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Fractional excitation in one-dimensional two species fermionic superfluids

Fei Ye

Published Dec 20, 2015• 0 citations
OpenAlex

Abstract

We study one-dimensional two-species fermionic superfluids with order parameter twisted by an angle $\varphi$ at the two ends. By solving the corresponding Bogoliubov-de-Gennes equation, we obtain the U(1) soliton state which turns out to carry $\varphi S/(2\pi)$ spin, where $S$ is the spin of single fermion. For $Z_2$ soliton with $\varphi=\pi$, the localized state carries only $1/2$ of a single fermion spin. To conserve the total spin of the system as an integal multiple of $S$, we demonstrate, the localized fractional spin $\varphi S/(2\pi)$ can be compensated by an opposite total spin $-\varphi S/(2\pi)$, which distributes uniformly in space and is thus undetectable locally in the thermodynamic limit.

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We study one-dimensional two-species fermionic superfluids with order parameter twisted by an angle φ\varphiφ at the two ends. By solving the corresponding Bogoliubov-de-Gennes equation, we obtain the U(1) soliton state which turns out to carry φS/(2π)\varphi S/(2\pi)φS/(2π) spin, where SSS is the spin of single fermion. For Z2Z_2Z2​ soliton with φ=π\varphi=\piφ=π, the localized state carries only 1/21/21/2 of a single fermion spin. To conserve the total spin of the system as an integal multiple of SSS, we demonstrate, the localized fractional spin φS/(2π)\varphi S/(2\pi)φS/(2π) can be compensated by an opposite total spin −φS/(2π)-\varphi S/(2\pi)−φS/(2π), which distributes uniformly in space and is thus undetectable locally in the thermodynamic limit.
We study the soliton modes carrying fractional quantum numbers in one-dimensional superfluids. In the sss-wave pairing superfluid with the phase of the order parameter twisted by opposite angles ±φ/2\pm φ/2±φ/2 at the two ends there is an emergent complex Z2Z_2Z2​ soliton mode carrying fractional spin number φ/(2π)φ/(2π)φ/(2π) if there is only one pairing branch. We demonstrate that in finite systems of length LLL, the spin density for one pairing branch in the presence of a single soliton mode consists of two terms, a localized spin density profile carrying fractional quantum number φ/(2π)φ/(2π)φ/(2π), and a uniform background −φ/(2πL)-φ/(2πL)−φ/(2πL). The latter one vanishes in the thermodynamic limit leaving a single soliton mode carrying fractional excitation, however it is essential to keep the total quantum number conserved in finite systems. This analysis is also applicable to other systems with fractional quantum numbers, thus provides a mechanism to understand the compatibility of the emergence of fractional charges with the integral quantization of charges in a finite system. For the ppp-wave pairing superfluid with the chemical potential interpolating between the strong and weak pairing phases, the Z2Z_2Z2​ soliton is associated with a Majorana zero mode. By introducing the dimension density, we argue that the Majorana zero mode may be understood as an object with 1/2 dimension of the single particle Hilbert space. We conjecture a connection of the dimension density of one-dimensional solitons with the quantum dimension of topological excitations.
We study the soliton modes carrying fractional quantum numbers in one-dimensional fermionic systems. In particular, we consider the solitons in the one-dimensional fermionic superfluids. For the s -wave order parameter with phase twisted by an angle φ , the complex Z 2 soliton may occur carrying a localized fractional quantum number φ /(2 π ). If the system is finite with length L , we show the existence of a uniform background − φ /(2 πL ) which, though vanishing in the thermodynamic limit, is essential to maintain the conservation of the total integer quantum number. This analysis is also applicable to other systems with fractional quantum numbers, thus providing a mechanism to understand the compatibility of the emergence of fractional quantum number in the thermodynamic limit of a finite system with only integer quantum numbers. For the p -wave pairing case, the Majorana zero mode may occur associated with a real Z 2 soliton, and the fractionalized quantum number is the dimension of the single particle Hilbert space, which turns out to be 1/2. Again for a finite system with length L , there is an accompanying uniform dimension density −1/(2 L ) to maintain the dimension of the Hilbert space invariant. We conjecture a connection of the dimension density of one-dimensional solitons with the quantum dimension of topological excitations.
