Summation of Divergent Series and Quantum Phase Transitions in Kitaev Chains with Long-Range Hopping

Abstract

We study the quantum phase transitions (QPTs) in extended Kitaev chains with long-range ($1/r^{\hat{I}}\pm$) hopping. Formally, there are two QPT points at $\hat{I}¼=\hat{I}{¼}_0(\hat{I}\pm )$ and $\hat{I}{¼}_{\ddot{I}}€(\hat{I}\pm )$ ($\hat{I}¼$ is the chemical potential) which correspond to the summations of ${\sum }_{m=1}^{\infty }{m}^{-\hat{I}\pm }$ and ${\sum }_{m=1}^{\infty }(-1{)}^{m-1}{m}^{-\hat{I}\pm }$, respectively. When $\hat{I}\pm \le 0$, both the series are divergent and it is usually believed that no QPTs exist. However, we find that there are two QPTs at $\hat{I}¼=\hat{I}{¼}_0(0)$ and $\hat{I}{¼}_{\ddot{I}}€(0)$ for $\hat{I}\pm =0$ and one QPT at $\hat{I}¼=\hat{I}{¼}_{\ddot{I}}€(\hat{I}\pm )$ for $\hat{I}\pm <0$. These QPTs are second order. The $\hat{I}{¼}_0(0)$ and $\hat{I}{¼}_{\ddot{I}}€(\hat{I}\pm \le 0)$ correspond to the summations of the divergent series obtained by the analytic continuation of the Riemann $\hat{I}\P$ function and Dirichlet $\hat{I}\cdot$ function. Moreover, it is found that the quasiparticle energy spectra are discontinue functions of the wave vector $k$ and divide into two branches. This is quite different from that in the case of $\hat{I}\pm >0$ and induces topological phases with the winding number $\ddot{I}‰:=\pm 1/2$. At the same time, the von Neumann entropy are power law of the subchain length $L$ no matter in the gapped region or not. In addition, we also study the QPTs, topological properties, and von Neumann entropy of the systems with $\hat{I}\pm >0$.