Observing Majorana bound states of Josephson vortices in topological superconductors Eytan Grosfelda,1 and Ady Sternb aDepartment of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801-3080; and bDepartment of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel Edited* by Michael H. Freedman, Redmond, WA, and approved June 3, 2011 (received for review January 26, 2011) In recent years there has been an intensive search for Majorana fermion states in condensed matter systems. Predicted to be loca- lized on cores of vortices in certain nonconventional superconduc- tors, their presence is known to render the exchange statistics of bulk vortices non-Abelian. Here we study the equations governing the dynamics of phase solitons (fluxons) in a Josephson junction in a topological superconductor. We show that the fluxon will bind a localized zero energy Majorana mode and will consequently behave as a non-Abelian anyon. The low mass of the fluxon, as well as its experimentally observed quantum mechanical wave-like nature, will make it a suitable candidate for vortex interferometry experiments demonstrating non-Abelian statistics. We suggest two experiments that may reveal the presence of the zero mode carried by the fluxon. Specific experimental realizations will be discussed as well. fluxons ∣ Josephson effect ∣ Majorana mode ∣ p-wave ∣ sine-Gordon on-Abelian statistics (1–3) has recently been the subject of Nintensive research driven both by its possibly profound impact on the field of quantum computation (4–6) and by the search for its manifestations (7, 8). Among all mechanisms giving – p Fig. 1. Aharonov Casher effect in a long circular Josephson junction. The rise to such statistics, the route via spin-polarized -wave super- junction traps a single fluxon that is traveling around the ring propelled fluidity may be the simplest one. It was previously argued (9–11) by a bias charge Q induced between the two ring-shaped superconductors. that an Abrikosov vortex in a p-wave superfluid can trap a zero The energy spectrum of the junction is periodic in Q with periodicity e when energy Majorana fermion, being a self-conjugate “half” fermion. Φ is increased to nucleate a vortex within the interior hole. Copper wires act A pair of Majorana modes constitute a regular fermion, and the as reservoirs of unpaired electrons. resulting nonlocal occupancies label a set of degenerate ground Aharonov–Casher (16) oscillations (Fig. 2), similar in spirit to states. Braiding of vortices results in mixing of these ground the one proposed in ref. 17. states, sometimes in a noncommutative fashion: it matters in which order multiple braidings are performed. The search for Results an explicit experimental signal of the resulting vortex exchange Hamiltonian of a Circular Josephson Junction. We start by considering statistics, as well as for the presence of Majorana modes on their a circular Josephson junction, made of two concentric supercon- cores, is currently on its way. ducting annuli, separated by a thin insulator. We assume that the In this paper we propose an experiment that probes Majorana hole at the center of the inner superconductor is of a size com- fermions in Josephson vortices (fluxons). Josephson vortices are parable to the superconducting coherence length, and encloses trapped in insulating regions between superconductors. For Nv vortices. The Hamiltonian governing the junction would be conventional superconductors, they are described as solitonic H Hϕ Hψ H Hϕ composed of three parts, ¼ þ þ tun. The first, solutions of the sine-Gordon equation moving with small inertial is related to the dynamics of the phase across the junction. mass (estimated to be smaller than the electron mass). In the case For a Josephson junction of height hz, this part of the dynamics of topological superconductors, we find that such vortices bind a is derived from the following Hamiltonian (see, for example, localized Majorana zero mode and would therefore behave as refs. 14 and 18) non-Abelian anyons, despite the fact that they lack a normal core. Z We show that the non-Abelian nature of these vortices manifests h2 1 1 1 Hϕ ℏ¯c β2 z n − σ 2 ∂ ϕ 2 1 − ϕ ; itself in measurable transport properties of the Josephson junc- ¼ ð Þ þ 2 ð x Þ þ 2 ð cos Þ x 2 β 2 λJ tions that house them. [1] We start by showing that Josephson vortices traveling in a Josephson junction in a topological superconductor bind a single 6 Majorana zero energy state (see Eq. ). We then discuss two Author contributions: E.G. and A.S. designed research, performed research, and wrote experiments that can be used to measure the presence of these the paper. Majorana fermions. The first probes a thermodynamical property The authors declare no conflict of interest. of a circular charge biased Jospephson junction (Fig. 1), by *This Direct Submission article had a prearranged editor. measuring the nonlinear capacitance induced by the persistent 1To whom correspondence should be addressed. E-mail: [email protected]. – motion of the vortex trapped in the junction (12 15). The This article contains supporting information online at www.pnas.org/lookup/suppl/ second is an interference experiment of fluxons demonstrating doi:10.1073/pnas.1101469108/-/DCSupplemental. 