Topological Flat Band Models and Fractional Chern Insulators

Abstract

Topological insulators and their intriguing edge states can be understood in a singleparticle picture and can as such be exhaustively classified. Interactions significantly complicate this picture and can lead to entirely new insulating phases, with an altogether much richer and less explored phenomenology. Most saliently, lattice generalizations of fractional quantum Hall states, dubbed fractional Chern insulators, have recently been predicted to be stabilized by interactions within nearly dispersionless bands with non-zero Chern number, C. Contrary to their continuum analogues, these states do not require an external magnetic field and may potentially persist even at room temperature, which make these systems very attractive for possible applications such as topological quantum computation. This review recapitulates the basics of tight-binding models hosting nearly flat bands with non-trivial topology, C = 0, and summarizes the present understanding of interactions and strongly correlated phases within these bands. Emphasis is made on microscopic models, highlighting the analogy with continuum Landau level physics, as well as qualitatively new, lattice specific, aspects including Berry curvature fluctuations, competing instabilities as well as novel collective states of matter emerging in bands with |C| > 1. Possible experimental realizations, including oxide interfaces and cold atom implementations as well as generalizations to flat bands characterized by other topological invariants are also discussed.

Exotic electronic states in the world of flat bands: From theory to material

“…A few nice reviews have already been written on different aspects of the FB, such as ferromagnetism [24] and FQH effects [25,26]. We intend not to repeat these previous efforts.…”

“…We limit our discussions to the most well-studied 2D FBs, and purposely avoid invoking abstract mathematical derivations in this article. With the intuitive pictures provided here, we encourage the readers to work on more specialized literature for the rigorous proof [24][25][26]. Also, we primarily focus on realizing electronic FBs in solid-state materials using the charge degree of freedom.…”

Interaction-Driven Spontaneous Quantum Hall Effect on a Kagome Lattice

“…Moreover, the lowest energy band on a kagome lattice is perfectly flat [38,[48][49][50]; thus, one could imagine a nearly flat band with a nonzero Chern number after the gap opening through interactions. This would be quite significant since it may provide a platform to realize the spontaneously fractional QAH phase or fractional Chern insulators [54][55][56][57][58][59][60][61] when such a flat band is partially filled. On the experimental side, based on the recent experimental development of artificial kagome systems [62], we anticipate activities for realizing and detecting the QAH state in ultracold atomic systems [63].…”

Quantum Anomalous Hall Insulator of Composite Fermions

“…Experimentally, it is demonstrated recently that a Cr-doped (Bi,Sb) 2 Te 3 thin film could be the first QAHI ever realized [8]. As a next logical development of theory, the possibility of finding a QAHI exhibiting the fractional quantum Hall effect in the presence of strong electronelectron interaction is extensively discussed in the context of the flat Chern band insulator [9][10][11][12][13], which is yet to be materialized.In this Letter, we propose a new class of QAHI that exhibits the fractional quantum Hall effect, i.e., quantum anomalous Hall insulator of composite fermions (CF-QAHI). The composite fermion (CF), which is defined as an electron binding with two or even number of quantized vortices, is not only the key theoretical apparatus for describing the extremely complex fractional quantum Hall states, but also a physical entity with well defined charge, spin, and statistics [14][15][16].…”