Mohammed M. Al Ezzi

Two-dimensional multilayer materials with an induced moir\'e pattern, either due to strain or relative twist between layers, provide a versatile platform for exploring strongly correlated and topological electronic phenomena. While these systems offer unprecedented tunability, their theoretical description remains challenging due to their complex atomic structures and large unit cells. A key step in constructing effective theories is determining the relevant symmetry group of these systems. Here, we use a combination of analytical methods, molecular dynamics simulations, and first-principles calculations to show that twisted atomic configurations with distinct microscopic symmetries converge to a universal moir\'e-scale symmetry that governs the low-energy physics. We apply this framework to three widely studied families of moir\'e materials---twisted bilayer graphene, twisted hexagonal boron nitride, and twisted transition metal dichalcogenides---and introduce quantitative criteria for structural and electronic equivalence grounded in numerical accuracy and experimental resolution. This emergent universality provides a robust foundation for symmetry-respecting models and offers insight into the role of commensurability in real twisted moir\'e systems.

Electrons in solids owe their properties to the periodic potential landscapes they experience. The advent of moiré lattices has revolutionized our ability to engineer such landscapes on nanometre scales, leading to numerous ground-breaking discoveries. Despite this progress, direct imaging of these electrostatic potential landscapes remains elusive. Here we introduce the atomic single electron transistor (SET), a new scanning probe that uses a single atomic defect in a van der Waals material as an ultrasensitive, high-resolution potential sensor. Built on the quantum twisting microscope (QTM) platform¹, this probe leverages the capability of the QTM to form a pristine, scannable two-dimensional interface between vdW heterostructures. Using the atomic SET, we present the first direct images of the electrostatic potential in a canonical moiré interface: graphene aligned to hexagonal boron nitride^{2,3,4,5,6,7,8,9,10}. The measured potential exhibits an approximate C₆ symmetry, minimal dependence on carrier density and a substantial amplitude of approximately 60 mV, even in the absence of carriers. Theory indicates that this symmetry arises from a delicate interplay of physical mechanisms with competing symmetries. The measured amplitude significantly exceeds theoretical predictions, suggesting that current understanding may be incomplete. With 1 nm spatial resolution and sensitivity to detect the potential of even a few millionths of an electron charge, the atomic SET enables ultrasensitive imaging of charge order and thermodynamic properties across a wide range of quantum phenomena, including symmetry-broken phases, quantum crystals, vortex charges and fractionalized quasiparticles.

By virtue of being atomically thin, the electronic properties of heterostructures built from two-dimensional materials are strongly influenced by atomic relaxation. The atomic layers behave as flexible membranes rather than rigid crystals. Here we develop an analytical theory of lattice relaxation in twisted moiré materials. We obtain analytical results for the lattice displacements and corresponding pseudo gauge fields, as a function of twist angle. We benchmark our results for twisted bilayer graphene and twisted WSe_{2} bilayers using large-scale molecular dynamics simulations. Our single-parameter theory is valid in graphene bilayers for twist angles θ≳0.7°, and in twisted WSe_{2} for θ≳1.6°. We also investigate how relaxation alters the electronic structure in twisted bilayer graphene, providing a simple extension to the continuum model to account for lattice relaxation.

Moiré-pattern-based potential engineering has become an important way to explore exotic physics in a variety of two-dimensional condensed matter systems. While these potentials have induced correlated phenomena in almost all commonly studied 2D materials, monolayer graphene has remained an exception. We demonstrate theoretically that a single layer of graphene, when placed between two bulk boron nitride crystal substrates with the appropriate twist angles, can support a robust topological ultraflat band emerging as the second hole band. This is one of the simplest platforms to design and exploit topological flat bands.