Edge Excitations In Fractional Chern Insulators

One of the most essential and fascinating topics in condensed matter physics is to explore and classify the various states of matter, among which the integer quantum Hall effect (IQHE) and the fractional quantum Hall effect (FQHE) have long been the major focus. Despite the transparent fact that all macroscopic systems are made of atoms, even the same matter transforms both its structure and property violently, when such environment, be it temperature or strain, changes. According to Landau’s sym-metry breaking theory, various states of matter correspond to various ways of atoms’ arrangement. The symmetry of arrangement always changes within phase transition, and thus every state can be characterized by some order parameter. When IQHE was first discovered experimentally in1980, which encounters plateau in Hall conductance whenever filling factor of electrons becomes integer, people started to realize that sym-metry breaking is not enough to characterize the course of phase transition. In fact, different plateaus correspond to different kind of topological orders, among which no symmetry are broken. Even more, the discovery of FQHE in1982opens up the gate to investigate fractional topological phases. The experimentally realized QHEs have been so far limited in the presense of Landau levels, which occurs only under high magnetic field and at extremely low temperature. To understand the deep mechanism in QHE, Haldane proposed a toy model in crystal lattice with staggered flux in1988, in which he realized IQHE without Landau levels. Such newly found quantum anomoulous Hall effect in lattice system, also dubbed as Chern insulator nowadays, does not impose the existence of uniform magnetic field, and may be observed at normal temperature in the future. When interacting particles are loaded into this energy band, however, no convincing signs of lattice version of FQHE show up.Recent theoretical works have demonstrated the realization of fractional quantum anomalous Hall states (also called fractional Chern insulators) in topological flat band lattice models without an external magnetic field. Such newly proposed lattice systems play a vital role to obtain a large class of fractional topological phases. Here we report the exact numerical studies of edge excitations for such systems in a disk geometry loaded with hard-core bosons, which will serve as a more viable experimental probe for such topologically ordered states. We find convincing numerical evidence of a series of edge excitations characterized by the chiral Luttinger liquid theory for the bosonic fractional Chern insulators in both the honeycomb disk Haldane model and the kagome-lattice disk model. We further verify these current-carrying chiral edge states by inserting a central flux to test their compressibility. With the aid of generalized Pauli principle, we can understand quite well these intriguing phenomena from numerical investigation. This work directly demonstrates edge excitations in fractional Chern insulator, which mimic the edge physics in Laughlin-like states. We hope this work could be a starting point to explore more intriguing phenomena in fractional Chern insulators, including high Chern number band and lattice specific effects. Part of this work has been published on Journal Physical Review B:Rapid Communications.