Majorana edge modes in one-dimensional Kitaev chain with staggered $p$-wave superconducting pairing

TL;DR

This work analyzes a 1D Kitaev chain with staggered $p$-wave pairing and reveals three distinct edge-state regimes: a topological phase with two Majorana zero modes at the ends, a trivial phase with four unprotected nonzero-energy edge modes, and a trivial phase with no edge modes. Using Bogoliubov–de Gennes formalism and a Majorana ladder mapping to two SSH-like legs coupled by the chemical potential, the authors derive the phase boundary $\mu^2 + 4 \Delta^2 \delta^2 = 4 t^2$ and identify a $Z$ topological invariant in class $BDI$ that captures the nontrivial regime. When both ladder legs are nontrivial, the zero modes hybridize into four nonzero-energy edge modes with energies that scale linearly with $\mu$, a mechanism clarified by perturbation theory giving $E = \pm |S| \mu$. Incorporating dissipation via an imaginary part in the chemical potential shows MZMs remain real and robust, while the nonzero-energy edge modes become complex and unstable. The study provides a new platform for Majorana edge physics driven by staggered pairing and suggests experimental routes in proximitized nanowires or coupled quantum-dot arrays to observe both Majorana and non-Majorana edge states.

Abstract

We introduce a new type of one-dimensional Kitaev chain with staggered $p$-wave superconducting pairing. We find three physical regimes in this model by tuning the $p$-wave pairing and the chemical potential of the system. In the topologically nontrivial phase, there are two Majorana zero modes localized at the opposite ends of the lattice, which are characterized and protected by nonzero topological invariants. More interestingly, we also find a regime where the system can hold four unprotected nonzero-energy edge modes in the trivial phase, which is analogous to a weak topological phase. The third regime is also trivial but holds no edge modes. The emergence of zero- and nonzero-energy edge modes in the system are analyzed by transforming the lattice model into a ladder consisting of Majorana fermions, where the competition between the intra- and inter-leg couplings leads to different phases. We further investigate the properties of edge modes under the influences of dissipation, which is represented by introducing a imaginary part in the chemical potential. Our work unveils the exotic properties induced by the staggered $p$-wave pairing and provides a new platform for further exploration of Majorana edge modes.

Figures (6)

• Figure 1: (Color online) Schematic of the 1D Kitaev chain with staggered $p$-wave SC pairing. The gray shaded area and blue dashed ellipse represent the different pairing term $\Delta(1+\delta)$ and $\Delta(1-\delta)$, respectively. The orange line denotes the hopping $t$ between the nearest-neighboring sites. Each unit cell of the lattice contains two sites, i.e., the $A$ and $B$ site, and the onsite potential is uniform for all sites.
• Figure 2: (Color online) The energy spectrum as a function of chemical potential $\mu$ for the 1D Kitaev chain with staggered $p$-wave SC pairing. The system shows different phases with or without edge modes by tuning the staggered modulation $\delta$ in the SC pairing term: (a) $\delta=-0.5$, (b) $\delta=-1.0$, (c) $\delta=-1.5$, (d) $\delta=0.5$, (e) $\delta=1.0$, and (f) $\delta=1.5$. The colorbar indicates in (a) the IPR value of the eigenstate. The system parameters are chosen as $t=\Delta=1$. The number of unit cell in the lattice is $N=50$.
• Figure 3: (Color online) The energy level of the eigenenergies of the system at $\mu=0.2$ with (a) $\delta=-0.5$ and (b) $\delta=-1.5$. Other system parameters are the same as in Fig. \ref{['fig2']}. The upper-left inset in (a) shows the distribution of the two MZMs in the 1D lattice. The zoom-in in the lower-right corner in (a) and (b) indicate that there are two and four edge modes in the band gap, respectively.
• Figure 4: (Color online) (a) Phase diagram for the 1D Kitaev chain with staggered $p$-wave pairing. The blue region represent the topologically nontrivial regime with $N=\pm 1$ for $\Delta>0$ or $\Delta<0$, respectively. The region outside the ellipse is trivial with $N=0$. (b) The spectrum of the system as a function of $\delta$. Other parameters: $t=\Delta = 1$, $\mu=1$, and $N=50$.
• Figure 5: (Color online) (a) Schematic of the Majorana fermion ladder by writing the model Hamiltonian in the Majorana fermion operators. Each leg of the ladder resembles the SSH model with staggered hopping amplitudes. The two legs in the ladder are coupled by the chemical potential $\mu$. (b) In the case with $|\delta|<1$, only one leg will be in the topologically nontrivial phase, while the other one is in the trivial phase. Here the lower leg is in the trivial phase and the upper leg is nontrivial. Thus the whole ladder holds only two zero-energy modes at the two ends. The Majorana fermions enclosed by the gray ellipse are paired with each other due to the staggered hopping. (c) When $\delta<-1$, both the legs are topologically nontrivial, each of them holds a zero-energy modes at each end of the ladder. The zero-energy modes at the same end are coupled by $\mu$, and their hybridization results to the non-zero energy edge modes.