Shot noise signatures identifying non-Abelian properties of Jackiw-Rebbi zero modes

TL;DR

This work addresses identifying non-Abelian braiding in Jackiw-Rebbi zero modes by linking braiding fidelity to an experimentally accessible transport signature. It derives a central relation $\mathcal{F} = 1/\sqrt{(M/\epsilon)^2 + 1}$ between braiding fidelity and the on-site energy difference $M$, and shows via a transport model that the resonant tunneling coefficient satisfies $T(E_0) = \mathcal{F}^2$ with a Fano factor at resonance obeying $F_0 = 1 - \mathcal{F}^2$. When $M=0$, JR zero modes effectively decompose into two independent Majorana channels, yielding noise suppression and unity fidelity, while $M\neq 0$ degrades braiding due to cross-coupling. Numerical simulations on SSH and Kitaev lattice realizations validate the analytical relations, demonstrating that shot-noise measurements provide a practical diagnostic for the quality of non-Abelian braiding in topological platforms. The results offer a feasible experimental route to certify topological quantum operations without performing explicit braiding, by monitoring current noise in SSH-based architectures.

Abstract

Jackiw-Rebbi zero modes were first proposed in 1976 as topologically protected zero-energy states localized at domain walls in one-dimensional Dirac systems. They have attracted widespread attention in the field of topological quantum computing, as they serve as non-superconducting analogs of Majorana zero modes and support non-Abelian statistics in topological insulator systems. %In the braiding process of the Jackiw-Rebbi zero modes, their braiding properties are closely related to the strength of disorder. However, compared to their Majorana cousins, the braiding properties of Jackiw-Rebbi zero modes are vulnerable to the on-site energy deviation between the modes involved in the experiment. In this work, we propose to estimate the braiding properties of Jackiw-Rebbi zero-modes through measurements of transport signatures, which are readily measurable in current experiments. We find that the fidelity of braiding operation reaches unity when the current noise is fully suppressed, while this braiding fidelity monotonously decreases with the increasing of the current noise. Based on these transport signatures, we further discuss the correspondence between Majorana and Jackiw-Rebbi zero modes, highlighting their similarity in supporting non-Abelian statistics.

Figures (5)

• Figure 1: (a) Two SSH chains hosting four Jackiw-Rebbi zero modes $\varphi_1$, $\varphi_2$ (in SSH chain 1) and $\varphi_3$, $\varphi_4$ (in SSH chain 2). The NOT gate is realized by swapping $\varphi_2$ and $\varphi_3$ twice in succession. (b) This braiding operation can be implemented in a cross junction via tuning tunneling amplitude $t_{14}$, $t_{24}$, $t_{34}$ between the terminal sites of different SSH chains, following Refs. amorim2015majoranawu2020double.
• Figure 2: Decomposition of one pair of Jackiw-Rebbi zero modes into two pairs of Majorana zero modes. Left: two Jackiw-Rebbi zero modes, $\varphi_1$ and $\varphi_2$, are coupled by $\epsilon$ and have on-site energies $\pm M$, respectively. Right: the system is decomposed into Majorana components with $\varphi_1 = (\gamma_1^a + i\gamma_1^b)/2$ and $\varphi_2 = (\gamma_2^b + i\gamma_2^a)/2$. Dashed lines indicate Majorana couplings: coupling terms $\tfrac{i\epsilon}{2}\gamma_1^a\gamma_2^a$ (and $-\tfrac{i\epsilon}{2}\gamma_1^b\gamma_2^b$) between MZMs of the same "flavor" $a$ or $b$, and cross coupling term $\pm\tfrac{iM}{2}\gamma_j^a\gamma_j^b$ ($j=1,2$) mixing MZMs with different "flavors".
• Figure 3: Numerical verification of the quantitative relation between the NOT gate (implemented by the braiding operations on the Jackiw-Rebbi zero modes) fidelity $\mathcal{F}$ and the Fano factor at resonant energy $F_{0}$. The braiding simulation is carried out on the SSH cross junction shown in Fig. \ref{['fig:braiding schemes']}(b). The Hamiltonian parameters are chosen such that the finite-size coupling between Jackiw-Rebbi zero modes are $\epsilon = 6.2 \times 10^{-5}$, with on-site energy difference $M$ added at comparable energy scales. Shot noise is calculated across the SSH chain 1 [see Fig. \ref{['fig:braiding schemes']}(b)] that the electrons are transported through the Jackiw-Rebbi zero modes $\varphi_1$ and $\varphi_2$. The numerical results (blue circles) verify the analytical relation (red line) $\mathcal{F} = \sqrt{1 - F_{0}}$, confirming the relationship between the NOT gate fidelity $\mathcal{F}$ and the Fano factor at resonant energy $F_{0}$.
• Figure 4: Transport through the two MZM channels $\gamma_j^a$ and $\gamma_j^b$ decomposed from a single pair of Jackiw-Rebbi zero modes. (a) The normal electron transport, where the two MZM channels act independently and each contributes to the total current. (b) The CAR process, in which charge conjugation converts the incoming electron into a hole. Due to charge conservation, a $2e$ charge Cooper pair is transferred between these two Majorana channels in the CAR, and the two MZM channels contribute the opposite current in the process, resulting in a net zero current being transmitted across the two channels.
• Figure 5: (a) Shot noise curves for $M = 0, 0.5\epsilon, 1.0\epsilon$ of the Jackiw-Rebbi zero modes system, showing the characteristic noise suppression at resonance. Right column: enlarged view of the noise dip. (b) Tunneling coefficient $T(E)$ as a function of energy for Kitaev chains (left) and SSH chains (right) with different coupling strengths between the central region and the lead $t_c = 0.2, 0.6, 1.0, 1.6$. The tunneling coefficient for MZMs is always half that of Jackiw-Rebbi modes as $T_{\mathrm{MZM}}(E) = \frac{1}{2}T_{\mathrm{JR}}(E)$, proving that the result shown in Eq. (\ref{['eq:halfcurrent']}) is independent of the coupling strength between the leads and the central region.