Edge states of the long-range Kitaev chain: an analytical study

TL;DR

This paper analytically investigates the edge states of a long-range Kitaev chain, revealing how their decay behavior depends on the decay exponents of tunneling and pairing, with results confirmed by numerical methods.

Contribution

It provides an analytical characterization of the decay properties of edge states in a long-range Kitaev chain, extending understanding beyond previous numerical studies.

Findings

Edge states decay exponentially when =eta and coefficients are equal.

Otherwise, decay is exponential at short distances and algebraic asymptotically.

Algebraic tail exponent is determined by the smaller of  and eta.

Abstract

We analyze the properties of the edge states of the one-dimensional Kitaev model with long-range anisotropic pairing and tunneling. Tunneling and pairing are assumed to decay algebraically with exponents $\alpha$ and $\beta$, respectively, and $\alpha,\beta>1$. We determine analytically the decay of the edges modes. We show that the decay is exponential for $\alpha=\beta$ and when the coefficients scaling tunneling and pairing terms are equal. Otherwise, the decay is exponential at sufficiently short distances and then algebraic at the asymptotics. We show that the exponent of the algebraic tail is determined by the smallest exponent between $\alpha$ and $\beta$. Our predictions are in agreement with numerical results found by exact diagonalization and in the literature.