论文标题
基于广义位置运算符和相互作用的水稻模型的拓扑不变性
Topological invariants based on generalized position operators and application to the interacting Rice-Mele model
论文作者
论文摘要
我们讨论了基于位置运算符鉴定相变的几个拓扑不变的不同属性和能力，并使用更准确的方法进行比较，例如越过激发能量水平和浆果阶段的跳跃。不变性具有$ \ text {im} \ text {ln} \ left \ langle \ exp \ left [i（2π/l）σ_{j} x_ { }\hat{n}_{j\downarrow }\right) \right] \right\rangle $, where $L$ is the length of the system, $x_{j}$ the position of the site $j$, $\hat{n}_{jσ}$ the operator of the number of particles at site $j$ with spin $σ$.我们表明，$ m_ {σ} $应该是整数，在某些情况下，大于1的幅度，以导致定义明确的期望值。对于相互作用的水稻模型（其中包含相互作用的Su-Schrieffer-Heeger和离子Hubbard模型作为特定情况），我们表明，三种不同的不变物提供了互补信息，并且足以构建相位图，以在不变的区域保护不变的区域中构造相位图。我们还讨论了泵送电荷和自旋的后果，以及Ising自旋旋转相互作用或交错磁场的影响。
We discuss different properties and the ability of several topological invariants based on position operators to identify phase transitions, and compare with more accurate methods, like crossing of excited energy levels and jumps in Berry phases. The invariants have the form $\text{Im} \text{ln} \left\langle \exp \left[ i(2π/L)Σ_{j}x_{j} \left( m_{_{\uparrow }}\hat{n}_{j\uparrow } +m_{\downarrow }\hat{n}_{j\downarrow }\right) \right] \right\rangle $, where $L$ is the length of the system, $x_{j}$ the position of the site $j$, $\hat{n}_{jσ}$ the operator of the number of particles at site $j$ with spin $σ$. We show that $m_{σ}$ should be integers, and in some cases of magnitude larger than 1 to lead to well defined expectation values. For the interacting Rice-Mele model (which contains the interacting Su-Schrieffer-Heeger and the Ionic Hubbard model as specific cases), we show that three different invariants give complementary information and are necessary and sufficient to construct the phase diagrams in the regions where the invariants are protected by inversion symmetry. We also discuss the consequences for pumping of charge and spin, and the effect of an Ising spin-spin interaction or a staggered magnetic field.
