Topological superconducting phase in a non-Hermitian Kitaev chain with staggered pairing imbalance

TL;DR

The paper investigates a one-dimensional non-Hermitian Kitaev chain with staggered $p$-wave pairing imbalance ($\gamma_1,\gamma_2$) and a uniform chemical potential $\mu$. By combining real-space BdG analysis and momentum-space dispersion, it reveals real–complex spectral transitions and real/imaginary line gaps, deriving analytical gap-closing conditions $\mu^2=4t^2+(\gamma_1-\gamma_2)^2$ and $\mu^2=-4\Delta^2+(\gamma_1+\gamma_2)^2$ that delimit topological phases. The system hosts a $\mathbb{Z}_2$ topological invariant $\nu=\mathrm{sgn}\{ Pf[A(0)] Pf[A(\pi)] \}$, and the nontrivial phase supporting Majorana zero modes can persist even for arbitrarily large $|\mu|$ when $\gamma_1\gamma_2\le 0$. Moreover, zero-energy Majorana modes can coexist with finite-energy edge modes, explained via a Majorana ladder picture where inter-leg coupling hybridizes edge Majoranas. These findings demonstrate constructive roles for non-Hermitian pairing imbalance in stabilizing topological superconductivity and point to experimental platforms such as proximitized 1D wires or quantum-dot arrays for exploring non-Hermitian Majorana physics.

Abstract

We introduce a one-dimensional non-Hermitian Kitaev chain with staggered imbalance in the $p$-wave superconducting pairing. By tuning the chemical potential and the pairing imbalance, we find that the eigenenergy spectrum undergoes real-to-complex transitions, and the spectral gap can change from a real to an imaginary line gap. The pairing imbalance significantly enlarges the parameter region supporting a topological superconducting phase. Remarkably, we show that a topologically nontrivial phase hosting Majorana zero modes can be induced by varying the pairing imbalance, even in the regime of strong chemical potential. The gap-closing points and phase boundaries are determined analytically, and the resulting phase diagrams are characterized by a nonzero topological invariant. Furthermore, we identify the existence of Majorana zero modes and finite-energy Majorana edge modes in this system. Our results reveal exotic phenomena arising from imbalanced pairing and establish a new platform for exploring topological superconductivity in non-Hermitian systems.

Figures (7)

• Figure 1: (Color online) Schematic illustration of the one-dimensional Kitaev chain with staggered $p$-wave superconducting pairing imbalance under open boundary conditions. Each unit cell contains two sites, labeled A and B, with a uniform chemical potential $\mu$. The orange lines denote nearest-neighbor hopping with amplitude $t$. The gray shaded regions indicate staggered superconducting pairing terms $\Delta \pm \gamma_{1,2}$ on the intra- and intercell bonds, respectively, which give rise to non-Hermiticity through imbalanced pair creation and annihilation.
• Figure 2: (Color online) Real and imaginary parts of the eigenenergy spectrum as a function of chemical potential $\mu$ for different values of $\gamma_1$ with $\gamma_2=0$. Panels (a)–(c) correspond to (a) $\gamma_1=1.5$, (b) $\gamma_1=2.0$, (c) $\gamma_1=2.5$. The color scale denotes the inverse participation ratio (IPR) of the eigenstates. Other parameters are $t=1$, $\Delta=1$, and system size $L=2N=200$.
• Figure 3: (Color online) Eigenenergy spectrum for the system with $\gamma_1=2.0$ and $\gamma_2=0$ for different system size: (a) $L=100$, (b) $L=400$, and (c) $L=800$ under open boundary conditions. Panel (d) shows the corresponding spectrum under periodic boundary conditions. Other parameters are $t=1$, $\Delta=1$.
• Figure 4: (Color online) Real and imaginary parts of the eigenenergy spectrum as a function of chemical potential $\mu$ for different values of $\gamma_2$ with $\gamma_1=0$. Panels (a)–(d) correspond to (a) $\gamma_2=0.5$, (b) $\gamma_2=1.5$, (c) $\gamma_2=2.0$, (d) $\gamma_2=2.5$. Panels (e) and (f) show the spatial probability distributions of the Majorana zero modes and finite-energy Majorana edge modes, respectively. Other parameters are $t=1$, $\Delta=1$, and system size $L=2N=200$.
• Figure 5: (Color online) Eigenenergy spectrum of the non-Hermitian Kitaev chain for finite pairing imbalance in both intra- and intercell pairing terms. Panels (a) and (b) show the real and imaginary parts of the spectrum, respectively, as functions of the chemical potential $\mu$ for $\gamma_1=0.5$ and $\gamma_2=1.5$. Panels (c) and (d) show the real and imaginary parts of the spectrum, respectively, as functions of the intercell pairing imbalance $\gamma_2$ for $\gamma_1=0.5$ and $\mu=2$. The system size is $L=200$.