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arXiv:cond-mat/9510075

One-dimensional fermi liquid and symmetry breaking in the vortex core

Yuriy Makhlin, Григорий Ефимович Воловик

Published Oct 13, 1995• 40 citations
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Abstract

Fermions localized within vortex cores can form one-dimensional Fermi liquids. The nonzero density of states in these Fermi-liquids can lead to instability of the symmetric structure of the vortex core. We consider a symmetry breaking which is obtained due to spontaneous admixture of the spin-triplet p-wave component of the order parameter in conventional s-wave vortex in the presence of magnetic field. This occurs at low enough temperature and leads to an asymmetric shape of the vortex core. Similar phenomenon of the symmetry breaking induced by the core fermions occurs in electroweak $Z$-strings (S.G. Naculich, Phys. Rev. Lett., 75 (1995), 998).

DOI
10.48550/arxiv.cond-mat/9510075

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Fermions localized within vortex cores can form one-dimensional Fermi liquids. The nonzero density of states in these Fermi-liquids can lead to instability of the symmetric structure of the vortex core. We consider a symmetry breaking which is obtained due to spontaneous admixture of the spin-triplet p-wave component of the order parameter in conventional s-wave vortex in the presence of magnetic field. This occurs at low enough temperature and leads to an asymmetric shape of the vortex core. Similar phenomenon of the symmetry breaking induced by the core fermions occurs in electroweak ZZZ-strings (S.G. Naculich, Phys. Rev. Lett., 75 (1995), 998).
We study several aspects of the structure of vortices in conventional s-wave type-II superconductors. It is well known that there are low-energy quasiparticles bound to the core of a vortex. We show that under certain conditions, these quasiparticles form a degenerate Fermi gas with a finite density of states at the Fermi energy. In three-dimensional superconductors, the result is a one-dimensional Fermi gas of quasiparticles bound to the core of a vortex line. As is usual in one-dimensional systems, interactions between the quasiparticles lead to Luttinger liquid behavior. This may be probed through scanning tunneling microscopy into the vortex core. We further suggest that a Peierls-type instability in the shape of the vortex line may develop due to interactions between the quasiparticle gas and fluctuations in the vortex line shape.
Chiral fermions are responsible for low-temperature properties of vortices in fermionic condensates, both superconducting (charged) and superfluid (neutral). One of the most striking consequences of this fact is that the core of a single-quantum vortex collapses at low temperatures, $T\ensuremath{\rightarrow}0$ (i.e., the Kramer-Pesch effect for superconductors), due to the presence of chiral quasiparticles in the vortex-core region. We show that the situation changes drastically for fermionic condensates confined in quasi-one-dimensional and quasi-two-dimensional geometries. Here quantum confinement breaks the chirality of in-core fermions. As a result, instead of the ultimate shrinking, the core of a single-quantum vortex extends at low temperatures, and the condensate profile surprisingly mimics the multiquantum vortex behavior. Our findings are relevant for nanoscale superconductors, such as recent metallic nanoislands on silicon, and also for ultracold superfluid Fermi gases in cigar-shaped and pancake-shaped atomic traps.
The Gross-Pitaevskii-type equation is solved for the charge Bose liquid in an external magnetic field at zero temperature. There is a vortex lattice with locally broken charge neutrality. Remarkably, there is no upper critical field at zero temperature, so the density of single flux-quantum vortices monotonously increases with the magnetic field up to $B=\ensuremath{\infty}$ and no indication of a phase transition. The size of each vortex core decreases at about ${B}^{\ensuremath{-}1∕2}$ keeping the system globally charge neutral. If bosons are composed of two fermions, a phase transition to a spin-polarized Fermi liquid at some magnetic field larger than the pair-breaking field is predicted.
Motivated by the recent paper of Mross and Senthil [Phys. Rev. B 84, 165126 (2011)], which provides a dual description for a Mott transition from a Fermi liquid to a quantum-spin liquid in two space dimensions, we extend their approach to higher-dimensional cases, and we provide explicit formalism in three space dimensions. Instead of the vortices driving a conventional Fermi liquid into quantum-spin-liquid states in two dimensions (2D), it is the vortex lines that lead to the instability of a Fermi liquid in three dimensions (3D). The extended formalism can result in rich consequences when the vortex lines condense in different degrees of freedom. For example, when the vortex lines condense in charge-phase degrees of freedom, the resulting effective fermionic action is found to be equivalent to that obtained by well-studied slave-particle approaches for Hubbard and/or Anderson lattice models, which confirm the validity of the extended dual formalism in 3D. When the vortex lines condense in spin-phase degrees of freedom, a doublon metal with a spin gap and an instability to the unconventional superconducting pairing can be obtained. In addition, when the vortex lines condense in both phase degrees, an exotic doubled U(1)U(1)U(1) gauge theory occurs, which describes a separation of spin-opposite fermionic excitations. It is noted that the first two features have been discussed in a similar way in 2D and that the last one has not been reported in previous papers. The present paper is expected to be useful in understanding the Mott transition happening beyond two space dimensions.
The rapid experimental progress of ultra-cold dipolar fermions opens up a whole new opportunity to investigate novel many-body physics of fermions. In this article, we review theoretical studies of the Fermi liquid theory and Cooper pairing instabilities of both electric and magnetic dipolar fermionic systems from the perspective of unconventional symmetries. When the electric dipole moments are aligned by the external electric field, their interactions exhibit the explicit d(r(2)-3z(2)) anisotropy. The Fermi liquid properties, including the single-particle spectra, thermodynamic susceptibilities and collective excitations, are all affected by this anisotropy. The electric dipolar interaction provides a mechanism for the unconventional spin triplet Cooper pairing, which is different from the usual spin-fluctuation mechanism in solids and the superfluid (3)He. Furthermore, the competition between pairing instabilities in the singlet and triplet channels gives rise to a novel time-reversal symmetry breaking superfluid state. Unlike electric dipole moments which are induced by electric fields and unquantized, magnetic dipole moments are intrinsic proportional to the hyperfine-spin operators with a Lande factor. Its effects even manifest in unpolarized systems exhibiting an isotropic but spin-orbit coupled nature. The resultant spin-orbit coupled Fermi liquid theory supports a collective sound mode exhibiting a topologically non-trivial spin distribution over the Fermi surface. It also leads to a novel p-wave spin triplet Cooper pairing state whose spin and orbital angular momentum are entangled to the total angular momentum J = 1 dubbed the J-triplet pairing. This J-triplet pairing phase is different from both the spin-orbit coupled (3)He-B phase with J = 0 and the spin-orbit decoupled (3)He-A phase.
We study the spectrum of fermion states localized within the vortex core of a weak-coupling p-wave superfluid. The low energy spectrum consists of two anomalous branches that generate a large density of states at the locations of the half cores of the vortex. Fermi liquid interactions significantly stretch the vortex structure, which leads to a Lifshitz transition in the effective Fermi surface of the vortex core fermions. We apply the results to the rotational dynamics of vortices in superfluid ^{3}He-B and find an explanation for the observed slow mode.