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Three-phase Majorana zero modes at tiny magnetic fields

Proposals for realizing Majorana fermions in condensed matter systems typically rely on magnetic fields, which degrade the proximitizing superconductor and plague the Majoranas' detection. We propose an alternative scheme to realize Majoranas based only on phase-biased superconductors. The phases (at least three of them) can be biased by a tiny magnetic field threading macroscopic superconducting loops, focusing and enhancing the effect of the magnetic field onto the junction, or by supercurrents. We show how a combination of the superconducting phase winding and the spin-orbit phase induced in closed loops (Aharonov-Casher effect) facilitates a topological superconducting state with Majorana end states. We demontrate this scheme by an analytically tractable model as well as simulations of realistic setups.

Introduction.-The realization of robust Majorana zero modes (MZMs) at the ends of quasi-one-dimensional (1D) p-wave superconductors (SCs) has been a longstanding goal in contemporary condensed matter physics [1]. These exotic quasiparticles, predicted to possess non-Abelian exchange statistics, signal the appearance of a novel phase of matter: a topological superconductor. Interest in realizing MZMs has been stimulated by the fundamental-physics quest of discovering new phases of matter, as well as by potential applications to topological quantum computation [2,3].
Generally, three ingredients are needed to realize topological superconductivity in one dimension: proximity coupling to a conventional s-wave superconductor (sufficiently thick to be free of phase fluctuations), a spinrotation mechanism, most commonly spin-orbit coupling (SOC), and a source of time-reversal-symmetry breaking. With a proper combination of these ingredients, the lowenergy band becomes effectively spinless while remaining susceptible to pairing, thus realizing a p-wave superconductor. Time-reversal symmetry is usually broken by an external Zeeman field or by internal magnetic phenomena, such as the exchange field of a nearby ferromagnet.
Realizations in which the proximitizing superconductor is subjected to a magnetic field have the drawback of degrading superconductivity [25]. In particular, all types of time-reversal-symmetry breakers -Zeeman field, exchange field, magnetic flux in the presence of conventional impurities, or magnetic impurities -lead to depairing of Cooper pairs and the formation of in-gap states. In extreme cases, a gapless superconductor is formed [26]. This makes MZMs fragile and renders their detection ambiguous. Moreover, one may wonder why a Zeeman or exchange field is necessary at all. Indeed, several proposals rely on controlling the phase of the SC order parameter or on applying a weak magnetic flux [6,9,14,15,24,27,28], which already break time-reversal symmetry.
In this manuscript, we show that in the presence of a winding superconducting phase, topological superconductivity arises without any Zeeman field or magnetic flux penetrating the sample. The distinction between opposite spins is generated by closed electron trajectories (loops) having gauge-invariant Aharonov-Casher phases [29]. Such gauge invariant phases arise when the loops encircle a net charge [29]. The winding can be obtained when the phases of at least three superconductors form a polygon on the unit circle surrounding the origin [30] [see Fig. 1(b)]. This alleviates the need for a Zeeman field, an exchange field, magnetic fluxes [24], or relatively large supercurrents [27]. The superconducting phases can be controlled by macroscopic superconducting loops, which focus the time-reversal-breaking element on the junction. Therefore, a tiny magnetic field, of less than a micro-tesla for a micron-size loop, can be used to achieve topological superconductivity. In this method, the superconductors remain free of pair-breaking perturbations and in-gap states.
Our main result is the phase diagram in Fig. 1(b) for the three-phase system depicted in Fig. 1(a). The phase diagram depends on the two phase differences φ 1 and φ 2 (φ 3 is set to zero), and periodically repeats the unit cell indicated by the black square [31]. To highlight the role of time-reversal symmetry, we plot the phase diagram as a function of θ = (φ 1 − φ 2 ) /2 and φ = (φ 1 + φ 2 ) /2. Then, similar to a single phase-biased planar Josephson junction [14,15], φ = π and θ = 0 is a timereversal-symmetric point (as are φ = π/2, θ = π/2 and φ = 3π/2, θ = π/2). In contrast to conventional Josephson junctions where a Zeeman field is needed to break time-reversal symmetry and to drive the system into a topological state, here this effect is achieved by the phase difference θ between the superconductors. Coupled-wires model.-To demonstrate our approach in a tractable model, we consider three spin-orbit-coupled wires in proximity to three s-wave superconductors with pair potentials of magnitude ∆ and phases φ 1 , φ 2 , φ 3 , as illustrated in Fig. 1(a).
