Haldane state with toroidal magnetic order

Abstract

We show that a simplified two-band model describing toroidal magnetic order in two-dimensional crystal is entirely equivalent to the well-known Haldane model of a honeycomb lattice in periodic internal magnetic field with zero total flux through the unit cell. Crystals with toroidal ordering can be considered as real physical systems which should exhibit topologically nontrivial state, predicted by Haldane, that is similar to the quantum Hall effect regime without any external magnetic field. Analogous description is formulated for quantum spin Hall effect without any spin-orbit interaction.