We derive the semiclassical Bloch dynamics with second-order Berry phase correction in the presence of the slow-varying scalar potential as perturbation. Our mathematical derivation is based on a two-scale WKB asymptotic analysis. For a uniform external electric field, the bi-characteristics systemafter a positional shift introduced by Berry connections agrees with the recent result in previous works. Moreover, for the case with a linear external electric field, we show that the extra terms arising in the bi-characteristics system after the positional shift are also gauge independent.

中文翻译：

---

具有二阶Berry相位校正的Bloch动力学

我们在存在缓慢变化的标量势作为扰动的情况下，通过二阶Berry相位校正推导了半经典Bloch动力学。我们的数学推导基于两尺度WKB渐近分析。对于均匀的外部电场，由Berry连接引入的位置偏移后的双特征系统与先前工作中的最新结果是一致的。此外，对于具有线性外部电场的情况，我们证明了在位置偏移后在双特征系统中出现的额外项也与规范无关。