  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 $p_x$ and $p_y$ 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 $m_0$ 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.
