Summation of Divergent Series and Quantum Phase Transitions in Kitaev Chains with Long-Range Hopping
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866908360983445504 |
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| author | Fu, Hao Tong, Peiqing |
| author_facet | Fu, Hao Tong, Peiqing |
| contents | We study the quantum phase transitions (QPTs) in extended Kitaev chains with long-range ($1/r^α$) hopping. Formally, there are two QPT points at $μ=μ_0(α)$ and $μ_π(α)$ ($μ$ is the chemical potential) which correspond to the summations of $\sum_{m=1}^{\infty}m^{-α}$ and $\sum_{m=1}^{\infty}(-1)^{m-1}m^{-α}$, respectively. When $α\leq0$, both the series are divergent and it is usually believed that no QPTs exist. However, we find that there are two QPTs at $μ=μ_0(0)$ and $μ_π(0)$ for $α=0$ and one QPT at $μ=μ_π(α)$ for $α<0$. These QPTs are second order. The $μ_0(0)$ and $μ_π(α\leq0)$ correspond to the summations of the divergent series obtained by the analytic continuation of the Riemann $ζ$ function and Dirichlet $η$ 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 $α>0$ and induces topological phases with the winding number $ω:=\pm1/2$. 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 $α>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_09566 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Summation of Divergent Series and Quantum Phase Transitions in Kitaev Chains with Long-Range Hopping Fu, Hao Tong, Peiqing Statistical Mechanics We study the quantum phase transitions (QPTs) in extended Kitaev chains with long-range ($1/r^α$) hopping. Formally, there are two QPT points at $μ=μ_0(α)$ and $μ_π(α)$ ($μ$ is the chemical potential) which correspond to the summations of $\sum_{m=1}^{\infty}m^{-α}$ and $\sum_{m=1}^{\infty}(-1)^{m-1}m^{-α}$, respectively. When $α\leq0$, both the series are divergent and it is usually believed that no QPTs exist. However, we find that there are two QPTs at $μ=μ_0(0)$ and $μ_π(0)$ for $α=0$ and one QPT at $μ=μ_π(α)$ for $α<0$. These QPTs are second order. The $μ_0(0)$ and $μ_π(α\leq0)$ correspond to the summations of the divergent series obtained by the analytic continuation of the Riemann $ζ$ function and Dirichlet $η$ 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 $α>0$ and induces topological phases with the winding number $ω:=\pm1/2$. 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 $α>0$. |
| title | Summation of Divergent Series and Quantum Phase Transitions in Kitaev Chains with Long-Range Hopping |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2312.09566 |