Analytical theory of lattice relaxation in moiré materials

The discovery of correlated and topological phases in moiré materials, such as twisted bilayer graphene, has opened a new frontier in condensed matter physics, where electronic properties can be engineered by stacking atomically-thin layers with a small twist or lattice mismatch. However, there is a wrinkle in this story: atomic layers are flexible membranes, not rigid lattices. As a result, the moiré system relaxes elastically which can qualitatively change its electronic properties. Motivated by this, we developed an analytical theory of moiré reconstruction in the long-wavelength limit.

In this talk, I will first present a symmetry-constrained perturbative expansion of the atomic displacement field that offers a general framework for lattice relaxation in a wide range of moiré materials and matches microscopic simulations up to 1 degree for twisted bilayer graphene and 3 degrees for twisted transition-metal dichalcogenides [1]. As an example, we apply this formalism to twisted bilayer graphene and obtain a relaxed electronic moiré theory with twist-angle dependent moiré potentials [1, 2]. Finally, I will consider the limit of marginal twist angles, where relaxation is nonperturbative, and present a one-parameter analytical theory for the triangular soliton network that applies to any twisted homobilayer with D3h (D3d) layers near 0 (60) degrees [3]. Moreover, this parameter can be obtained from benchmarking to computationally inexpensive microscopic calculations in the perturbative limit. I will conclude by showing how our methodology can be extended to other moiré systems with lattice mismatch or square symmetry.

[1] Analytical Model for Atomic Relaxation in Twisted Moiré Materials, M. M. Al Ezzi, G. N. Pallewela, C. De Beule, E. J. Mele, and S Adam, Phys. Rev. Lett. 133, 266201 (2024).

[2] Elastic Screening of Pseudogauge Fields in Graphene, C. De Beule, R. Smeyers, W. N. Luna, E. J. Mele, and L. Covaci, Phys. Rev. Lett. 134, 046404 (2025).

[3] Theory for Lattice Relaxation in Marginally Twisted Bilayers, C. De Beule, G. N. Pallewela, M. M. Al Ezzi, E. J. Mele, and S. Adam, arXiv:2503.19162.