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arXiv:1509.01803 · Physical Review Letters

Traveling Majorana Solitons in a Low-Dimensional Spin-Orbit-Coupled Fermi Superfluid

Peng Zou, Joachim Brand, Xia-Ji Liu, Hui Hu

Published Nov 23, 2016• 19 citations
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Abstract

We investigate traveling solitons of a one- or two-dimensional spin-orbit-coupled Fermi superfluid in both topologically trivial and nontrivial regimes by solving the static and time-dependent Bogoliubov-de Gennes equations. We find a critical velocity v_{h} for traveling solitons that is much smaller than the value predicted using the Landau criterion due to spin-orbit coupling. Above v_{h}, our time-dependent simulations in harmonic traps indicate that traveling solitons decay by radiating sound waves. In the topological phase, we predict the existence of peculiar Majorana solitons, which host two Majorana fermions and feature a phase jump of π across the soliton, irrespective of the velocity of travel. These unusual properties of Majorana solitons may open an alternative way to manipulate Majorana fermions for fault-tolerant topological quantum computations.

DOI
10.1103/physrevlett.117.225302

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We investigate traveling solitons of a one- or two-dimensional spin-orbit-coupled Fermi superfluid in both topologically trivial and nontrivial regimes by solving the static and time-dependent Bogoliubov-de Gennes equations. We find a critical velocity v_{h} for traveling solitons that is much smaller than the value predicted using the Landau criterion due to spin-orbit coupling. Above v_{h}, our time-dependent simulations in harmonic traps indicate that traveling solitons decay by radiating sound waves. In the topological phase, we predict the existence of peculiar Majorana solitons, which host two Majorana fermions and feature a phase jump of π across the soliton, irrespective of the velocity of travel. These unusual properties of Majorana solitons may open an alternative way to manipulate Majorana fermions for fault-tolerant topological quantum computations.
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 theoretically study trapped one-dimensional Fermi gases in the presence of spin-orbit coupling induced by Raman lasers. The gas changes from a conventional (nontopological) superfluid to a topological superfluid as one increases the intensity of the Raman lasers above a critical chemical-potential-dependent value. Solving the Bogoliubov-de Gennes equations self-consistently, we calculate the density of states in real and momentum space at finite temperatures. We study Majorana fermions (MFs) which appear at the boundaries between topologically trivial and topologically nontrivial regions. We linearize the trap near the location of a MF, finding an analytic expression for the localized MF wave function and the gap between the MF state and other edge states.
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.
Abstract By solving the Bogoliubov–de Gennes equations analytically, we derive the fermionic zero-modes satisfying the Majorana property that exist in vortices of a two-dimensional s-wave Fermi superfluid with spin–orbit coupling and Zeeman spin-splitting. The Majorana zero-mode becomes normalisable and exponentially localised to the vicinity of the vortex core when the superfluid is topologically non-trivial. We calculate the energy splitting due to Majorana hybridisation and identify that the s-wave Majorana vortices obey non-Abelian statistics.
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.
We show that a single dark soliton can exist in a spin-orbit-coupled Fermi gas with a high spin imbalance, where spin-orbit coupling favors uniform superfluids over nonuniform Fulde-Ferrell-Larkin-Ovchinnikov states, leading to dark soliton excitations in highly imbalanced gases. Above a critical spin imbalance, two topological Majorana fermions without interactions can coexist inside a dark soliton, paving a way for manipulating Majorana fermions through controlling solitons. At the topological transition point, the atom density contrast across the soliton suddenly vanishes, suggesting a signature for identifying topological solitons.