Engineering 2D Square Lattice Hubbard Models in 90° Twisted <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>GeX</mml:mi> <mml:mo>/</mml:mo> <mml:mi>SnX</mml:mi> </mml:math> ( <mml:math xmlns:mml="http://www.w3.org/1

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>GeX</a:mi> <a:mo>/</a:mo> <a:mi>SnX</a:mi> </a:math> ( <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi mathvariant="normal">X</c:mi> <c:mo>=</c:mo> <c:mi mathvariant="normal">S</c:mi> </c:math> , 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.