11810–11814 ∣ PNAS ∣ July 19, 2011 ∣ vol. 108 ∣ no. 29 www.pnas.org/cgi/doi/10.1073/pnas.1101469108 Downloaded by guest on October 6, 2021 ψ 1ðxÞ, ψ 2ðxÞ therefore depend on Nv, the number of vortices enclosed by the two superconducting annuli. When the two superconducting islands are brought into close † proximity, tunneling terms of the form c1ðxÞc2ðxÞ translate to Z H 2im dxψ x ψ x ϕ∕2 ; [4] tun ¼ 1ð Þ 2ð Þ cosð Þ with ϕðxÞ ≡ ϕ2ðxÞ − ϕ1ðxÞ and m being a tunneling amplitude. Writing Eqs. 2 and 4 compactly as a matrix equation, the Hamil- Hψ Hψ H tonian 1 þ 2 þ tun becomes Z ivψ ∂x im cosðϕ∕2Þ H ¼ dxΨT ðxÞ ΨðxÞ; [5] −im cosðϕ∕2Þ −ivψ ∂x T where Ψ ¼ðψ 1;ψ 2Þ is a spinor composed of the two counterpro- pagating Majorana modes. The Hamiltonian possesses a symme- try under ϕ → ϕ þ 2π and Ψ → σzΨ. – Bound Majorana Mode on the Background of a Soliton. A solitonic Fig. 2. Vortex interferometry experiment based on the Aharonov Casher 1 effect (17) adapted to Josephson vortices. A superconducting wire creates solution of Eq. , also known as a fluxon or a Josephson vortex, a circulating magnetic field acting as a source for the entrance of Josephson is a finite energy solution that interpolates between two minima vortices into the sample. An applied supercurrent drives the vortex along one of the periodic potential described by the Josephson term. For a of two paths circumventing an island toward the top of the sample. A charge long Josephson junction (L ≫ λJ ) it acquires the form Q enclosed in the island controls the interference term via the Aharonov– x−x0 ϕsðxÞ¼4 arctan½expð Þ where x0 is the position of the soliton Φ λJ Casher effect. When the flux nucleates a vortex in the central region, 5 the interference term would be obliterated. (see, for example, refs. 14 and 18). In the following we solve Eq. in the background of a single soliton, explicitly plugging ϕs into ϕ, ϕ x x−x ϕ n and using cos sð Þ − tanh 0 . This would in turn result in a where is the phase difference across the junction, is the two ð 2 Þ¼ ð λJ Þ ϕ dimensional (2D) density of Cooper pairs on, say, the inner plate, tunneling amplitude m cosð sÞ whose sign is different on the two and σ is the 2D density of the externally induced charge. The first 2 sides of the soliton. In light of the Jackiw–Rebbi mechanism, part of the Hamiltonian is the capacitive energy, the second the Eq. 5 will now bind a zero energy mode at the position of the magnetic energy, and the third is the Josephson energy. The re- soliton x0, sulting equation of motion is the sine-Gordon equation, with the ¯2 2 d Z typical speed of light reduced to c ¼ c d 2λ (here d is the width þ L γ dxf x ψ x m ψ x : [6] of the insulating barrier and λL is the London penetration length). J ¼ ð Þ½ 1ð Þþsgnð Þ 2ð Þ The Josephson vortex is a soliton described by this equation, with its typical size set by the Josephson penetration length λJ .We In the limit of a long junction, L ≫ λJ and L ≫ vψ ∕m, the shape shall assume throughout that the circumference of the junction, of the Majorana mode is described by the localized function L, is much larger than λJ , and that λL ≪ hz ≪ λJ . The para- x−x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0 −jmjλJ ∕vψ 2 e2 fðxÞ¼N ½coshð λ Þ , with N a normalization factor and meter β ≡ 16π dð2λL þ dÞ∕hz. J ℏc x γ The second part of the Hamiltonian, Hψ , originates from the 0 the center of the Josephson vortex. The operator J is a loca- J neutral protected edge modes of the topological superconductor, lized Majorana fermion and the subscript indicates that this which give rise to its quantized thermal Hall conductance. The mode is bound to a Josephson vortex. This mode satisfies ψ ψ ψ γ† γ Hamiltonian governing these neutral modes is H ¼ H1 þ H2 J ¼ J . Indeed, the entire low energy spectrum of bound states with can be extracted. Plugging the solitonic solution into Eq. 5 and Z rotating the spinors according to Hψ iv dxψ x ∂ ψ x ; [2] 1;2 ¼ ψ 1;2ð Þ x 1;2ð Þ 1 11 Ψ → pffiffiffi Ψ; 2 −11 where 1 and 2 refer to the two counterpropagating Majorana ψ x ψ † x ψ x x edge modes.