Nonlinear electromagnetic response functions have reemerged as a crucial tool for studying quantum materials, due to recently appreciated connections between optical response functions, quantum geometry, and band topology. Most attention has been paid to responses to spatially uniform electric fields, relevant to low-energy optical experiments in conventional solid state materials. However, magnetic and magnetoelectric phenomena are naturally connected by responses to spatially varying electric fields due to Maxwell’s equations. Furthermore, in the emerging field of moiré materials, characteristic lattice scales are much longer, allowing spatial variation of optical electric fields to potentially have a measurable effect in experiments. In order to address these issues, we develop a formalism for computing linear and nonlinear responses to spatially inhomogeneous electromagnetic fields. Starting with the continuity equation, we derive an expression for the second-quantized current operator that is manifestly conserved and model independent. Crucially, our formalism makes no assumptions on the form of the microscopic Hamiltonian and so is applicable to model Hamiltonians derived from tight-binding or ab initio calculations. We then develop a diagrammatic Kubo formalism for computing the wave vector dependence of linear and nonlinear conductivities, using Ward identities to fix the value of the diamagnetic current order by order in the vector potential. We apply our formula to compute the magnitude of the Kerr effect at oblique incidence for a model of a moiré-Chern insulator and demonstrate the experimental relevance of spatially inhomogeneous fields in these systems. We further show how our formalism allows us to compute the (orbital) magnetic multipole moments and magnetic susceptibilities in insulators. Turning to nonlinear response, we use our formalism to compute the second-order transverse response to spatially varying transverse electric fields in our moiré-Chern insulator model, with an eye toward the next generation of experiments in these systems.

中文翻译：

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超越均匀性的电荷守恒：周期性固体中的空间不均匀电磁响应

由于最近认识到光学响应函数、量子几何和能带拓扑之间的联系，非线性电磁响应函数已重新成为研究量子材料的重要工具。大多数注意力都集中在与传统固态材料的低能光学实验相关的空间均匀电场的响应上。然而，根据麦克斯韦方程，磁和磁电现象通过对空间变化电场的响应自然地联系在一起。此外，在新兴的莫尔材料领域，特征晶格尺度更长，使得光电场的空间变化可能在实验中产生可测量的影响。为了解决这些问题，我们开发了一种计算空间不均匀电磁场的线性和非线性响应的形式。从连续性方程开始，我们推导出第二量化电流算子的表达式，该表达式明显守恒且与模型无关。至关重要的是，我们的形式主义对微观哈密顿量的形式没有做出任何假设，因此适用于从紧束缚或从头计算导出的模型哈密顿量。然后，我们开发了一种图解 Kubo 形式，用于计算线性和非线性电导率的波矢量依赖性，使用 Ward 恒等式来固定矢量势中的抗磁电流的值。我们应用我们的公式来计算莫尔-陈绝缘体模型斜入射时克尔效应的大小，并证明了这些系统中空间不均匀场的实验相关性。我们进一步展示了我们的形式如何使我们能够计算绝缘体中的（轨道）磁多极矩和磁化率。转向非线性响应，我们使用形式主义来计算莫尔-陈绝缘体模型中空间变化横向电场的二阶横向响应，着眼于这些系统中的下一代实验。