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

TL;DR

This paper proposes a lattice model approach to realize the topological Hopf term in two-dimensional spin systems by tuning spin-fermion couplings, overcoming fermion-doubling challenges in condensed matter physics.

Contribution

It introduces a novel lattice model with orbital degrees of freedom to generate a quantized Hopf term in spin systems, addressing fermion-doubling issues.

Findings

Successfully generates a $ heta=2\pi$ Hopf term in a honeycomb lattice model.

Shows the Hopf term's quantization can be affected by small fermion gaps or spin-orbit coupling.

Discusses the physical implications and response of the spin system with the Hopf term.

Abstract

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.