Double-frequency Aharonov-Bohm effect and non-Abelian braiding properties of Jackiw-Rebbi zero-mode

Abstract Ever since its first proposal in 1976, Jackiw-Rebbi zero-mode has been drawing extensive attention for its charming properties including charge fractionalization, topologically protected zero-energy and possible non-Abelian statistics. We investigate these properties through the Jackiw-Rebbi zero-modes in quantum spin Hall insulators. Though charge fractionalization is not manifested, Jackiw-Rebbi zero-mode's zero-energy nature leads to a double-frequency Aharonov-Bohm effect, implying that it can be viewed as a special case of Majorana zero-mode without particle-hole symmetry. Such relation is strengthened for Jackiw-Rebbi zero-modes also exhibiting non-Abelian properties in the absence of superconductivity. Furthermore, in the condition that the degeneracy of Jackiw-Rebbi zero-modes is lifted, we demonstrate a novel non-Abelian braiding with continuously tunable fusion rule, which is a generalization of Majorana zero-modes’ braiding properties.

The Hamiltonian describing an AB ring with a Jackiw-Rebbi zero-mode embedded in one arm has the form of: (0) (S1) where i = 1, 2 is the lead index. The creation operator for the conducting mode (in the metal lead) and the Jackiw-Rebbi zero-mode are denoted as ψ † i (x) and φ † (0), respectively. From left to right, the four terms in Eq. (S1) are the kinetic energy of the metal leads (v f the Fermi velocity), direct hopping term (with strength t d ) between two metal leads, hopping term (with strength t 0 ) between the Jackiw-Rebbi zero-mode and the metal leads, and the on-site energy (denoted by ϵ 0 ) of the Jackiw-Rebbi zero-mode, respectively.
The Hamiltonian Eq. (S1) is derivated as following. Assuming both these two metal leads in the AB ring contain only one conducting mode per moving direction. Hence the Hamiltonian of the first lead can be written as [S1]: where v f is the Fermi velocity, ϵ denotes the left-/right-moving mode, and σ is the spin index. Assuming ψ 1Lσ (x) = ψ 1Rσ (−x) for x > 0 and suppressing the right-moving index ϵ = R, Eq. (S2) is simplified as [S1]: The Hamiltonian for the second lead H L2 can be dealed with in the same way. Besides, in the AB ring, the two hopping paths between the tip of these two leads have the form of: Both hopping strength t d and t 0 are assumed to be real, ξ σ are complex numbers with |ξ σ | = 1 and ϕ is the magnetic flux inclosed. Operating a unitary transformation and dropping ψ ′ i for not participating in the interference (only contributing a conductance constant), finally we get the full Hamiltonian whose form is exactly Eq. (S1) by combining H L1 , H L2 , H T and the on-site energy of the zero-mode. Adopting the celebrated Heisenberg's equation of motion (EOM) i∂ tÔ = [Ô, H], we can write down the EOMs for ψ 1 , ψ 2 and φ with real space and time variables. Operating the Fourier transform (where i = 1, 2) S5) and then integrating the EOMs around are inserted}, finally two independent EOMs can be written in a matrix form as . The operator at x = 0 − (x = 0 + ) is explained as the incoming (outgoing) mode, since the conducting mode of the lead at x = 0 − is mapped from the left-moving mode. Therefore, the S-matrix defined as has the explicit form of (where λ ≡ it d e iϕ ) and T 12 is the modulus square of the non-diagonal element of the S-matrix as T 12 = |S 12 | 2 , which is exactly Eq. (2) in the main text. As shown in Fig. S1, the numerical results of T 12 (obtained by the Green's function) could be fitted by the analytic formula as [where c 0 , c 1 , and ϕ ′ are constants].

DERIVATION OF THE S-MATRIX FOR MAJORANA ZERO-MODE'S AHARONOV-BOHM EFFECT
The Hamiltonian describing an AB ring with a Majorana zero-mode (MZM) embedded can be obtained by substituting the last two terms of Eq. (S1) by: where η(0) is the MZM operator. The first term of Eq. (S8) is the coupling (with strength t M ) between MZM and the metal leads, as the second term is the on-site energy of the MZM.
The derivation of the S-matrix describing MZM's AB effect is in the same procedure as the Jackiw-Rebbi zeromode's case. An important difference lies in that the EOMs for ψ i and ψ † i are coupled due to the presence of MZM. Therefore the S-matrix relates the incoming mode and outgoing mode has the definition of (where 1, 2 for lead indices, and e, h for electron and hole, respectively) in the Bogoliubov-de Gennes (BdG) basis. The explicit form of the S-matrix is shown to be: With the MZM, the conductance G i is defined as the derivative of the current inside the ith lead [S2, S3] with respect to the bias V : The conductance in Eq. (S11) can be decomposed into two parts as ii | 2 } is induced by the current which flows out of lead i and then flows back into lead i, and is proportional to the current which flows out of lead j and then flows into lead i. Apart from the conductance between two leads G ij [Eq. (3) in the main text], the explicit form of the conductance G i is It is easy to see that G 1 and G 2 are in an anticorrelated fashion [S3], and the total conductance is quantized at 2e 2 h in the zero-energy condition [S3]. The total conductance G decays in the manner of 1 1+E 2 for non-zero energy, and the oscillation term ] is always in the period of 2π.

