Exotic States at the Edge: Majorana Fermions and Parafermions

Jelena Klinovaja, Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
Peter Stano, Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan
Daniel Loss, Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland

 

The current hunt for Majorana fermions in condensed matter physics resembles the one for the Higgs boson in high energy physics. Majorana fermions are different to all other known fermions in that they are their own antiparticles. Though the possibility for their existence was shown already in 1937 [1], as of today we still do not know whether they exist in Nature as elementary particles. Much more recent is the discovery that such particles might be produced in condensed matter laboratories, as collective excitations in specially designed semiconductor heterostructures.

Apart from being fundamentally different to electrons and thus interesting by itself, Majorana fermions also find applications in topologically protected quantum computation schemes, which has raised extraordinary interest in these and other exotic particles such as fractional fermions, parafermions, or Fibonacci anyons. A further hallmark of these particles is that, when confined to a two-dimensional plane, they obey non-Abelian braid statistics, which means that it matters in which order we move them around each but not which precise path we choose. It is the latter property which makes the results of such braiding or exchange operations topologically stable.

One of the structures suggested to host Majorana fermions is a semiconducting nanowire with spin orbit interaction and proximity-coupled to a superconductor - a 'super-semi' hybrid system [2]. Applying a magnetic field drives it through a topological phase transition, upon which Majoranas are predicted to appear as zero-energy bound states at the nanowire ends. A recent experiment showing signatures of such a transition has caused a lot of excitement but the results remain inconclusive [3]. The difficulty in the stabilization of the Majorana phase lies in the stringent requirements on system parameters and their precise tuning. Namely, the spin-orbit interaction in currently available nanowires is unfavourably weak, resulting in Majorana fermions spreading out too far inside the nanowire, by which it can fuse with another Majorana fermion from the other end into a trivial electron of finite energy. In addition to that, tuning of the chemical potential is a challenging task due to the screening of electric fields by the nearby superconductor.

Recently we suggested how to overcome the weak spin-orbit interaction limitation by replacing the inherent material property by what we call a synthetic spin-orbit interaction [4]. The idea originates from the simple observation that a spatially rotating (helical) magnetic field mimics spin-orbit effects in one-dimensional systems [5]. Creating such periodically varying magnetic fields, for example by nanomagnets as shown in Fig. 1, allows one to generate effective spin-orbit interactions much stronger than is available in currently used nanowires. Moreover, it suggests wider possibilities for spintronics applications in materials with weak intrinsic spin-orbit interaction, such as graphene [6] or molybdenum disulfide [7].

 

FIG. 1. The super-semi hybrid system, consisting of a spin-orbit coupled nanowire (green cylinder) on top of a superconductor (red slab) with a uniform magnetic field B (gray arrow) applied along the wire. Periodically arranged nanomagnets (blue bars) produce a spatially varying magnetic field Bn (gray arrows).
Phys. Rev. Lett.: Figure taken from Ref. [4].
FIG. 2. Wire (yellow cylinder) in contact with a superconductor (green slab). Localized magnetic moments (colored arrows) are ordered into a spin helix by RKKY interaction mediated by wire electrons.
Phys. Rev. Lett: Figure taken from Ref. [9].

The synthetic spin-orbit interaction can also arise spontaneously in one-dimensional nanowires containing localized magnetic moments [8–10]. At low temperatures, itinerant electrons that interact with these moments organize them into a helix via the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, see Fig. 2. Semiconducting nanowires usually contain localized spins in atomic nuclei of the constituent material but can be also intentionally doped with magnetic impurities. Alternatively, one can work with atomic chains on a superconducting surface [11, 12]. Concerning Majorana fermions, the most important feature of such RKKY systems is the self-tuning property. Namely, the relation between the Fermi wave length and the effective spin-orbit wavelengths produced by the impurity spin helix is, somewhat miraculously, exactly the one required to stabilize the Majorana phase. This completely removes the need for fine-tuning of parameters [2, 9, 10].

The potential of the 'super-semi' hybrid structures reaches beyond Majorana fermions. This has been indicated by our previous finding that by combining the synthetic spin-orbit interaction and external uniform magnetic field, the topological transition can transform a Majorana fermion into a fractionally charged fermion (charge e/2) [4]. These exotic many-body excitations were postulated long ago in polyacetylene [13] (preceded by a field theory model [14]). However, there the half-charge is masked as they appear in pairs due to spin degeneracy. With the spin SU(2) symmetry broken by the rotating field, this degeneracy is absent and thus the fractional charge becomes amenable to direct experimental observation via transport [15].

More recently, some of us proposed a nanowire-based structure, which hosts fractionalized Majorana fermions (parafermions) [16], which obey fractional exchange statistics enforced by strong interaction effects. These parafermions allow for a wider class of topologically protected gate operations due to a more general braid statistics. Whereas Majorana fermions, being Ising anyons, enjoy the protection only for two out of the four universal qubit gates, the parafermions enable one to go one step further, raising the number of topologically protected gates to three. One can even speculate further that in an array of coupled nanowires a Fibonacci anyon phase, which enables the complete set of universal gates, might be accessible, reaching the holy grail of topological quantum computation.

With experiments under way, the 'super-semi' hybrid system [3] and atomic chains [11] remain currently one of the most promising candidates for fundamentally novel particles in condensed matter systems.

 

[1] E. Majorana, Nuovo Cimento 14, 171 (1937).
[2] S. R. Elliott and M. Franz, arXiv:1403.4976.
[3] V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Science, 336, 1003 (2012).
[4] J. Klinovaja, P. Stano, and D. Loss, Phys. Rev. Lett. 109, 236801 (2012).
[5] B. Braunecker, G. I. Japaridze, J. Klinovaja, and D. Loss, Phys. Rev. B 82, 045127 (2010).
[6] J. Klinovaja and D. Loss, Phys. Rev. X 3, 011008 (2013).
[7] J. Klinovaja and D. Loss, Phys. Rev. B 88, 075404 (2013).
[8] B. Braunecker, P. Simon, and D. Loss, Phys. Rev. B 80, 165119 (2009).
[9] J. Klinovaja, P. Stano, A. Yazdani, and D. Loss, Phys. Rev. Lett. 111, 186805 (2013).
[10] B. Braunecker and P. Simon, Phys. Rev. Lett. 111, 147202 (2013).
[11] A. Yazdani, Presentation at conference ”Majorana Physics in Condensed Matter”, Erice, Italy (2013).
[12] M. Menzel, et al., Phys. Rev. Lett. 108, 197204 (2012).
[13] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979).
[14] R. Jackiw and C. Rebbi, Phys. Rev. D 13, 3398 (1976).
[15] D. Rainis, A. Saha, J. Klinovaja, L. Trifunovic, and D. Loss, arXiv:1309.3738.
[16] J. Klinovaja and D. Loss, arXiv:1312.1998.

 

 

[Released: May 2014]