Observing Majorana Bound States of Josephson Vortices in Topological Superconductors

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 localized on cores of vortices in certain nonconventional superconductors, 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. PHYSICS

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.

www.pnas.org/cgi/doi/10.1073/pnas.1101469108 PNAS Early Edition ∣ 1of5 Downloaded by guest on September 28, 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 counterpropagating Majorana modes. The Hamiltonian possesses a symmetry 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. Here ið Þ is a Majorana field, i ð Þ¼ ið Þ, is the Hamiltonian can be written conveniently as v the coordinate running along the Josephson junction, and ψ is the velocity along the edge. In terms of the electron’s creation 0 A† A −iv ∂ − iW x † H ; ¼ ψ x ð Þ : [7] and annihilation operators ci x , c x and the phase fields of ¼ with † ð Þ i ð Þ A 0 A ¼ −ivψ ∂x þ iWðxÞ the two superconductors on the two sides of the junctions ϕiðxÞ the Majorana fields may be expressed as The spectrum of this Hamiltonian possesses a supersymmetry x † with a superpotential W x m tanh . In particular, the zero iϕiðxÞ∕2−iπx∕L −iϕiðxÞ∕2þiπx∕L ð Þ¼ ðλJ Þ ψ iðxÞ¼e ciðxÞþe ci ðxÞ: [3] mode is annihilated by either A or A† depending on the sign In this explicit form, the equation holds true for a spin-polarized of the mass, whereas the other operator will not have a normal- p-wave superconductor. For a general topological superconduc- izable eigenfunction at zero energy. The rest of spectrum is dou- † tor, it is still the case that the Majorana field will acquire a minus bly degenerate with A and A connecting the states of the doublet. sign going around the edge, in addition to one minus sign per Squaring the Hamiltonian in Eq. 7 and linearizing the potential each vortex enclosed in its path (i.e., one minus sign for every we get a shifted harmonic oscillator, and the spectrum of excita- 2π winding of the phase ϕi). The boundary conditions of the fields tions above the zero energy mode is

2of5 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1101469108 Grosfeld and Stern Downloaded by guest on September 28, 2021 sffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffi bolas of n and n þ 1, it becomes energetically favorable to tunnel ℏvψ mp E ∼ 2n; n 0;1;2;… [8] a Cooper pair to decrease the charging energy, thus crossing to n λ with ¼ J the next parabola. The matrix element required for this tunneling is generated by a small amount of disorder. The velocity of the Majorana fermions have to come in pairs, due to the properties moving fluxon is proportional to the derivative of the ground E Q E Q Q of the BdG equations. The second Majorana fermion will be on state energy gsð Þ¼minn∈Z nð Þ with respect to . In turn, one of the superconductor edges, and will be denoted by γe, with e the moving vortex generates a measurable voltage via the Joseph- standing for edge. Its position will depend on the number of vor- son relation between the inner ring and outer ring of the device, tices Nv in the inner hole. A zero mode is found on every edge which will be oscillating as function of Q with a periodicity of 2e. that encloses an odd number of vortices. Hence, when Nv is odd, The periodicity of 2e is quite expected, given the energy cost γe will be localized on the edge separating the inner superconduc- needed to break a Cooper pair. None of these results depend tor from the vacuum. When Nv is even, γe will be localized on the on the value of Nv. edge separating the outer superconductor from the vacuum. The The same spectrum shows a strikingly different behavior when two Majorana fermions, γe and γJ , cannot shift from zero energy the junction is made of two topological superconductors. In par- without hybridizing, and the superconductor that separates them ticular, this difference manifests itself in the dependence of the prevents that from happening. Note that in the absence of spectrum on the charge Q and Nv. The fermionic part of the electron tunneling across the junction, the presence of these Hamiltonian now has two degenerate eigenstates, which are two Majorana modes is protected by an index theorem. Their wa- two eigenstates of the operator iγJ γe, with eigenvalues 1. When vefunction will be spread evenly around the two edges of the it comes to the spectrum of the Josepshon vortex, there is a cru- superconductor. When we next turn on electron tunneling, the cial difference between the case where γe is localized at the edge Majorana mode γJ cannot disappear without hybridizing with of the outer superconductor (even Nv) and that in which it is the distant Majorana mode on the edge of the sample. Therefore localized at the center of the inner superconductor (odd Nv). it necessarily persists in the junction. Due to the effect of tunnel- As we saw before, in the latter case the winding of the soliton ing it gets localized around the center of the soliton as described around the junction multiplies γJ by −1. It is easy to see that by Eq. 6. it multiplies γe by −1 as well. These two operator transformations iα Due to the presence of the soliton in the junction, the are implemented by the unitary transformation U ¼ e iγJ γe on

