Topological Superfluid and Majorana Zero Modes in Synthetic Dimension

Recently it has been shown that multicomponent spin-orbit-coupled fermions in one-dimensional optical lattices can be viewed as spinless fermions moving in two-dimensional synthetic lattices with synthetic magnetic flux. The quantum Hall edge states in these systems have been observed in recent experiments. In this paper we study the effect of an attractive Hubbard interaction. Since the Hubbard interaction is long-range in the synthetic dimension, it is able to efficiently induce Cooper pairing between the counterpropagating chiral edge states. The topological class of the resultant one-dimensional superfluid is determined by the parity (even/odd) of the Chern number in the two-dimensional synthetic lattice. We also show the presence of a chiral symmetry in our model, which implies Z classification and the robustness of multiple zero modes when this symmetry is unbroken.

This interaction is apparently long-range in the synthetic dimension, thus it is quite capable of pairing the counterpropagating modes at the opposite edges (near = m F and = − m F, respectively) in the synthetic dimension. Before proceeding to a quantitative study of H I , we would like to discuss the physical picture of possible topological superfluidity in this model. Since the hopping lacks translational symmetry along the synthetic dimension, the bulk Chern number as an integral of Berry curvature in the two-dimensional Brillouin zone cannot be defined, however, its manifestation as the number of chiral edge modes is well-defined. Suppose that the "bulk Chern number" C = 1, i.e. there is a single pair of chiral edge modes in the bulk gap [see Fig. 2, in which the purple dotted lines intersect with the chiral modes]. If there is a small Cooper pairing between these two edge modes, the system is a one-dimensional topological superfluid. This can be inferred using Kitaev's Z 2 topological invariant 2 , which essentially counts the parity (even/odd) of the number of Fermi points within π , /a [0 ] in the absence of pairing. In the case = C 2, i.e. there are two pairs of chiral edge modes, as illustrated by the curves intersecting with the blue dotted line in Fig. (2b), the superfluid resulting from pairing the edge states is topologically trivial. Provided that the pairing is small, the superfluid (or superconductor) is nontrivial(trivial) when the bulk Chern number is odd(even), namely, where ν = , 0 1 (mod 2) is the Z 2 topological number of 1D superconductor or superfluid 2 . In the rest part of this paper we shall present a quantitative study of the picture outlined above.
Cooper pairing in self-consistent mean-field. At the mean-field level the Hubbard interaction can be decomposed as Since the basic physics is the pairing of chiral edge modes with opposite momenta, it is natural to consider Cooper pairing with zero total momentum, namely that ∆ ≡ ∆ , ′ ′ n mm m m is independent of n. We determine these ∆ ′ mm self-consistently. Whenever there are several sets of self-consistent solutions of ∆ ′ { } mm , we compare their mean-field energies and pick up the ground state. This calculation can be carried out for all possible values of F. Hereafter we take F = 7/2 as an example.
The pairings as functions of Hubbard U are shown in Fig. (3) for two values of chemical potential μ. In Fig. (3a) we take µ = − .  To illustrate the robustness of the present picture, we study a set of quite different parameters, shown in Fig. (4). The behavior of pairings is qualitatively the same as that for the previous parameters.
A few remarks are in order. First, although we have taken the strength of hopping along the synthetic dimension as Ω , g F m , we have also checked that the result is qualitatively the same when it is m-independent. Second, when the 2D bulk is metallic, topological superfluidity can still emerge, though there is no clear criteria using Chern number. We shall not focus on details about this.
Majorana zero modes. The hallmark of 1D topological superconductor or superfluid is the emergence of topological zero modes localized near the two ends of an open chain.
We have solved the BdG mean-field Hamiltonian Eq.   , which are the mean-field pairing obtained at = . U 1 3 ( see also Fig. (3a)). Other ∆ ′ mm 's are much smaller and thus neglected. The two zero modes have the same profile of ψ ψ ( ) ≡ ∑ ( , ) x xm m 2 As a comparison, we also present the zero mode solutions at µ = − . 0 95 2 , for which the free Hamiltonian have two pairs of chiral edge modes (C = 2), indicating that the superfluid phase at small U should be Z 2 topologically trivial (see Eq. (4)). In the numerical calculation with open boundary condition, we find two Majorana zero modes at each end (see Fig. 6), which means that the superfluid is Z 2 trivial. Therefore we see again that (− 1) C determines the Z 2 topological classification of the 1D superfluid in synthetic dimension.
One may wonder why there is no hybridization between the two zero modes, which may open a gap for them. We shall explain the reason as follows. In fact, the BdG Hamiltonian has a time-reversal symmetry and a particle-hole symmetry, which can be combined into a chiral symmetry 48 . If the Cooper pairing ∆ ′ mm are real, then we can check that the Hamiltonian satisfies This matrix is written in the BdG basis of ( , , , , Due to these symmetries, the BdG Hamiltonian can be classified as BDI 48,49 , whose classification in 1D is Z. The Z 2 topologically trivial phase is nontrivial according to the Z classification of BDI class, which is the reason why zero modes appear at the edge of Z 2 trivial states. We have checked that, if we break the symmetries, e.g. by giving a phase factor to ∆ / ,− / 7 2 7 2 (with other ∆ ′ mm s unchanged), then these zero modes will be shifted to nonzero energies.
According to Fidkowski and Kitaev's work 50 , in the presence of interaction, the classification of BDI-class topological superconductors in 1D is Z 8 instead of Z. Because of the flexible tunability to topological superconductors with large topological number (say 8) in our system, the Z 8 classification can be tested experimentally. If we tune the bulk Chern number of our system to 8, the eight nominally-zero-modes will be shifted away from zero energy due to the (beyond-mean-field) interaction effects of the Hubbard term. If observed, this will be an experimental test of Fidkowski and Kitaev's Z 8 classification.
To make a closer connection to experiment, we also study the existence of Majorana zero modes in a system with soft boundary created by a harmonic trap V(x). In the presence of V(x), the chemical potential becomes µ µ . We take µ = − . ( / ) V x x a 0 00025 2 , such that the center of the system is topologically nontrivial. Since µ ( ) x is not constant, ∆ ′ mm should also be x-dependent. To incorporate this effect, we numerically calculate the functions µ ∆ ( ) ′ mm at = − . U 1 3, which is then used to produce the mean-field BdG Hamiltonian in harmonic trap. In the solution to this BdG Hamiltonian, the zero modes can be clearly seen, as shown in Fig. (7), though the quantitative details are different from the case of hard boundary.