NUMERICAL SIMULATION FOR THE BRAIDING OF JACKIW-REBBI ZERO-MODES
The Hamiltonian describing QSHI constriction [Eq.
(1)] in a square lattice has the form of where r i stands for the location of the ith lattice site. T 0 , T x , and T y are the on-site energy, hopping term along the x-direction, and hopping term along the y-direction, respectively. Each of the four arms in the cross-shaped junction . |⟨ϕ(t = 6T )|ψ 12 − ⟩| 2 +|⟨ϕ(t = 6T )|ψ 12 + ⟩| 2 is significantly smaller than 1 in (d), as the non-Abelian braiding is destructed for strong disorder destroying the topological gap.
[ Fig. 3 (a) in the main text] can be described by Eq. (S14), while the hopping term near the crossing controlled by the gate voltages has the form of where r i,α denotes the ith lattice site in the αth arm (α = 1, 2, 3, 4), r j,c denotes the jth lattice site at the crossing point, and ⟨i, j⟩ means the nearest neighbour. In the numerical simulation, gate voltages G1, G2, G3 are turned on (off) linearly, therefore g α (α = 1, 2, 3) in Eq. (S15) is approximated as step functions g α = 1 − n/N (g α = n/N ) with n = 0, 1, 2, ..., N (N a large integer). The whole braiding Hamiltonian H t = H 0 + H gate is time-dependent and the time-evolution operator in the form of U (t) =Te i ∫ dt·H(t) (T is the time-ordering operator) is approximated as U (t) ≈ ∏ n e iδt·Ht due to the step-function approximation. The eigenstate of the junction evolves as |ϕ(t)⟩ = U (t)|ϕ(t = 0)⟩ where |ϕ(t = 0)⟩ is the initial eigenstate (before braiding). As the braiding protocol stated in the main text, each braiding step takes time of T . The adiabatic condition is satisfied when the excitation energy ∼ 1/T will not give rise to energy level transition. There are two energy scales in the QSHI cross-shaped junction, the topological gap ∆ b , and the coupling between Jackiw-Rebbi zero-modes ϵ 12 , ϵ 34 . In both Fig. 3 Fig. S2, ∆ b ≈ 0.2 and ϵ 12 , ϵ 34 ≈ 7×10 −5 , so we choose δ t = 0.1 and N = 1000, hence the time cost in each braiding step T = 2 × N δ t = 200 satisfies the adiabatic condition as ∆ b ≫ 1/T ≫ ϵ 12 [S6].

BRAIDING PROPERTIES OF MAJORANA ZERO-MODES IN THE PRESENCE OF DISORDER
MZMs' braiding is performed with the same protocol and in the same shape of junction as Jackiw-Rebbi zero-modes, where the difference is that the cross-shaped junction here is composed of p ± ip-wave SC supporting MZMs. The Hamiltonian of p ± ip-wave SC in the BdG basis possesses PH symmetry as −H(−p) = PH T (p)P −1 with P = σ x , and hence is in the D symmetry class (the same symmetry class as Kitaev's chain with complex SC pairing). Similar to the Jackiw-Rebbi case, there are six MZMs (denoted as γ i=1,2,...,6 ) in the cross-shaped junction, and the effective Hamiltonian describing the coupling energy (ϵ 2i−1,2i ) and a "fictitious" energy deviation (∆ 2i−1,2i ) of the MZMs reads [S4]: FIG. S4: Numerical simulation results of ⟨ϕ(t = 6T )|ψ 12 + ⟩ in a fixed PH-symmetry-breaking disorder profile with different disorder strength W and braiding time T . The coupling energy between MZMs ϵ12, ϵ34 ≈ 1 × 10 −9 , and the SC gap ∆SC ≈ 2.6. Too long (T = 1 × 10 8 ) or too short (T = 0.1, 1) braiding time will violate the adiabatic condition ∆SC ≫ 1/T ≫ ϵ12, ϵ34. The fitting curve for the adiabatic results (T = 1 × 10 2 , 1 × 10 5 ) by Eq. (S20) is shown in black. The blue (red) shaded region indicates W ∼ ϵ12, ϵ34 (W ∼ ∆SC).
In analogy with the case of Jackiw-Rebbi zero-mode, in the condition that the disorder has a PH-symmetryconserved form as , MZMs' non-Abelian properties remain unchanged until the SC gap is destructed by strong disorder W ∼ ∆ SC [ Fig. S3 (c), (d)]. This is quite reasonable since H P dis imposes disorder of opposite signs on the electron band and hole band of the p ± ip-wave SC, therefore the energy deviation of MZMs (∆ 2i−1,2i ) vanishes and the non-Abelian braiding properties maintain integrity.