We will discuss recent results on the theoretical and experimental investigations on solitons in a superfluid Fermi gas along the BCS-BEC crossover. On the theory side, by solving the Bogoliubov de Gennes equations and looking for real and odd solutions for the order parameter, it has been shown that a dark soliton at unitarity posses a large density contrast and fermionic bound states. The superfluid gap is found to be significantly quenched by the presence of the soliton due to the occurrence of Andreev fermionic bound states localized near the nodal plane of the order parameter. By solving the time dependent Bogoliubov de Gennes equations, also the decay and collisions of dark solitons has been investigated. Recently, a cascade of Solitonic Excitations in a Superfluid Fermi gas has been experimentally observed.
Exact self-consistent soliton dynamics based on the Bogoliubov--de Gennes (BdG) formalism in unconventional Fermi superfluids/superconductors possessing an SU(d)\text{SU}(d)SU(d)-symmetric two-body interaction is presented. The derivation is based on the ansatz having the similar form as the Gelfand-Levitan-Marchenko equation in the inverse scattering theory. Our solutions can be regarded as a multicomponent generalization of the solutions recently derived by Dunne and Thies [Phys. Rev. Lett. 111, 121602 (2013)]. We also propose superpositions of occupation states, which make it possible to realize various filling rates even in one-flavor systems, and include Dirac and Majorana fermions. The soliton solutions in the d=2d=2d=2 systems, which describe the mixture of singlet sss-wave and triplet ppp-wave superfluids, exhibit a variety of phenomena such as rotating polar phases by soliton spins, SU(2)-DHN breathers, Majorana triplet states, $s\ensuremath{-}p$ mixed dynamics, and so on. These solutions are illustrated by animations, where order parameters are visualized by spherical harmonic functions. The full formulation of the BdG theory is also supported, and the double-counting problem of BdG eigenstates and NNN-flavor generalization are discussed.
A fully analytical theory of a traveling soliton in a one-dimensional fermionic superfluid is developed within the framework of time-dependent self-consistent Bogoliubov--de Gennes equations, which are solved exactly in the Andreev approximation. The soliton manifests itself in a kinklike profile of the superconducting order parameter and hosts a pair of Andreev bound states in its core. They adjust to the soliton's motion and play an important role in its stabilization. A phase jump across the soliton and its energy decrease with the soliton's velocity and vanish at the critical velocity, corresponding to the Landau criterion, where the soliton starts emitting quasiparticles and becomes unstable. The ``inertial'' and ``gravitational'' masses of the soliton are calculated and the former is shown to be orders of magnitude larger than the latter. This results in a slow motion of the soliton in a harmonic trap, reminiscent of the observed behavior of a solitonlike texture in related experiments in cold fermion gases [T. Yefsah et al., Nature (London) 499, 426 (2013)]. Furthermore, we calculate the full nonlinear dispersion relation of the soliton and solve the classical equations of motion in a trap. The strong nonlinearity at high velocities gives rise to anharmonic oscillatory motion of the soliton. A careful analysis of this anharmonicity may provide a means to experimentally measure the nonlinear soliton spectrum in superfluids.
We use the Bogoliubov-de Gennes formalism to analyze the ground state phases of harmonically trapped two-species fermion mixtures with unequal masses. In the weakly attracting limit and around unitarity, we find that the superfluid order parameter is spatially modulated around the trap center, and that its global maximum occurs at a finite distance away from the trap center where the mixture is locally unpolarized. As the attraction strength increases towards the molecular limit, the spatial modulations gradually disappear while the Bardeen-Cooper-Schrieffer (BCS) type nonmodulated superfluid region expands until the entire mixture becomes locally unpolarized.