boundary conditions are different for the two Majorana fields the state of the system (2). The phase α cannot be determined PHYSICS N and depend on Nv: ψ 1ðxÞ¼−ð−1Þ v ψ 1ðx þ LÞ and ψ 2ðxÞ¼ by this consideration, and is not important for what follows. N ð−1Þ v ψ 2ðx þ LÞ. A rotation of the soliton around the junction The two ground states are eigenstates of U, and thus the applica- shifts x by L and ϕ1ðxÞ by 2π. These shifts have two effects. First, tion of U multiplies them by two phases, that differ by π. The N N they multiply ψ 1 by −ð−1Þ v and ψ 2 by ð−1Þ v due to the bound- phase accumulated by the soliton encircling the ring affects its ary conditions. Second, they multiply the off-diagonal term of [5] spectrum. When the energy [9] is viewed as the energy of a π by −1 and hence multiply ψ 1ðxÞ in Eq. 6 by −1. Combining these charged particle on a ring threaded by a magnetic flux, a phase N two effects together, we see that γJ is multiplied by ð−1Þ v . In view shift corresponds to the introduction of half a quantum of flux. N 9 of this, the transformation γJ → ð−1Þ v γJ may be understood as a The spectrum [ ] is then changed to be consequence of the winding of one Majorana fermion around 2e 2 Q0 1 2 another for odd Nv, and the absence of such winding for even ð Þ En;f ðQÞ¼ − n − f ; [10] Nv. Remarkably, we find below that the capacitance of the 2C 2e 4 junction in the two cases is different. where the f ¼1 correspond to the two eigenstates of U for Manifestation of the Majorana Mode in Thermodynamics and α ¼ 0. The charge Q0 may be shifted with respect to the induced Transport. We now consider the energy of the long circular Joseph- charge Q by a nonuniversal number that depends on α. The spec- son junction when it is biased by an external charge Q ¼ 2eσLhz. trum is now periodic with a period of e, rather than the period of For a “conventional” Josephson junction made of s-wave super- 2e as was the case for the nontopological superconductor, as well conductors this problem was studied in ref. 14. In that case the as for the topological superconductor with an even Nv. Hamiltonian consists of Eq. 1 only. The magnetic and Josephson For the two sets of parabolas to be observed in an experiment energies constitute the rest mass of the vortex. The charging when Nv is odd, the system has to be able to switch between the energy constitutes its kinetic energy, and is the only component two values of f, namely the two states iγJ γe ¼1. This switch re- that depends on Q. This kinetic/charging energy may be under- quires the total number of electrons to change by one (7, 19, 20). stood in two ways. First, by viewing the vortex as a particle of mass To allow for that to happen, the junction should be weakly 2 ℏ hz M ¼ 2 in a one dimensional ring, subjected to a vector coupled to a reservoir of single electrons; i.e., to a metal. The 2πe dλJ E Q E Q Q 2πℏ ground state energy would be gsð Þ¼minn∈Z;f∈Z2 n;f ð Þ. potential 2e L , with a set of energy eigenstates The end result is that the voltage measured between the inner and outer edges of the superconductor would be periodic as func- 2πℏ 2 Q 2 E Q ð Þ n − ; [9] tion of Q (or V 0) with the periodicity now being e for odd Nv and nð Þ¼ 2 2ML 2e 2e for even Nv. Whereas this experiment probes a thermodynamical property, with n being the quantum number that quantifies the momentum the next one is an interference experiment that probes electronic of the vortex. Second, by writing this energy as a capacitive energy transport. The fluxons are generated near the bottom of the in- ð2eÞ2 Q 2 terferometer (Fig. 2), and driven by a supercurrent driven from of the form, EnðQÞ¼ 2C ð2e − nÞ where now C is the effective capacitance (which for a short junction coincides with the geo- left to right. The fluxon beam is split into two partial waves n enclosing an island with externally imposed charge Q. Due to metric capacitance). The quantum number is now identified – as the number of Cooper pairs charging the Josephson junction, the Aharonov Casher effect, the magnitude of the vortex current 2 2 oscillates according to and the capacitance is C ¼ Mð2eLÞ ∕ð2πℏÞ . The spectrum of [9] is described by a set of parabolas, each n Q Q characterized by a different integer value of . When is in- Jv ¼ Jv0 1 þ ζ cos 2π ; [11] creased so that the spectrum reaches a crossing point of two para- 2e

