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arXiv:1512.06366

Fractional excitations in one-dimensional fermionic superfluids

Ye, Fei, P. A. Marchetti

Published Dec 20, 2015• 0 citations
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Abstract

We study the soliton modes carrying fractional quantum numbers in one-dimensional superfluids. In the $s$-wave pairing superfluid with the phase of the order parameter twisted by opposite angles $\pm φ/2$ at the two ends there is an emergent complex $Z_2$ soliton mode carrying fractional spin number $φ/(2π)$ if there is only one pairing branch. We demonstrate that in finite systems of length $L$, 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 $φ/(2π)$, and a uniform background $-φ/(2πL)$. 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 $p$-wave pairing superfluid with the chemical potential interpolating between the strong and weak pairing phases, the $Z_2$ 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.

DOI
10.48550/arxiv.1512.06366

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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 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.
Dark solitons in quantum fluids are well-known nonlinear excitations that are usually characterized by a single length scale associated with the underlying background fluid. We show that in the presence of spin-orbit coupling and a linear Zeeman field, superfluid Fermi gases support two different types of nonlinear excitations featured by corresponding length scales related to the existence of two Fermi surfaces. Only one of these types, which occurs for finite spin-orbit coupling and a Zeeman field, survives to the topological phase transition, and is therefore capable of sustaining Majorana zero modes. At the point of the emergence of this soliton for varying Zeeman field, the associated Andreev bound states present a minigap that vanishes for practical purposes, in spite of lacking the reality condition of Majorana modes.
We theoretically investigate the behavior of dark solitons in a one-dimensional spin-orbit coupled atomic Fermi gas in harmonic traps by solving self-consistently the Bogoliubov--de Gennes equations. The dark soliton---to be created by phase imprinting in future experiments---is characterized by a real order parameter, which changes sign at a point node and hosts localized Andreev bound states near the node. By considering both cases of a single soliton and multiple solitons, we find that the energy of these bound states decreases to zero when the system is tuned to enter the topological superfluid phase by increasing an external Zeeman field. As a result, two Majorana fermions emerge in the vicinity of each soliton, in addition to the well-known Majorana fermions at the trap edges associated with the nontrivial topology of the superfluid. We propose that the soliton-induced Majorana fermions can be directly observed by using spatially resolved radio-frequency spectroscopy or indirectly probed by measuring the density profile at the point node. For the latter, the deep minimum in the density profile will disappear due to the occupation of the soliton-induced zero-energy Majorana fermion modes. Our prediction could be tested in a resonantly interacting spin-orbit coupled 40K{}^{40}\text{K}40K Fermi gas confined in a two-dimensional optical lattice.
We find the coexistence of two kinds of fractionalized modes, Majorana modes at the geometric ends and Jackiw–Rebbi-type bound state (JRBS) at a domain-wall, in a new topological superconducting phase in one-dimensional (1D) systems. Each localized JRBS carries a new fractional quantity, half of the parity of the fermion number. This induces a topological protected crossing at the zero energy for its eigenenergy. For a chain embedded with a JRBS, it is possible to switch between the occupied and the empty states of the Majorana zero energy state (MZES) by varying the strength of the external magnetic field across that crossing point. This enables a way to encode a quantum qubit into one MZES without breaking parity conservation. We propose that such JRBS and Majorana fermions may appear in two 1D models. One may be implemented in an artificial lattice with staggered hopping, staggered spin-orbital interaction and staggered superconducting pairing in cold fermion atoms. The other is a 1D semiconductor chain sandwiched between s-wave superconductor and antiferromagnet.
We investigated the core-filling properties of dark solitons in one-dimensional spin-orbit-coupled Fermi gases. Majorana fermions can exist at the edges of the topological superfluid, when the spin-orbit coupling drives this system from the superfluid to the topological superfluid. Our results show that when dark solitons participate, the state of the system and the existence of Majorana zero-energy modes at the edges of topological superfluid correlate with the filling status of the core of the dark soliton. Furthermore, this dark soliton can actually accommodate two extra Majorana excitations without interactions in its core, when it is filled with highly imbalanced spins in a specific region of parameters. These features can be utilized to manipulate Majorana fermions in the topological superfluid.