We present a complete analytical solution of the mean-field Hubbard model of undoped and doped graphene rectangulenes. These are non-chiral ribbons of arbitrary length and width, whose dimensions range from simple short acene molecules all the way up to the bulk limit. We rewrite the Hubbard model in the basis of bulk and edge non-interacting eigen-states, and provide explicit expressions for the Coulomb matrix elements. We present a general mean-field decoupling of the Hamiltonian, and discuss in detail the paramagnetic, ferromagnetic and antiferromagnetic mean-field solutions. We calculate the eigen-energies, occupations, spin densities and addition energies of rectangulenes with lengths and widths ranging from a nanometer to several hundreds of them. We rewrite the exact mean-field tight-binding Hamiltonian back in the site-occupation basis, that can be used to model electronic, thermo-electric, transport and optical properties of experimental-size graphene flakes.

We present a computational study to explore the potential of different experimental approaches to tune the magnetic interactions in two-dimensional van der Waals magnets. We selected CrGeSe$_3$, CrGeTe$_3$, and Janus Cr$_2$Ge$_2$(Se,Te)$_3$ monolayers as case studies and calculated the full exchange tensors among all relevant atomic pairs and analyze their dependence on different external parameters, such as biaxial and uniaxial strain, as well as gate voltage. We pay special attention to interactions that emerge or vanish due to changes of the symmetry of the system. We find that biaxial and uniaxial strains significantly modify isotropic exchange couplings, which can lead to a transition from a ferromagnetic to an antiferromagnetic phase, while a gate voltage induces Dzyaloshinskii-Moriya interactions, forming a vortex pattern whose chirality is determined by the sign of the electric field. The electric dipole moment of the Janus material is large, raising the possibility of multiferroic behaviour. The polarizability is similar for the three compounds.