Grosfeld and Stern PNAS Early Edition ∣ 3of5 Downloaded by guest on September 28, 2021 where ζ is the visibility of the oscillations and Jv0 the average cur- tization should be deposited, resulting in a Zeeman term of the † rent. By the Josephson relation, a measurable voltage difference form Mψ σzψ in the surface Hamiltonian. This breaks the time- is created between the left and right ends of the superconductor, reversal symmetry at the edges and by that chooses a single chir- which would oscillate with changing Q. If the flux Φ on the central ality for the flow of the neutral chiral edge modes. The dynamics island is increased so that another non-Abelian vortex is nu- of the soliton will be largely determined by the s-wave supercon- cleated in the central hole, the interference pattern would be ducting layer, whereas a zero energy Majorana mode will be zero, ζ ¼ 0. These results are sensitive to effects of decoherence, trapped by the soliton on the surface state of the topological in- which would be reflected in a reduction of the value of ζ as the sulator. Other than these material specific details, the rest of the temperature is increased. However, the fluxons are quite light, arguments in the paper can be applied with no further changes. and it was predicted (21) and experimentally demonstrated To estimate the experimental parameters we take typical va- (15, 22) that they can exhibit quantum behavior (23, 24). Also, lues λL ¼ 0.1 μm and λJ ¼ 30 μm for an s-wave superconducting the gap to the next fermionic state on the background of the layer with hz ¼ 0.5 μm and an insulating region of width d ¼ soliton has different parametric dependence as compared to 20 Å. For these parameters, the dimensionless parameter β2 is Abrikosov vortices. approximately 0.01, and the velocity of light is ¯c ¼ 0.1c. We take 5 For a soliton in a topological superconductor there are two the neutral edge velocity to be vψ ¼ 10 m∕s (27) (which coin- main energy scales that control decoherence effects. The first cides with the surface state Fermi velocity close to the Dirac Hϕ is related to the bosonic part , whereas the second stems from point), and the tunneling between Majorana edge states to be Hψ the fermionic part . 0.025 meV. The plasmon gap Ep is approximately 5 K (25), The spectrum of the bosonic Hamiltonian Hϕ contains nonto- whereas the intracore gap En given by direct substitution to pological excitations with which the fluxon can interact. These are Eq. 8, is approximately 120 mK. The junction charging energy quantized phase oscillations, or plasmons. Written in terms of 2 2 Ec ¼ð2eÞ ∕2C is approximately 250 ðλJ ∕LÞ mK. To observe collective coordinates for the fluxon, the Hamiltonian would quantum phenomena, we need the temperature to be smaller be similar to the Caldeira–Leggett Hamiltonian of a quantum T E ;E ;E – than all these energy scales, < minf n c pg. The operating particle interacting with a bath. However, unlike the Caldeira temperatures are therefore in the range of 10–100 mK. Leggett mechanism, here the plasmon spectrum is gapped, with ℏω ℏ¯c∕λ The implementation of the Josephson junction in a hybrid the plasmon gap given by p ¼ J . At energy scales below structure of semiconductor quantum well coupled to an s-wave this gap (estimated to be a few Kelvin) the coupling to plasmons superconductor and a ferromagnetic insulator (26) is similarly is exponentially suppressed, and the internal dephasing thus straightforward. The superconducting layer is deposited as in mostly avoided (25) over lengths much larger than the Josephson Fig. 1, whereas the ferromagnetic insulator layer will be deposited penetration length. This lies at the origin of the prediction of throughout with no restrictions, breaking the time-reversal sym- quantum behavior of Josephson vortices in long Josephson junc- metry for the quantum well. Another option is to use a quantum tions. In particular, scattering of the fluxon on static inhomogene- well with both Rashba and Dresselhaus spin-orbit coupling, and ities in the junction would be mostly elastic due to the presence of deposit the superconducting layer as before, with an external this plasmon gap (21). Tunneling of a fluxon through barriers was magnetic field applied in the in-plane direction (28). Both cases predicted to be enhanced by the presence of the plasmons (21) and experimentally observed (22). will result in a superconductor-normal-superconductor type The second energy scale is unique to topological superconduc- Josephson junction. tors, and is related to the presence of higher energy fermionic The implementation using SRO is different in several respects. states above the zero energy mode carried by the soliton, as First, SRO is not spin-polarized. As a consequence, in bulk SRO Hψ 8 Majorana modes are carried by half-vortices, that are made of a π described by and the fermionic tunneling terms, see Eq. . d π The presence of these states would result in decoherence of the rotation of the pairing -vector glued together with a phase winding of the order parameter. These half-vortices were recently non-Abelianpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mechanism. The gap to the first state would be observed (29) in mesoscopic samples of the type that may be use- ℏωn ¼ 2ℏvψ m∕λJ . To avoid these effects, the temperature ful for the present context. Similarly, Majorana modes in a long should be lower than this gap as well. Josephson junction would require half-fluxons. Second, SRO is a three dimensional material, made of two dimensional layers. The Proposed Realizations. There are several systems in which the ideas physics described above for the Majorana modes carried by semi- presented above may be implemented in practice. The most re- levant for our purpose is a topological insulator whose surface fluxons decouples into different layers, one Majorana edge per s layer. The multitude of Majorana modes does not affect the even states become superconducting by proximity to an -wave super- N N γ γ U conductor, and are driven into an effective p-wave pairing for a Qv case. In the case of odd v, the factor e J in is replaced by Lz γ γ i L single fermionic species. The second is a hybrid structure of three i¼1 e;i J;i, with the index numbering the layers, and z being materials: a semiconductor with a magnetic material and a super- the number of layers. The unitary transformation still has two ei- conductor layer placed on top of it, as discussed in ref. 26. The genvalues that differ by a minus sign, and a transfer of a single third is the perovskite material Sr2RuO4 (SRO) that becomes electron facilitates a transition between the two states. Thus, as superconducting at temperatures below 1.5 K, and for which long as the zero energy states do not split, the multilayer case will there is growing evidence that it realizes an unconventional pair- not be different from the single layer one. ing of a px ipy (spin-triplet) form. Tunnel coupling between the layers may split the Majorana The surface state of a topological insulator is described by zero energy modes. However, a tunneling term between the i, † T j the Hamiltonian H0 ¼ vFψ z^ · σ × ð−i∇Þψ (where ψ ¼ðψ ↑;ψ ↓Þ layers is of the form Z is a spinor composed of the electronic operators associated with H 2it dxψ x ψ x ϕ − ϕ ∕2 ; [12] the spin components). The chemical potential is tuned to the coupling ¼ ið Þ jð Þ sin½ð i jÞ Dirac point. Superconductivity is induced on the surface state by the proximity effect (27), represented by an extra term with i, j being the layers involved in the tunneling. This tunneling † † Δψ ↑ψ ↓ þ h:c: to the Hamiltonian. The superconducting layer is suppressed when the phase difference between the layers of height hz is deposited as described in Fig. 1. At the thin insu- vanishes. Fluctuations of ϕ are massive due to the presence of lating layer between the two annuli, as well as in the inner and the interlayer Josephson coupling, and thus we may expect the outer holes, an insulating magnetic material of constant magne- Majorana modes not to split (see also ref. 30).

