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

It is known that a two-dimensional spin system can acquire topological Hopf term by coupling to massless Dirac fermions whose energy spectrum has single cone. But it challenging realize the in condensed matter physics due fermion-doubling low-energy spectrum. In this work we propose scenario lattice models. The central aim tuning between spins and such terms contributed two cones do not cancel each other. To end, consider $p_x$ $p_y$ orbitals for on honeycomb there are totally four bands. By utilizing orbital degrees of freedom, $\theta=2\pi$ successfully generated after integrating out fermions. If have small gap $m_0$ or if spin-orbit considered, then $\theta$ no longer quantized, but may flow multiple $2\pi$ under renormalization. ground state physical response having discussed.