# Exact solutions of the strongly coupling Hubbard model and the proof of Hund’s rules

**Project level:** PhD, Masters, Honours, Summer

The Hubbard model is the simplest possible model of systems of interacting electrons and is believed to describe the basic physics of many materials from high-temperature superconductors, to organic spin-liquids, to cold atoms in optical traps. Yet it has only been solved exactly in three situations: one-dimension; infinite dimensions; or when the interactions are infinitely and some other conditions are met. The last result is known as Nagaoka’s theorem and proves that the magnetisation of the system takes its maximal possible value. We have recently shown that signed graph theory to relax the conditions required for Nagaoka’s theorem to hold. Hund’s rules tell us what how to fill up degenerate orbitals in atoms, molecules and solids. The rules are (1) maximise spin, S; (2) maximise orbital angular momentum, L; and (3) maximise total angular momentum, J=L+S. Hund discovered these rules experimentally and to date only heuristic derivations have been given. In this project you will investigate whether a modified version of the most modern proof of Nagaoka’s theorem can be used to rigorously derive Hund’s rules for models of molecular systems. If this is possible it would represent a shocking result as Hund’s rules and Nagaoka’s theorem are usually thought to represent entirely different physical effects. This will involve studying the models and results discussed above and extensions of these models that deal with relativistic quantum mechanics and spin-orbit coupling.