Abstract
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.
- Received 25 September 2023
- Revised 23 February 2024
- Accepted 1 March 2024
DOI:https://doi.org/10.1103/PhysRevX.14.011058
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Most everyday materials obey Ohm’s law, which says the current through a material is proportional to the voltage across it. In topological materials, violations of this law can be related to the underlying physics of the material. Much experimental and theoretical work in this area has focused on studying current that does not vary in space across the material. But advances in nanofabrication have opened the door to exploring the effects of spatially varying currents and voltage profiles in quantum materials. Here, we develop a theoretical formalism for computing nonlinear conductivities in quantum materials.
By making use of conservation laws, we develop a framework that can be applied to a wide variety of effective models of quantum materials. We apply our results to compute linear and nonlinear optical properties of model topological moiré systems, multilayer materials in which a moiré interference pattern arises from adjacent layers being slightly out of alignment. Such materials are also known to host a variety of exotic quantum states.
Our work will hopefully open doors to the next generation of experiments in 2D topological materials. Combining our results with recent advances in both moiré materials and nanoscale gate patterning, we provide the theoretical footing for studying electromagnetic responses in these systems. Additionally, the connection between our work and recent advances in the study of quantum geometry in nonlinear responses is a fruitful area for future research, both theoretical and experimental.
