Recent Developments in Fractional Chern Insulators

Abstract

Fractional Chern insulators (FCIs) are lattice generalizations of the conventional fractional quantum Hall effect (FQHE) in two-dimensional (2D) electron gases. They typically arise in a 2D lattice without time-reversal symmetry when a nearly flat Bloch band with nonzero Chern number is partially occupied by strongly interacting particles. Band topology and interactions endow FCIs exotic topological orders which are characterized by the precisely quantized Hall conductance, robust ground-state degeneracy on high-genus manifolds, and fractionalized quasiparticles. Since in principle FCIs can exist at zero magnetic field and be protected by a large energy gap, they provide a potentially experimentally more accessible avenue for observing and harnessing FQHE phenomena. Moreover, the interplay between FCIs and lattice-specific effects that do not exist in the conventional continuum FQHE poses new theoretical challenges. In this chapter, we provide a general introduction of the theoretical model and numerical simulation of FCIs, then pay special attention on the recent development of this field in moiré materials while also commenting on potential alternative implementations in cold atom systems. With a plethora of exciting theoretical and experimental progress, topological flat bands in moiré materials such as magic-angle twisted bilayer graphene on hexagonal boron nitride have indeed turned out to be a remarkably versatile platform for FCIs featuring an intriguing interplay between topology, geometry, and interactions.

Fractional Quantum Anomalous Hall Phase for Raman Superarray of Rydberg Atoms

“…Recently, the bosonic 1∕2-Laughlin state, [26] a hallmark of the FQHE, has DOI: 10.1002/qute.202300356 been demonstrated in photonic [27] and ultracold atomic systems in the few-body regime. [28] Another family of systems that exhibit FQHE-like behavior are fractional Chern insulators (FCIs), [29][30][31] which are lattice analogues of FQH phase with [32][33][34][35][36][37] or without LLs. [38][39][40][41][42][43][44][45][46] The latter are also called fractional quantum anomalous Hall (FQAH) insulator, and can be achieved by replacing fractionally filled LLs by fractionally filled topological flat bands that have zero net flux.…”

Topological phase transitions and flat bands on an islamic lattice

“…TFBs with non-zero Chern number play the similar role as the Landau levels, and they can effectively quench the kinetic energy and enhance interaction. When interacting particles fill into TFB models, several numerical and analytical studies have systematically revealed the existence of fractional CI (FCI) states [14,17,18,[26][27][28][29][30][31][32][33][34][35]. Therefore, finding TFB models becomes meaningful for the realization of FCIs.…”

Three-body problem in a multiband Hubbard model

“…Exploring and discovering exotic phases of interacting matter such as fractional Chern and topological Mott insulators remains a primary objective in this field [5,6]. However, due to the complexity of interacting manybody problems, a bottom-up approach examining the exactly solvable two-body problem in a multiband Hubbard model can sometimes be useful [7].…”