微扰非线性量子输运理论II：希尔伯特空间截断、规范不变性和空间均匀时变电场中的二阶输运
Theory of perturbatively nonlinear quantum transport II: Hilbert space truncation, gauge invariance, and second order transport in a spatially uniform, time-varying electric field
摘要：
这篇文章是三部曲的第二篇，讨论了一般量子系统的微扰响应，可能具有非平凡的基态几何，超越线性顺序。在这里，我们建立了简洁的，一般的公式，二阶响应的空间均匀，随时间变化的电场的速度计是$\textit{manifestly}$免费的静态极限杂散。我们首先讨论弯曲空间中的一般量子演化，然后详细说明这种情况是希尔伯特空间截断的自然副产品，并指出与所得有限曲率相关的关键微妙之处。然后，我们提出了一个几何的角度来看，两个流行的规范经常用于量子输运理论，速度规范和长度规范，并讨论他们如何，考虑截断诱导曲率效应，自然导致相同的结果，尽管截断。我们强调微妙的文学形式上的差异。最后，我们提供了一个通用的计划，用于消除静态极限杂散的速度计$\textit{without}$频率扩展和目前简洁和全面的格林函数公式的响应到二阶。作为我们的理论的具体方面的应用，在选定的情况下，二阶电荷电流响应进行了分析，在第一和第三部分。
英文摘要：
This article is the second of a trilogy that addresses the perturbative response of general quantum systems, with possibly nontrivial ground state geometry, beyond linear order. Here, we establish concise, general formulae for second order response to a spatially uniform, time-varying electric field in the velocity gauge that are $\textit{manifestly}$ free of static limit spurious divergences. We first discuss general quantum evolution in a curved space, then detail how such a situation is a natural byproduct of Hilbert space truncation, and point out crucial subtleties associated with the resulting finite curvatures. We then present a geometric perspective of the two popular gauges often used in quantum transport theories, the velocity gauge and the length gauge, and discuss how they, taking truncation-induced curvature effects into account, naturally lead to the same results in spite of the truncation. We highlight subtle formal discrepancies in the literature. Finally, we provide a general scheme for removing static limit spurious divergences in the velocity gauge $\textit{without}$ frequency expansions and present concise and comprehensive Green's function formulae for responses up to second order. As an application of specific aspects of our theory, second order charge current responses in selected cases are analyzed in parts I and III.
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