Realization of the topological Hopf term in two-dimensional lattice models

It is known that a two-dimensional spin system can acquire a topological Hopf term by coupling to massless Dirac fermions whose energy spectrum has a single cone. But it is challenging to realize the Hopf term in condensed matter physics due to the fermion-doubling in the low-energy spectrum. In this work we propose a scenario to realize the Hopf term in lattice models. The central aim is tuning the coupling between the spins and the Dirac fermions such that the topological terms contributed by the two cones do not cancel each other. To this end, we consider px and py orbitals for the Dirac fermions on the honeycomb lattice such that there are totally four bands. By utilizing the orbital degrees of freedom, a theta = 2 pi Hopf term is successfully generated for the spin system after integrating out the Dirac fermions. If the fermions have a small gap m0 or if the spin-orbit coupling is considered, then theta is no longer quantized, but it may flow to multiple of 2 pi under renormalization. The ground state and the physical response of a spin system having the Hopf term are discussed.

Automated summary

This work proposes a scenario to realize the Hopf term in lattice models by tuning the coupling between spins and Dirac fermions. By utilizing the orbital degrees of freedom, a theta = 2 pi Hopf term is successfully generated for the spin system on the honeycomb lattice.