Zero and Nonzero Energy Majorana Modes in an Extended Kitaev Chain

TL;DR

This work introduces an extended Kitaev chain with three sites per unit cell, formed by coupling a trimer SSH model to the Kitaev chain, which yields a hexamer Majorana lattice supporting both zero-energy Majorana modes and nonzero-energy Majorana edge modes. Through momentum-space analysis, a winding-number invariant is established to characterize MZMs, and special parameter limits yield analytic insights into the topological phase transitions. Real-space computations under open boundaries confirm multiple edge modes, with distinct edge profiles for zero- and nonzero-energy modes, and the edge states remain robust under a range of perturbations and disorder. The model offers a potential route to experimentally detect Majorana modes via finite-energy edge states and provides a platform for exploring enriched topological superconductivity with enhanced tunability and signatures.

Abstract

This paper studies an extended Kitaev chain with three sublattices per unit cell. This extended version is obtained by hybridizing a modified Su-Schrieffer-Heeger model featuring trimerized unit cells with the standard Kitaev chain, resulting in a hexamer structure on the Majorana basis. Due to the interplay between the sublattice configuration and the $p$-wave superconducting pairing, a rich structure of edge modes beyond the expected Majorana zero modes is obtained. The various Majorana edge modes are further found to demonstrate considerable robustness against some generic perturbations and disorder. The presence of the non-zero Majorana edge modes in our system has the potential advantage that they could, in principle, be more unambiguously detected as compared to their zero energy counterparts, the detection of which remains an open problem. Therefore, while our system does not directly solve this open problem, it potentially offers a route to an intermediate solution that involves unambiguously detecting non-zero energy Majorana edge modes instead.

Figures (11)

• Figure 1: Schematic diagram of two adjacent unit cells of the proposed model. Each pair of Majorana sites is represented by a different color for the sake of clarity. (Top) shows the intracell hopping, intercell hopping, and chemical potential. (Bottom) shows the intracell and intercell pairings.
• Figure 2: (a) Schematic diagram of the first and last unit cells in a chain of N=100 unit cells at $\mu=J_1=\delta=0$. Note that there are two Majorana modes decoupled at each end of the chain. (b) Schematic diagram of the first and last unit cells in a chain of N=100 unit cells at $J_1=\delta=0$. Note that there are two Majorana modes coupled at each end of the chain, but they are disconnected from the rest of the chain.
• Figure 3: Some representative energy spectra corresponding to Eq. (\ref{['eq:eqH(k)']}) at $J_1 = 0.5$, $J_2 = 2.5$, and $\delta =\Delta = 0.4$. (a) is at $\mu= 4.24$. (b) is at $\mu=2.59$. (c) at is $\mu=3.5$.
• Figure 4: (a) and (c) are the energy spectrum as a function of $\mu$ and $\Delta$, respectively, for $N=100$ unit cells. (b) and (d) show the numerically calculated winding number associated with (a) and (c), respectively. (a) and (b) are evaluated at $J_1 = 0.5$, $J_2 = 2.5$, and $\delta=\Delta=0.4$. (c) and (d) are evaluated at $J_1 = 0.5$, $J_2 = 2.5$, $\delta=0.4$, and $\mu=2$. (e) and (f) show zoomed-in views of the MZMs from (a) and (c), respectively. The black vertical lines mark the locations of the band touching points anticipated by Eq. (\ref{['mu']}). The red vertical dashed line indicates the value used in Fig. \ref{['fig:figwf']}.
• Figure 5: Wave function profiles associated with the system's edge states for $N=100$ unit cells at $J_1 = 0.5$, $J_2 = 2.5$, $\delta=\Delta=0.4$, and $\mu=3.5$. Note that each subplot displays one left-localized edge state and one right-localized edge state.