Moving Majorana Fermions Around
HUI HU
CENTRE FOR QUANTUM AND OPTICAL SCIENCE
SWINBURNE UNIVERSITY OF TECHNOLOGY, AUSTRALIA
Majorana fermions, also referred to as Majorana particles, are particles that are their own antiparticles. They were hypothesized by Ettore Majorana in 1937 and were considered only as a mathematical possibility in theoretical physics for more than 70 years. About ten years ago, however, it was realized that Majorana's concept is ubiquitous, and Majorana fermions are central to the work not only in neutrino physics (as Majorana originally proposed), supersymmetry and dark matter, but also for some exotic states of ordinary matter [1]. If realized and manipulated, their exotic behaviour  or, to put it in technical language, the nonabelian statistics  is anticipated to make Majorana fermions ideally suited to be faulttolerant quantum bits of information process in a quantum computer [2]; if the unit information is carried nonlocally by two Majorana fermions, the computer should be robust to any local decoherence.
There are already a number of interesting proposals on how to use Majorana fermions to perform fundamental quantum gate operations [3], which inevitably involve the movement of Majorana fermions. Unfortunately, while the properties of Majorana fermions at rest are understood to some extent, their fate (i.e., existence) while moving/traveling is less well known. Writing in Physical Review Letters [4], Peng Zou, XiaJi Liu and Hui Hu from the Centre for Quantum and Optical Science at Swinburne University of Technology, Australia, and Joachim Brand from the DoddWalls Centre for Photonic and Quantum Technologies at Massey University, New Zealand, discuss what happens if one moves Majorana fermions around.
Fig. 1: Top: Sketch of an experimental configuration for creating Majorana fermions. A bundle of onedimensional Fermi gases can be realized in coldatom experiments. Bottom: In the presence of spinorbit coupling (engineered via two Raman laser beams) and sufficiently large Zeeman field h, at low temperature a Fermi gas turns into a topological superfluid that supports Majorana fermions at the superfluidnormal interface, such as the two edges (see the inset, where x = ±x_{F}),
or at the soliton core (not shown).
The underlying system considered in their work is an interacting ultracold atomic Fermi gas with both spinorbit coupling and large Zeeman field, confined in one dimension (see Fig. 1, the upper panel). At incredibly low temperatures, i.e., a billionth of a degree Kelvin, the Fermi gas is expected to condense into a superfluid state. It behaves like a completely frictionless fluid with additional nontrivial properties due to its topological nature. The groundwork for understanding topological materials was laid by the recipients of the 2016 Nobel Prize in Physics. In particular, the topological superfluid can host Majorana fermions at the boundaries where it fails to be superfluid (Fig. 1, the inset in the lower panel). As recently suggested in two works [5,6], Majorana fermions also appear at the position of stationary/dark solitons, where the Fermi gas locally becomes normal. This suggestion is useful since in coldatom experiments dark solitons can be generated by phase imprinting techniques. The problem now is, if one boosts the dark solitons to let them travel, what happens to the Majorana fermions located at the soliton core?
Dark solitons are generally characterized by a sharp phase jump π and a vanishing order parameter at the soliton core. When solitons are allowed to move, the phase jump decreases from π and the order parameter becomes nonzero. In other words, solitons can be greyed by their motion. This turntogrey feature is typical for nontopological solitons in superfluids, including dark solitons in atomic BoseEinstein condensates (BECs) with and without optical lattices, as well as dark solitons in interacting Fermi gases at the crossover from BECs to BardeenSchriefferCooper (BCS) superfluids. If this feature continues to hold in topological superfluids, then, one must anticipate that the soliton core is filled by superfluid when the solitons travel and therefore Majorana fermions should disappear at infinitely small velocity of travel, since they can exist only at the superfluidnormal interface. This would definitely be a disaster for potential applications of Majorana fermions, as they are destroyed immediately by any movement/manipulation.
Fig. 2: Upper panel: time evolutions of the magnitude (left) and the phase (right) of the order parameter for a Majorana soliton oscillating in a trapped topological superfluid. Lower panel: The energy of the lowestlying vortex state of the Majorana soliton (i.e., the energy of Majorana fermions) as a function of the traveling velocity. The inset shows the phase jump of the Majorana fermions at different traveling velocities.
Luckily, in their work, Peng et al. found that a finite velocity of travel does not grey the soliton in topological superfluids [4]. Both timeindependent calculations (in the comoving frame) and timedependent simulations (in the laboratory frame) reveal a universal, constant π phase jump across the soliton, irrespective of the velocity of travel (see Fig. 2, the top right figure and the inset in the bottom figure). Correspondingly, Majorana fermions at the soliton core survive up to a threshold velocity, which is smaller than the Landau critical velocity, at which the soliton decays. These unusual properties of the traveling soliton in topological superfluids (referred to as a Majorana soliton in the work) may provide a new way to manipulate Majorana fermions for faulttolerant topological quantum computations. Further calculations and simulations in different setups, such as twodimensional spinorbit coupled Fermi superfluids and pwave Fermi superfluids, also confirm the existence of Majorana solitons [4], indicating that it is indeed a universal state of matter ensured by the nontrivial topological properties of topological superfluids.
References
[1] F. Wilczek, Nature Phys. 5, 614 (2009).
[2] C. Nayak, S. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008).
[3] J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher, Nature Phys.
7, 412 (2011).
[4] P. Zou, J. Brand, X.J. Liu, and H. Hu, Phys. Rev. Lett. 117, 225302 (2016).
[5] Y. Xu, L. Mao, B. Wu, and C. Zhang, Phys. Rev. Lett. 113, 130404 (2014).
[6] X.J. Liu, Phys. Rev. A 91, 023610 (2015).

Hui Hu is an associate professor at the Centre for Quantum and Optical Science at Swinburne University of Technology. He was educated at Tsinghua University in China (BSc  1996; PhD  2001) and has worked as a postdoctoral research fellow at the International Centre for Theoretical Physics (Trieste, 20012003), the Scuola Normale Superiore di Pisa (20032004) and the University of Queensland (2005). He was a chair professor of atomic, molecular, and optical physics at Renmin University of China (20062008). In addition, he was an Australian Research Council Queen Elizabeth II Fellow (20092013) and a Future Fellow (20142017) at Swinburne University of Technology. Hu became an elected fellow of the American Physical Society in 2016. His recent research interests focus on ultracold atoms and molecules, and the theory of quantum fluids. 
