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The electronic properties of van der Waals heterostructures can be dramatically modified by the formation of moiré patterns arising from lattice mismatch or rotational misalignment. In recent years, moiré materials such as twisted bilayer graphene, graphene on hexagonal boron nitride, and twisted transition-metal dichalcogenides have emerged as a versatile platform for engineering electronic band structures and correlated quantum phenomena.

In this talk, I will provide an introduction to the theoretical description of electronic states in moiré systems.
For small twist angles, the moiré period is much longer than the atomic lattice constant, enabling an elegant long-wavelength continuum theory [1,2] that has successfully explained a wide range of phenomena in twisted bilayer graphene, graphene on hexagonal boron nitride, and twisted transition-metal dichalcogenides. In the first half of the talk, I will introduce the basic theoretical framework of continuum moiré theory [3] and review its applications to these representative materials.

The conventional continuum approach, however, breaks down at large twist angles, where the moiré wavelength becomes comparable to the atomic scale. In the second half, I will discuss how the theory can be extended to this regime, focusing on 30° twisted bilayer graphene as a prototype quasiperiodic system [4]. I will present recent theoretical results on its electronic structure and Landau levels in magnetic fields [5], highlighting the distinctive physics that emerges beyond the conventional moiré paradigm.

[1] R. Bistritzer and A. H. MacDonald, *Proc. Natl. Acad. Sci. USA* 108, 12233 (2011).
[2] P Moon, M Koshino, Phys. Rev. B 87, 205404 (2013).
[3] M. Koshino, New J. Phys.17, 015014 (2015).
[4] P. Moon, M. Koshino, and Y.-W. Son, Phys. Rev. B 99, 165430 (2019).
[5] M. Hitomi, T. Kawakami, and M. Koshino, arXiv:2604.18854 (2026).

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