for the electrons to exhibit another kind of behavior: This is known as charge transfer and results in charge-transfer insulators. Unlike Mott–Hubbard insulators electron transfer happens only within a unit cell. Both of these effects may be present and compete in complex ionic systems. Numerical treatment The fact that the Hubbard model has not been solved analytically in arbitrary dimensions has led to intense research into numerical methods for these strongly correlated electron systems. One major goal of this research is to determine the low-temperature phase diagram of this model, particularly in two-dimensions. Approximate numerical treatment of the Hubbard model on finite systems is possible via various methods. One such method, the Lanczos algorithm, can produce static and dynamic properties of the system. Ground state calculations using this method require the storage of three vectors of the size of the number of states. The number of states scales exponentially with the size of the system, which limits the number of sites in the lattice to about 20 on 21st century hardware. With projector and finite-temperature auxiliary-field Monte Carlo, two statistical methods exist that can obtain certain properties of the system. For low temperatures, convergence problems appear that lead to an exponential computational effort with decreasing temperature due to the so-called fermion sign problem. The Hubbard model can be studied within dynamical mean-field theory (DMFT). This scheme maps the Hubbard Hamiltonian onto a single-site impurity model, a mapping that is formally exact only in infinite dimensions and in finite dimensions corresponds to the exact treatment of all purely local correlations only. DMFT allows one to compute the local Green's function of the Hubbard model for a given and a given temperature. Within DMFT, the evolution of the spectral function can be computed and the appearance of the upper and lower Hubbard bands can be observed as correlations increase. Simulator Stacks of heterogeneous 2-dimensional transition metal dichalcogenides (TMD) have been used to simulate geometries in more than one dimension. Tungsten diselenide and tungsten sulfide were stacked. This created a moiré superlattice consisting of hexagonal supercells (repetition units defined by the relationship of the two materials). Each supercell then behaves as though it were a single atom. The distance between supercells is roughly 100x that of the atoms within them. This larger distance drastically reduces electron tunneling across supercells. They can be used to form Wigner crystals. Electrodes can be attached to regulate