Unification of valley and anomalous Hall effects in a strained lattice

Two dimensional lattices are an important stage for studying many aspects of quantum physics, in particular the topological phases. The valley Hall and anomalous Hall effects are two representative topological phenomena. Here we show that they can be unified in a strained honeycomb lattice, where the hopping strengths between neighboring sites are designed by mimicking those between the Fock states in a three-mode Jaynes-Cummings model. Such a strain induces an effective magnetic field which results in quantized Landau levels. The eigenstates in the zeroth Landau level can be represented by the eigenstates of a large pseudo-spin. We find that the valley Hall current and the chiral edge current in the Haldane model correspond to the spin precession around different axes. Our study sheds light on connection between seemingly unrelated topological phases in condensed matter physics.