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Engineering 2D Square Lattice Hubbard Models in 90° Twisted GeX/SnX (X=S, Se) Moiré Superlattices

Qiaoling Xu1,2,3,*, Ammon Fischer4,*, Nicolas Tancogne-Dejean4, Tao Zhang5,3, Emil Viñas Boström4, Martin Claassen6,†, Dante M. Kennes7,4,‡, Angel Rubio4,8,§, and Lede Xian2,3,4,∥

  • *These authors contributed equally to this work.
  • †Contact author: claassen@sas.upenn.edu
  • ‡Contact author: Dante.Kennes@rwth-aachen.de
  • §Contact author: Angel.Rubio@mpsd.mpg.de
  • ∥Contact author: xianlede@sslab.org.cn
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Phys. Rev. X 15, 041049 – Published 15 December, 2025

DOI: https://doi.org/10.1103/wcbz-lbr1

Abstract

Because of the large-period superlattices emerging in moiré two-dimensional (2D) materials, electronic states in such systems exhibit low energy flat bands that can be used to simulate strongly correlated physics in a highly tunable setup. While many investigations have thus far focused on moiré flat bands and emergent correlated electron physics in triangular, honeycomb, and quasi-one-dimensional lattices, tunable moiré realizations of square lattices subject to strong correlations remain elusive. Here we propose a feasible scheme to construct moiré square lattice systems by twisting two or more layers of 2D materials in a rectangular lattice by 90°. We demonstrate the concept with twisted GeX/SnX (X=S, Se) moiré superlattices and calculate their electronic structures from first principles. We show that the lowest conduction flat band in these systems can be described by a square lattice Hubbard model with parameters which can be controlled by varying the choice of host materials, number of layers, and external electric fields. In particular, twisted double bilayer GeSe realizes a square lattice Hubbard model with strong frustration due to the next-nearest-neighbor hopping that could host unconventional superconductivity, in close analogy to the Hubbard model for copper-oxygen planes of cuprate high-temperature superconductors. The presented scheme uses 90° twisted 2D materials with rectangular unit cells as a promising platform for realizing the physical phenomena of square lattice Hubbard models, establishing a new route for studying its rich phase diagram of magnetism, charge order, and unconventional superconductivity in a highly tunable setting.

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Corrections

28 January, 2026

Correction: A grant number in the Acknowledgments contained an error and has been fixed.

Popular Summary

One of the main goals of twistronics—the study of stacked 2D materials rotated slightly with respect to one another—has been to reproduce and better understand the unusual electronic behaviors found in copper oxide, or cuprate, superconductors. Early excitement centered on twisted bilayer graphene, but that system turned out to behave differently and more intricately than expected. We propose a clearer route toward realizing cupratelike physics: rotating two or more rectangular-lattice 2D layers by 90°. This simple rotation produces a square moiré pattern that traps electrons in flat, low-energy bands, allowing their interactions to dominate and give rise to rich collective behavior.

Through first-principles calculations, we show that twisted stacks of GeX or SnX (where X represents sulfur or selenium) naturally form these square lattices. In such structures, electrons can move between lattice sites but also strongly repel each other when they come too close. This balance between movement and repulsion—captured in theoretical models known as Hubbard models—provides a framework for studying how electronic correlations lead to magnetism and superconductivity. The relative strength of these effects can be tuned by changing the material, the number of layers, or by applying electric fields. In double-bilayer GeSe, we find that competing pathways for electron motion may destabilize simple magnetic order and encourage the kind of superconductivity seen in cuprates.

This 90° twist strategy offers a practical and versatile platform for investigating how interacting electrons organize themselves into complex phases. By fabricating and tuning these stacked materials, researchers can experimentally explore magnetism, charge order, strange metals, and superconducting states in a controlled environment.

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Phys. Rev. X 15, 041049– Published 15 December, 2025
  • Received 26 March 2024
  • Revised 17 December 2024
  • Accepted 31 October 2025
  • Corrected 28 January 2026
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