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

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.