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Researchers have developed a theoretical framework for the systematic analysis of nonlinear responses of materials, which would not only be useful for a deeper understanding of the mechanism but also future material design.

Materials show responses to external stimuli. For instance, metals generate electric current under an electric field, and magnets show magnetization under a magnetic field. One of the ultimate goals of materials science is to develop materials that exhibit desired responses. The linear response theory developed by Ryogo Kubo has been widely used to understand the mechanisms of such responses in each material in the linear regimes of external stimuli [1]. In recent years, however, a variety of nonlinear responses, such as the nonlinear Hall effect (NLHE) and nonlinear magnetoelectric effect, have garnered considerable attention. While the group theory and the Ginzburg–Landau theory are useful for qualitative analysis, these phenomenological approaches are not capable of predicting such nonlinear responses at the quantitative level. Meanwhile, ab initio calculations can be used to compute the actual values of responses based on the material parameters, but they are still insufficient for achieving a microscopic understanding of the relevant mechanism and the essential parameters. For obtaining deeper insights into the nonlinear phenomena and the exploration of more efficient materials, it is highly desirable to develop a theory that bridges the gap between the phenomenological approaches and ab initio calculations.

Recently, Oiwa and Kusunose reported a new theoretical framework for the analysis of linear and nonlinear responses of materials [2]. Their framework not only enables one to predict the possible responses but also extract essential parameters in a systematic manner. As it exploits the symmetry information and can be applied to a generic form of the tight-binding model, the framework can be a useful tool that complements the existing theoretical approaches.

Specifically, their framework is implemented as follows. First, a model Hamiltonian is prepared with a set of input (external field) and output (response). While the Hamiltonian is limited to a one-body form, it may include the effect of many-body interactions at the level of the mean-field approximation. One can also construct the model using the Wannier functions from ab initio calculations. Next, the response tensor, which describes the relationship between the input and output in the tensor form, is decomposed into model-independent and model-dependent parts. This procedure is conducted using the Keldysh Green functions and Chebyshev polynomial expansion [3]. Then, by expanding the model-dependent part in terms of the model parameters and analyzing the low-order contributions, the essential parameters for the response are extracted.

For demonstration, Oiwa and Kusunose applied the framework to a ferroelectric monolayer of SnTe, for which the ab initio calculation and model analysis predict the NLHE [4]. Based on the analysis of a simplified tight-binding Hamiltonian, it was concluded that the second-neighbor hopping is essential for the NLHE, whereas the atomic spin–orbit coupling is not. Furthermore, the NLHE in this material was interpreted to be a result of two consecutive processes: the orbital magneto-current effect (MCE) and linear anomalous Hall effect (AHE) triggered by the induced orbital magnetization (Fig. 1). This insight on the mechanism of the NLHE was obtained by taking full advantage of the proposed framework.

2.

Decomposition of (a) the NLHE characterized by the Berry curvature dipole (BCD) using a simplified model for a ferroelectric monolayer of SnTe into two components: (b) the orbital MCE and (c) the linear AHE. The essential model parameters in each figure are extracted from the low-order expansion of the model-dependent part of the nonlinear response tensor. The figure is taken from Fig. 8 of Ref.The framework developed in Ref. 2 would aid both theorists and experimentalists to obtain deeper insights into the mechanisms of linear and nonlinear responses. Table I in Ref. 2, which summarizes the relationship between the nonlinear tensors, symmetry of the system, and essential parameters, is particularly useful in the analysis. Moreover, it can provide guidelines for future material design, beyond those obtained by the existing phenomenological approaches and ab initio calculations. As the theoretical procedures are generic, the proposed framework could be a versatile tool for the systematic analysis of nonlinear phenomena.

## References

- [1] R. Kubo, J. Phys. Soc. Jpn.
**12**, 570 (1957). 10.1143/JPSJ.12.570 Link, Google Scholar - [2] R. Oiwa and H. Kusunose, J. Phys. Soc. Jpn.
**91**, 014701 (2022). 10.7566/JPSJ.91.014701 Link, Google Scholar - [3] S. M. João and J. M. V. P. Lopes, J. Phys.: Condens. Matter
**32**, 125901 (2020). 10.1088/1361-648X/ab59ec Crossref, Google Scholar - [4] J. Kim, K.-W. Kim, D. Shin, S.-H. Lee, J. Sinova, N. Park, and H. Jin, Nat. Commun.
**10**, 3965 (2019). 10.1038/s41467-019-11964-6 Crossref, Google Scholar

## Author Biographies

Yukitoshi Motome obtained his B.Sc. (1993), M.Sc. (1995), and D.Sc. (1998) degrees from the University of Tokyo. He subsequently spent two years as a JSPS research fellow at Tokyo Institute of Technology, two years as a research associate at Institute for Material Science, Tsukuba University, one year as a JST-ERATO researcher, and three years as a research scientist in RIKEN. He was appointed as an associate professor at Department of Applied Physics, the University of Tokyo in 2006, and has been a professor since 2015. He has worked on theoretical studies of condensed matter physics, in particular metal–insulator transitions, magnetism, spin–orbit coupling, dynamics, and transport properties in strongly correlated electron systems.