiltonians in general, as will be discussed in the next section.…”

“…The counter-intuitive charge current can be attributed to the previously mentioned "unpaired electrons" which arise from an implicit pair breaking effect 19,20 . To see this, we again consider a continuum model for simplicity, which can be verified at low filling.…”

“…where {E n (k y )} are the eigenvalues of the BdG Hamiltonian H BdG 19,20,38,39 . η is called spectral asymmetry and it is essentially determined by the Fermi wavenumber of the surface/DW modes, because formations of surface/DW modes can cause spectral flow where some eigenvalues of H BdG change the signs depending on the model parameters, as shown in Fig.…”

“…c ′ -operators are given by superpositions of c-operators and F nn ′ (k y ) can be calculated from the unitary matrix which diagonalizes the BdG Hamiltonian 19,20,45 . The exponential factor describes the paired electrons, while the additional c ′ † c ′ † corresponds to the unpaired electrons.…”

“…We propose that, as a result of the non-cancelation between the DW and surface currents, a Fulde-Ferrell (FF) like stripe superconducting state 17,18 can be stabilized in an anisotropic sample of a ferromagnetic chiral superconductor in the Meissner state in the presence of FM domain walls. The microscopic physical origin of the FF-like state is also discussed in terms of hidden normal (non-superconducting) state components, "unpaired electrons", which are an important key for understanding paired states 19,20 .…”