features of cuprate superconductors, such as the AF and $d$-wave pairing, as well as stripe formation.[@scalapino94; @Zhang97; @Millis98; @Maier00; @Capone07; @Wu15] The AF-singlet transition and strongly renormalized effective electronic mass in the PAM, and its strong coupling limit (the Kondo lattice model) helps to explain different ground states in heavy-fermion materials.[@stewart84; @Hirsch86; @Fye90; @Fye91; @Costi88; @stewart01] Manganites[@dagotto01] and iron-pnictide superconductors[@si16] are also materials for which appropriate multi-orbital models have been useful for developing understanding of magnetism, pairing, charge order, and transport.

Geometries which can be regarded as arising from regular ‘depletions’ of the square lattice HM have also been explored, both to answer fundamental questions about types of magnetic order and for the understanding of specific materials. An example of the former is Lieb’s theorem[@lieb89], which rigorously demonstrates that a 1/4-depleted square lattice possesses ground state ferrimagnetic behavior. Instances of the latter are the 1/5-depleted square lattice which can explain spin liquid behavior in CaV$_4$O$_9$ [@troyer96; @khatami14], and the 1/3-depleted square lattice which shed light into the properties of layered nickelates such as La$_{4}$Ni$_{3}$O$_{8}$[@Guo17; @Botana16; @Zhang16]. In these situations, the depletion converts the initial single band nature to a model with multiple bands. In the case of the Lieb lattice, 