Non-local edge mode hybridization in the long-range interacting Kitaev chain

TL;DR

The paper addresses how long-range electron interactions affect Majorana edge modes in the Kitaev chain by formulating a self-consistent long-range Kitaev chain (seco-LRKC). Using a mean-field BCS framework, it derives a self-consistent gap matrix $\Delta_{x,x'}$ and a correlated BdG Hamiltonian, revealing a bimodal gap structure: a short-range translationally invariant band and exponentially localized long-range end clusters that scale as $n^{- u}$. These end clusters couple edge modes non-locally, lifting ground-state degeneracy and giving a finite Majorana mass $|E_{ m MZM}| \sim a\,n^{- u}$ for all $\nu>0$, while preserving massless Majorana modes in the thermodynamic limit. The topology reduces to two phases—trivial and topological—with a winding number independent of $\nu$, contrasting with non-self-consistent models that exhibit richer phase structure. The findings have implications for realizing robust Majorana modes in mesoscopic systems where long-range interactions are present, and point to future explorations in higher dimensions and dynamical regimes.

Abstract

In one-dimensional p-wave superconductors with short-range interactions, topologically protected Majorana modes emerge, whose mass decays exponentially with system size, as first shown by Kitaev. In this work, we extend this prototypical model by including power law long-range interactions within a self-consistent framework, leading to the self-consistent long-range Kitaev chain (seco-LRKC). In this model, the gap matrix acquires a rich structure where short-range superconducting correlations coexist with long-range correlations that are exponentially localized at both chain edges simultaneously. As a direct consequence, the topological edge modes hybridize even if their wavefunction overlap vanishes, and the edge mode mass inherits the asymptotic scaling of the interaction. In contrast to models with imposed power law pairing, where massive Dirac modes emerge for exponents $ν< d$, we analytically motivate and numerically demonstrate that, in the fully self-consistent model, algebraic edge mode decay with system size persists for all interaction exponents $ν> 0$, despite exponential wave function localization. While the edge mode remains massless in the thermodynamic limit, finite-size corrections can be experimentally relevant in mesoscopic systems with effective long-range interactions that decay sufficiently slowly.

Figures (2)

• Figure 1: Self-consistent solution for $U_0=\tau/2,\ \nu=1/2,\ \mu=\tau/2,\ n=256$. (a) Self-consistent gap $\Delta_{x,x'}$ as a function of $x$ and $x'$. (b) Eigenvalues of the BdG matrix $\mathcal{H}$ with self-consistent gap. The inset shows the separation of the edge modes. The bulk is gapped and its largest value is $E=\tau+|\mu|$. For $E=\pm |\mu|$ we observe a Van-Hove singularity. (c) Eigenstates of both coupled edge modes, where only the the annihilation part is shown. The inset shows the phase diagram as a function of chemical potential $\mu$ and long-range exponent $\nu$. Yellow indicates a winding number of $w=0$, while violet denotes $|w|=1$.
• Figure 2: (a) Absolute value of the edge mode eigenvalue as a function of system size $n$ for different interaction exponents $\nu$, contrasting self-consistent and non-self consistent solutions with parameters $U_0=\tau/2$ and $\mu=0$. The dashed lines represent a power law fit to the data. (b) Decay exponent $\gamma$ of the edge mode eigenvalue (obtained by the fit in (a)) as a function of the interaction exponent $\nu$. $E_{\mathrm{MZM},\infty}$ is a fitting parameter for the mass in the thermodynamic limit.