Strong electronic interactions in quantum materials are responsible for phenomena such as high-Tc superconductivity or the Mott insulating phase. The Fermi-Hubbard model provides a low-energy description of electrons and is believed to have the ingredients to explain such exotic behaviors. The model consists of a lattice of sites where electrons are situated, allowed hopping between nearest- neighbor sites, and an interaction term representing the Coulomb repulsion between electrons as well as electron-ion interaction, which can be long-range. However, solutions of the model in its generic form are not accessible to current theory or numerics. It has recently been shown that arrays of dopant quantum dots in Silicon effectively simulate the Hubbard model. With the ability to place atoms with atomic precision using scanning tunneling microscopes, it is possible to make arbitrary lattices and tune hopping parameters and interaction parameters for electrons.

We analyze a two-by-two array of quantum dots with four gates to vary the chemical potential landscape, and source and drain leads to allow electrons to tunnel onto or off the array. The primary focus is to establish a connection between device parameters and those of the Hubbard model by comparing a classical simulation of the quantum dot device to experimental data. However, we also explore the effects of disorder in the system; this is important because fabrication can be imprecise. We use the open-source Python package QmeQ to construct the Hamiltonian of the system and by finding its ground states, map out the electronic configurations as the gate voltages are varied. QmeQ gives us a testbed to tune parameters such that the experiment and simulation agree.