在普通SHH模型的格点加上位能(on-site potential) $\Delta$ $\begin{aligned} H=&H_{S S H}+\sum_{j}(-1)^{i} \Delta c_{i}^{\dagger} c_{i} \\=&H_{S S H}+\sum_{j} \Delta c_{j}^{\dagger} c_{j}-\Delta d_{j}^{\dagger} d_{j} \end{aligned}$ 傅里叶变换 $c_{j}=\frac{1}{\sqrt{N}} \sum_{k} e^{i j k} c_{k}, \quad d_{j}=\frac{1}{\sqrt{N}} \sum_{k} e^{i j k} d_{k}$ 新增加的项变为 $\begin{aligned} \Delta \sum_{j} c_{j}^{\dagger} c_{j}-d_{j}^{\dagger} d_{j} &=\frac{\Delta}{N} \sum_{j, k, k^{\prime}} e^{i j\left(k^{\prime}-k\right)}\left(c_{k}^{\dagger} c_{k^{\prime}}-d_{k}^{\dagger} d_{k^{\prime}}\right) \\ &=\Delta \sum_{k}\left(c_{k}^{\dagger} c_{k}-d_{k}^{\dagger} d_{k}\right), \\ &=\sum_{k}\left(c_{k}^{\dagger} d_{k}^{\dagger}\right)\left(\begin{array}{cc} \Delta & 0 \\ 0 & -\Delta \end{array}\right)\left(\begin{array}{l} c_{k} \\ d_{k} \end{array}\right) \\ &=\sum_{k}\left(c_{k}^{\dagger} d_{k}^{\dagger}\right) h^{\prime}(k) \cdot \sigma\left(\begin{array}{c} c_{k} \\ d_{k} \end{array}\right) \end{aligned}$ 其中 $h ^{\prime}(k)=(0,0, \Delta)$ 最后哈密顿量为 $h = h _{S S H}+ h ^{\prime}=(v+w \cos k, w \sin k, \Delta)$ 本征能量 $E_{\pm}=\pm| h |=\pm \sqrt{\Delta^{2}+v^{2}+w^{2}+2 v w \cos k}$