4of5 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1101469108 Grosfeld and Stern Downloaded by guest on September 28, 2021 Third, SRO may take two possible forms of p-wave pairing, a localized Majorana zero mode. Consequently, these solitons commonly denoted by px ipy. The analysis above assumes would constitute anyons with non-Abelian exchange statistics. that both superconductors are of the same pairing form. If the Exploiting the quantum nature of these solitons we suggest converse is true, the shape of the Majorana zero mode is differ- two experiments that can reveal the presence of the Majorana ent, but the rest of the analysis is unaffected. We analyze this case modes. One experiment involves voltage oscillations of a topolo- SI Text in . gical superconducting capacitor, realized by a circular Josephson We note that SRO suffers from the presence of a small minigap 2 junction hosting a single fluxon. The other experiment involves approximately Δ ∕EF for quasi-particle excitations in the core interference effects of a fluxon beam. of bulk vortices and in edge states. This sets a constraint of about 10 mK on the maximum temperature. For the case of a topological insulator tuned to the Dirac point this problem is ACKNOWLEDGMENTS. We thank P. Bonderson, E. Fradkin, M. Freedman, A. Ludwig, R. Lutchyn, B. Seradjeh, and S. Vishveshwara for useful discussions. avoided. We are grateful for the hospitality of the Aspen center for physics during which part of this work was carried out. E.G. thanks the Institute for Con- Conclusion densed Matter Theory fellowship program for support. A.S. acknowledges In summary, we predict that phase solitons in a long Josephson support from the United States-Israel Binational Science Foundation, junction embedded in a topological superconductor would carry Minerva Foundation, and Microsoft Corporation.

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