In the continuum limit, the topological properties are already encoded in the spectrum for zero momentum along the wires, k = 0 [4]. In this case, the Hamiltonian of the three-wire model takes the form n,s c n,s + t ⊥ e iλnσz ss c † n,s c n+1,s + H.c.
where c n,s annihilates an electron in wire n with k = 0 and spin projection s along z, t ⊥ is the inter-wire hopping amplitude, µ is the chemical potential, ∆ is the induced SC pair potential, and λ n is the SOC angle accumulated between the neighboring wires n and n + 1. Here, we assume periodic boundary conditions, c N +1,s = c 1,s . As we will see, it is crucial that electrons acquire an Aharonov-Casher phase [29], which will conspire with the SC phase winding to eliminate one spin species at the Fermi level.
Equation (1) is written for a general number of wires N ; for simplicity, we will focus on the minimal value to create a phase winding, N = 3. Notice that the gauge transformation c n,s → c n,s e iφn/2 eliminates the phases from the SC terms and changes the hopping term . This resembles but is not equivalent to magnetic flux: unlike magnetic flux, the phases φ n can be gauged away when ∆ = 0.
To identify phase transitions in the parameter space of our model, we search for gap closures by equating the determinant of the Hamiltonian Eq. (1) to zero: The SC phases appear in the determinant through a single parameter −3/2 ≤ f ≤ 3, which has a simple geometric interpretation: for −3/2 ≤ f ≤ −1 the phases wind, i.e., when plotted as complex numbers e iφn on the unit circle, the triangle connecting them contains the origin [32]. Solving the quadratic equation det H k = 0 = 0 for Λ, we find that a real solution is possible only for f ≤ −1 [32], and therefore phase winding is a necessary condition for the existence of a zeroenergy state, in agreement with the results of Ref. [30].
Assuming that the SC phase winds, we still have to determine the regions in the three-dimensional parameter space spanned by µ, ∆, and Λ (choosing units such that t ⊥ = 1) for which the system is topological. An optimal situation occurs when the values of the three parameters are such that the determinant Eq. (2) is zero already for f = −1. Then, the system is topological for the maximal range of −3/2 ≤ f < −1. Setting f = −1 in Eq. (2) we find that the optimal situation occurs when (µ, ∆, Λ) are points on a circle C parametrized by µ, 1 − µ 2 , µ , see [32] and Fig. S1.
(2) defines a surface in the parameter space; when (µ, ∆, Λ) lie on this surface, topological superconductivity occurs for −3/2 ≤ f < f crit , see Fig. 1(b). Hence we conclude that topological superconductivity is obtained for all (µ, ∆, Λ) points within the bulk of the shape defined at f crit = −3/2 (see Fig. S1 of the Supplemental Material [32]), with optimal values on the circle C.
To find the energy gap in the topological state, we analyze the full spectrum of the system away from k = 0. Belonging to symmetry class D [33][34][35], the full Hamiltonian is characterized by the Z 2 topological invariant [4,36] where Pf is the Pfaffian and P is the particle-hole operator. Q = 1 indicates the trivial phase, whereas Q = −1 in the topological phase, where the system supports MZMs [37]. The energy gap must be calculated for all values of k .
The numerically calculated [38] topological phase diagram of the system is shown in Fig. 2(a), for parameters chosen on the optimal manifold. Remarkably, the model supports a topological phase with an excitation gap of about 0.3∆, with the application of only a phase difference and without any applied Zeeman or orbital field in the sample. We note that at θ = π 3 , φ = π and θ = 2π 3 , φ = 0 -perfect vortices with equal phase differences forming an equilateral triangle -the system becomes C 3 -symmetric and turns out to be gapless, due to a non-topological gap closure at finite k , see Fig. 2 In Fig. 2(c), we demonstrate that the gap opens when the C 3 symmetry is broken [39]. In addition, the topological gap is bounded from above by the minimum of ∆ (the induced SC gap) and ∆ SO ∼ t ⊥ sin 2 1 2N n λ n (the SOC splitting energy).
Further confirmation for the existence of the topological phase is given in Fig. 2(d), where we show the Majorana wavefunctions deep in the topological phase. These are obtained by diagonalizing the Hamiltonian on a finite lattice. The appearance of two localized Majorana modes with near zero energy at the edges of the system signals its topological nature.
Quantum-well model.-Having established the possibility of realizing a 1D topological superconductor based on phase bias alone, we now turn to exemplifying this concept in a realistic system comprising readily available ingredients. Specifically, our proposal relies on a spinorbit-coupled 2DEG proximitized by three thick SCs. As we have seen, the topological transition requires an Aharonov-Casher phase and thus, our proposal does not easily lend itself to an all-planar geometry. Instead, we propose to use a 2DEG with two (or more) layers giving rise to several subbands, see Fig. 3(a). If the Rashba SOC parameter α is different in the two subbands, there are closed loops in which electrons acquire a non-zero Aharonov-Casher phase, mimicking the periodic boundary conditions in the simplified model we previously studied.
