Researchers at the Tsientang Institute for advanced Study (TIAS) and the Max Planck Institute for the Structure and Dynamics of Matter (MPSD) together with international collaborators, have developed a general and experimentally feasible strategy to realize square-lattice moiré materials. By rotating two-dimensional semiconductors with rectangular unit cells by 90 degrees, the team demonstrates that square-symmetric moiré superlattices with flat and well-isolated electronic bands emerge naturally.

This geometric approach provides a new route to engineer moiré systems that realize a tunable square-lattice Hubbard model, a cornerstone theoretical framework for understanding magnetism and high-temperature superconductivity. The method applies to a broad class of materials and offers a clean, gate-controllable platform for exploring strongly correlated electronic phases.

Moiré physics has so far been dominated by triangular and honeycomb lattices, largely because naturally occurring square two-dimensional crystals are rare. The new study circumvents this limitation by exploiting the relative abundance of rectangular-lattice 2D materials. The researchers propose stacking materials such as GeX and SnX monochalcogenides and rotating the layers by 90 degrees. This rotation produces a well-defined lattice mismatch along both in-plane directions, giving rise to a long-wavelength moiré pattern with nearly ideal square symmetry.

Using large-scale first-principles simulations of twisted GeS, GeSe, SnS, and their multilayer variants, the team shows that the spatial modulation of interlayer registry generates narrow, isolated flat bands near the conduction-band edge. These electronic states are accurately captured by an effective square-lattice Hubbard model with tunable parameters, including nearest- and next-nearest-neighbor hopping amplitudes, providing a minimal and transparent description of the low-energy physics.

The extreme flatness of these bands strongly enhances electron–electron interactions. Based on interaction strengths derived from first-principles calculations, the researchers predict that a half-filled flat band realizes a Mott insulating state with Néel antiferromagnetic order. Notably, the magnetic moments are associated with moiré-scale orbitals rather than individual atomic sites. The platform further offers exceptional tunability: material choice and layer number control magnetic frustration, while applied electric displacement fields continuously tune band anisotropy. Together, these knobs enable systematic exploration of antiferromagnetism, stripe order, pseudogap phenomena, and potentially unconventional superconductivity.

MPSD Director and co-author Angel Rubio highlights the elegance of the approach: “The surprising thing is how simple the idea turned out to be. By twisting two rectangular layers by 90 degrees, a clean square lattice appears almost automatically. Once we saw how robust this mechanism was, we realized it could open a large new direction in moiré research.”

Co-author Lede Xian from TIAS emphasizes the broader implications for the field: “We hope this work encourages the community to look more widely at two dimensional materials. Rectangular lattices are more common, and with a simple 90-degree twist they can become a powerful playground for strongly interacting electrons.”

The research was led by Professor Lede Xian of the Tsientang Institute for Advanced Study, Zhejiang, and carried out in collaboration with researchers from Sichuan Normal University, Songshan Lake Materials Laboratory, the Max Planck Institute for the Structure and Dynamics of Matter, RWTH Aachen University, and the University of Pennsylvania. The results were published on December 15, 2025, in Physical Review X (https://doi.org/10.1103/wcbz-lbr1).

Schematic illustration of the quantum simulation of square lattice Hubbard model using 90˚ twisted 

bilayer rectangular two-dimensional materials