Towards Extremal Topological Phases In Periodically Driven Systems

Topological insulators are materials possessing insulating bulks and conductive boundaries, which are classified with the notion of topological order. They are capable to realise the quantum Hall effect with no symmetries broken. All these peculiar properties come from the topological insulators’ gapless edge states,whose numbers are determined by the topology of the bands. Because the topology is insensitive to smooth changes in material parameters, the physical properties(such as Hall conductance and the number of gapless boundary modes) are relatively stable. In addition to quantum Hall effects, topological insulators are also closely related to the fields such as Majorana fermions, Weyl semimetals and quantum computaion, which makes it a hot topic in recent years.The topology of topological insulators is characterised by corresponding topological invariant. Because the topological invariant of two dimensional systems is the Chern number, they are also called the Chern insulators. Although extremal topological phases such as large-Chern-number phases is proved to be indispensable in many fields, the experimentally realized Chern numbers are only ±1. Theory shows that the realisation of large-Chern-number phases requires long-range interactions or multi-layered structures in materials, which are difficult to realise.We engage in the exploration of extremal topological phases in the periodically driven systems where long-range interactions and multi-layered structures are both absent. Basing on the Floquet theory, we analytically establish the theory of the phase transitions induced by periodic driving in two-band systems.Our theory shows the criteria and the rules of phase transitions. In the theory, we reveal the mechanism of generating and engineering multiple Dirac cones in periodic driven systems which makes the large-Chern-number phases possible. Then we apply the periodic driving to the N3 Haldane model to study the topological phase transitions. After inducing the periodic driving, we obtain the large Chern numbers ±7 in the N3 Haldane model, which only has Chern numbers ±2 at most in static case. We also analysis the topological phase diagram and check our analytic theory numerically.Our work not only show that the diverse topological phases with a widely tunable Chern numbers can be induced by periodic driving, but also greatly relaxes the experimental difficulty in material fabrication, and opens an avenue to generate extremal topological states of matter.