The system is described by the continuum Hamiltonian where the Pauli matrices τ , ρ act in particle-hole and layer space, respectively, t ⊥ is the interlayer hopping amplitude and τ ± = (τ x ± iτ y ) /2 (ρ + is defined similarly). We assume that the Rashba SOC parameter is opposite in the two layers. To be specific, we consider an InSb 2DEG with m * = 0.014m e and α = 15 meV nm [40], corresponding to a SOC length SO ≈ 360 nm. We take an induced SC gap of ∆ = 1 meV, appropriate for, e.g., Nb and Pb [41], in only one layer. The widths of the SCs (normal regions between them) are chosen to be W SC = 70 nm (W N = 40 nm.) The typical length W is chosen roughly according to the relation SO ∆ SO = W ∆. This rule of thumb, which is derived in the Supplemental Material [32], provides a way to approximate favorable dimensions of the system given the material's parameters [42]. The Hamiltonian Eq. (4) was investigated by discretizing it on a lattice of spacing a = 10 nm. The topological phase diagram, calculated by the Pfaffian formula Eq. (3) (now with k = k y ), is shown in Fig. 3(b). The system indeed becomes a topological superconductor in the relevant region of phases. The topological phase constitutes 17% of the displayed θ-φ section, compared to 25% on the optimal manifold of the coupled-wires model [cf. Fig. 1(b)], implying that further optimization is possible. The maximal topological gap is of order ∆ SO , which is reasonable: for the chosen materials ∆ SO is the smallest energy scale. Using materials with larger ∆ SO will lead to a larger topological gap.
As seen in Fig. 3(b), the gap is small compared to ∆ SO in some parts of the topological region. By inspecting the Bogoliubov-de-Gennes spectrum, we find that the small gap originates from the presence of low-energy high-k modes. Semiclassically, these modes result from long trajectories that hardly encounter the SCs, which is a common problem in such systems [27]. Perturbations that eliminate these trajectories, such as non-standard geometries [27,43] or disorder [44], lead to an increased topological gap. We have verified that adding a chemical potential modulation along the x, y directions may significantly increase the topological gap [32].
Discussion.-In contrast to previous schemes, the topological phase in our proposal is induced solely by phase winding in the SC, which is proximity coupled to semiconductors with strong spin-orbit coupling such as InAs, InSb, or HgTe. SC phases can be manipulated using large external loops, through which magnetic flux is threaded, or by application of supercurrent. The applied magnetic field (or the supercurrent), being very small and removed from the sample itself, should have only a mild effect on the parent SC. Therefore in-gap states, which may mask the MZMs, are unlikely to appear. We illustrated our scheme by an analytically accessible toy model and introduced a realistic setup in which these ideas can be implemented. Beyond these settings, we expect that the concept presented here -relying exclusively on SC phase bias and on the spin-dependent phase acquired in closed loops (the Aharonov-Casher phase [29]) -may be harnessed in other systems, as well. For example, it might be possible to realize the wire model experimentally by contacting three of the six facets of an InAs nanowire with three thick phase-biased SCs. The role of disorder deserves a separate treatment. Disorder eliminates trajectories that do not encounter the super- conductors [44] and therefore increases the topological gap. We expect that under the right conditions it also gives rise to non-retro Andreev reflection, thereby facilitating the existence of the relevant closed trajectories.
Finally, a desirable goal for all Majorana platforms is an extension to networks to implement quantum information processing [45]. In our proposal, the experimental challenge is to establish control over a larger number of superconducting phases. At the same time, engineering aspects may be significantly simplified by the absence of a need for a Zeeman field, which requires careful alignment and induces harmful in-gap states.

FURTHER ANALYSIS OF THE PHASE DIAGRAM
In this section we provide further details of the phase diagram in the coupled-wires model. The derivations are based on Eq. (2) of the main text, which determines the phase boundaries.
