Universal Rule for Topological Hopf Term via Dirac-Spin Coupling

Abstract

It is known that the topological Hopf term in two-dimensional (2D) spin systems can be derived by coupling to massless Dirac fermions. We establish a universal rule governing the generation of Hopf terms in 2D quantum spin systems coupled to Dirac fermions. The key insight identifies the Hopf coefficient as the oriented volume in the $\mathfrak{su}(2)$ Lie algebra space formed by Dirac cone matrix bases. This geometric interpretation allows direct determination of Hopf term without path integral computations. Applying this framework, we demonstrate nontrivial Hopf term in tailored checkerboard lattice models and recover known results in graphene-based systems.