Conclusions and Discussions
We have studied the pairing between counterpropagating chiral edge modes in the quantum Hall strip in synthetic dimension. This picture has several merits. Creation of magnetic flux in the synthetic dimension by Raman beans is easier than in physical dimensions. The spatial separation of left and right moving chiral edge states in the synthetic dimension effectively prevents the backscattering between them, which implies their robustness. Meanwhile, the Hubbard interaction is not suppressed by this spatial separation: It is infinite-range in the synthetic dimension, therefore, it can pair the two edge modes quite efficiently. As we have shown, the resultant states are topological superfluid carrying Majorana zero modes. If the chiral symmetry of our model is unbroken, the classification is Z and multiple zero modes are stable; on the other hand, if this symmetry is broken, the classification is Z 2 .
Finally, we remark that quantum fluctuations of the phase factor of pairing in 1D is generally strong. One can put the 1D system in proximity to a 3D supefluid to suppress these fluctuations 20 . Moreover, it has been shown 51,52 that long-range superconducting order is not a necessary condition for the existence of Majorana zero modes. The zero modes persist even when the long-range superconducting order is replaced by algebraic order (i.e. the correlations of pairings decay by power-law). In our system this conclusion applies.

Methods
Mean-field calculations. The mean-field calculation is carried out by the standard procedure of decomposing the Hubbard interaction as fermion bilinear terms, leading to Eq. (6). The Cooper pairing is calculated from Eq.(6) in a self-consistent manner. All self-consistent solutions for the Cooper pairing are obtained. In the case that there are more than one self-consistent solutions, the one with lowest mean-field energy is selected. Figure 7. Majorana zero modes in the harmonic trap. The parameters are t = 1, Ω = 0.1, and γ π = /3, which are the same as used in Fig. (2b). The harmonic trap is ( ) = .
( / ) V x x a 0 00025 2 , where x = 0 is the center of the chain with size N = 150. The two zero modes have the same profile of ψ ( , ) x m 2 , thus we only show one. The inset of (a) shows several energies close to E = 0.