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

We study the quantum phase transitions (QPTs) in extended Kitaev chains with long-range ($1/r^{\alpha}$) hopping. Formally, there are two QPT points at $\mu=\mu_0(\alpha)$ and $\mu_\pi(\alpha)$ ($\mu$ is the chemical potential) which correspond to the summations of $\sum_{m=1}^{\infty}m^{-\alpha}$ and $\sum_{m=1}^{\infty}(-1)^{m-1}m^{-\alpha}$, respectively. When $\alpha\leq0$, both the series are divergent and it is usually believed that no QPTs exist. However, we find that there are two QPTs at $\mu=\mu_0(0)$ and $\mu_\pi(0)$ for $\alpha=0$ and one QPT at $\mu=\mu_\pi(\alpha)$ for $\alpha<0$. These QPTs are second order. The $\mu_0(0)$ and $\mu_\pi(\alpha\leq0)$ correspond to the summations of the divergent series obtained by the analytic continuation of the Riemann $\zeta$ function and Dirichlet $\eta$ 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 $\alpha>0$ and induces topological phases with the winding number $\omega:=\pm1/2$. 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 $\alpha>0$.