赝自旋-1量子自旋霍尔系统中的Holstein极化子：一阶和二阶拓扑相变
Holstein polaron in a pseudospin-$1$ quantum spin Hall system: first and second order topological phase transitions
AI总结：
本文在α-T3晶格赝自旋-1体系中研究电子-声子耦合驱动的量子自旋霍尔相和二阶拓扑相变，利用Lang-Firsov变换得到有效哈密顿量，揭示临界耦合处的一阶相变及磁场诱导的鲁棒角模式和高阶拓扑相变。
摘要：
我们从理论上提出，在α-T3晶格上的赝自旋-1费米子体系中，电子-声子（e-p）耦合可以驱动量子自旋霍尔（QSH）相和二阶拓扑相变（TPT）的发生。我们的模型按照Kane-Mele模型的思路构建，并由Holstein哈密顿量进行修正。采用Lang-Firsov方法在反绝热（高频）极限下合理地描述极化子，并得到有效的电子哈密顿量。研究表明，在临界e-p耦合λc以下，体系具有拓扑非平庸相，其特征是在一定α范围内存在螺旋QSH边缘态以及非零的Z2不变量。拓扑相在λc之后消失，并伴随λc处的体能隙闭合转变，表现为一个TPT。对于较大的α值，我们观察到更有趣的现象：体系表现出由λc1和λc2处两个不同能隙闭合转变所支持的TPT，而在略低值处的一个狭窄区域内，在λc1以下呈现出半金属特征。随后，为了探索更复杂的特征，我们引入破坏时间反演对称性的磁场以触发二阶拓扑相的形成。该磁场通过构造导致边缘态的边界依赖性能隙打开，从而在定制的开放边界条件下产生鲁棒的角模式。我们通过采用合适的不变量，即投影自旋陈数，来证明高阶相的形成。最后，我们表明e-p耦合对角模式（以及实空间能带结构）有显著影响，证实了当我们在给定α值下将λ调至超过临界值时会发生高阶TPT。
英文摘要：
We theoretically propose the occurrence of a quantum spin Hall (QSH) and a second order topological phase transition (TPT) driven by electron-phonon (e-p) coupling in a pseudospin-$1$ fermionic system on an $α$-$T_3$ lattice. Our model is formulated in the spirit of the Kane-Mele model modified by the Holstein Hamiltonian. The Lang-Firsov approach is employed to describe polarons reasonably well in the anti-adiabatic (high frequency) limit and to obtain an effective electronic Hamiltonian. It is shown that the system possesses topologically nontrivial phases up to a critical e-p coupling, $λ_c$ and are characterized by the helical QSH edge states along with a non-zero $\mathbb{Z}_2$ invariant for a certain range of $α$. The topological phase vanishes beyond $λ_c$ and is accompanied by a bulk gap closing transition at $λ_c$, manifesting a TPT. We observe a more intriguing phenomenon for higher values of $α$, where the system exhibits TPTs supported by two distinct gap closing transitions at $λ_{c_1}$ and $λ_{c_2}$, while a slim region at slightly lower values hosts a semi-metallic signature below $λ_{c_1}$. Subsequently, to explore more intricate features, we introduce a time reversal symmetry breaking magnetic field to trigger the formation of a second order topological phase. The magnetic field, by construction causes a boundary dependent gapping out of the edge states, consequently giving rise to robust corner modes in a tailored open boundary conditions. We justify the formation of the higher order phase by employing an appropriate invariant, namely the projected spin Chern number. Finally, we show that the e-p coupling significantly influences the corner modes (and also the real space energy bandstructure), corroborating a higher order TPT as we tune $λ$ beyond a critical value for a given value of $α$.
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