We begin by setting t ⊥ = 1, and writing the solution of Eq. (2) which is a quadratic equation for Λ: In order to have a real solution, the argument of the square root must be non-negative. Treating the argument of the square root as a second-order polynomial in µ 2 , we find that the condition for a real solution is Since −3/2 ≤ f ≤ 3, the first factor is negative whereas the last factor is positive. Therefore, this inequality is fulfilled only when f ≤ −1. This result, combined with our previous proof that f ≤ −1 corresponds to phase winding, is in agreement with Ref. [30]. The manifold in µ, ∆, Λ parameter space defined by Eq. (S5) is shown in Fig. S1 for several values of f . As explained in the main text, the parameters µ, ∆, Λ are said to be "optimal" if they support a solution of Eq. (2) for f = −1, i.e., if the necessary condition f ≤ −1 is also sufficient. Solving again for Λ, we obtain The only way to make this expression real is demanding µ 2 + ∆ 2 = 1. In this case the two solutions Λ ± are identical and equal to µ/t ⊥ . This gives us the optimal curve C -the circle µ, 1 − µ 2 , µ . It is worth noting that along this circle, one can make a simple connection to the continuum description, thus finding an optimal condition for topological superconductivity in experimental system parameters. If we take ∆/t ⊥ to be small (which means µ ≈ t ⊥ ), we obtain λ ≈ ∆/t ⊥ (assuming λ 1 = λ 2 = λ 3 = λ). In our minimal three-wires tight-binding description, the effective lattice spacing is the typical distance between two superconductors W . Using a continuum description of the tight-binding model along the circumference, we get ∆ SO ≈ t ⊥ λ 2 . Since λ is the spin-dependent angle accumulated when hopping between nearest neighbors, and SO is the distance where a phase of 2π is accumulated, we have W/ SO ≈ λ. Therefore, we obtain the condition which expresses the ideal geometry as a function of the continuum parameters only. Let us exemplify the practical use of this relation. In the main text, we assumed a superconducting gap of ∆ = 1 meV, which is appropriate for e.g. Nb and Pb [41]. Let us now take ∆ = 0.5 meV, which is appropriate for e.g. Sn and V [41]. Using the relation Eq. (S8) above, we simulate a larger system compared to that of Fig. 3(b), with W SC = 100 nm, W N = 100 nm. Fig. S2 shows the resulting topological phase diagram, which indeed exhibits topological regions with a topological gap comparable to ∆ SO , but the topological region is smaller, indicating that further optimization might be necessary. , and plotted in a log-log scale. The dashed lines are the asymptotic forms at ∆ ∆SO, which is √ 2∆SO∆, i.e., the topological gap may be parametrically larger than ∆SO. (b) Same for the quantum-well model Eq. (4), using the same parameters of Fig. 3(b). The maximal topological gap is of order ∆SO, and since ∆ ∆SO for the parameters we used, it is much smaller than ∆.

BOUNDS ON THE TOPOLOGICAL GAP
Here we discuss the bounds limiting the topological gap, in order to justify the choice of comparing it to ∆ SO , which we made in Fig. 3(b) of the main text.
To set the stage, we study the topological nanowire model [7,8] where B is the applied Zeeman field and u is the SOC parameter. For simplicity, we focus on µ = 0 where the condition for a topological phase is B > ∆. The two relevant energy scales are ∆ and ∆ SO = mu 2 /2, and the question is whether or not they both set a bound on the energy gap in the topological phase. At finite B and ∆ there are two minima of the gap in the spectrum as a function of the momentum k, one at k = 0 and the other near the Fermi momentum. The topological gap of the system is determined by the smallest of the two, when B > ∆. It is maximized at B = B * > ∆, for which the gap at k = 0 is equal to the gap near the Fermi momentum. A closed-form expression for B * is hard to obtain, but it is straightforward to find it numerically given the values of the other parameters. Fig. S3(a) shows the maximal topological gap as a function of ∆/∆ SO , normalized by ∆ and by ∆ SO . For InAs/InSb nanowires proximitized by Al, ∆ and ∆ SO are of the same order of magnitude. However, for a InAs/InSb 2DEG such as the one we studied, ∆ ∆ SO and therefore we analyze the asymptotic behavior of the maximal topological gap in this limit-see the dashed lines in Fig. S3(a). By fitting the asymptotes, which can be obtained numerically or analytically, we find that the maximal topological gap in this limit is ∼ √ 2∆ SO ∆. Therefore, the gap can be parametrically larger than ∆ SO . However, it is evident and also seen in Fig. S3(a) that the gap cannot exceed ∆.
The situation is qualitatively different for the quantum-well model studied here, see Eq. (4) and Fig. 3. We demonstrate this by using the same parameters as in Fig. 3(b), with the phases optimally chosen, and vary the ratio ∆/∆ SO . The results are shown in Fig. S3(b). It is clear from this figure that for our system, the maximal gap in the topological region is of order ∆ SO (at the optimal configuration), but it is much smaller than ∆.