A Graph-Theoretic Approach to Free-Fermion Solvability - Lecture II

Abstract: The Jordan-Wigner transformation represents a profound connection between the physics of many-body spin systems and the physics of fermionic systems. In the setting where the effective fermions are noninteracting, the Jordan-Wigner transformation gives an exact solution method for an otherwise apparently complicated spin model. I will describe a graph-theoretic framework which captures mappings to free fermions under a unified characterization, yielding new exact solutions to spin models. Remarkably, the relationships between exact-solution methods in this framework reflect the relationships between families of graphs. This suggests a promising approach to understanding the physics of many-body spin models through graph theory.