Fractional Topological Insulators: A Pedagogical Review

Abstract

Fractional topological insulators are electronic systems that carry fractionally charged excitations, conserve charge and are symmetric to reversal of time. In this review we introduce the basic essential concepts of the field, and then survey theoretical understanding of fractional topological insulators in two and three dimensions. In between, we discuss the case of "two and a half dimensions", the fractional topological insulators that may form on the two dimensional surface of an unfractionalized three dimensional topological insulator. We focus on electronic systems and emphasize properties of edges and surfaces, most notably the stability of gapless edge modes to perturbations. CONTENTS

Surfaces and slabs of fractional topological insulator heterostructures

“…Alternatively it can acquire a many-body interacting mass while preserving both symmetries, and exhibit long-ranged entangled surface topological order [5][6][7][8] . On the other hand, fractional topological insulators (FTI) [9][10][11][12][13][14][15] are long-range entangled topologically ordered electronic phases in (3 + 1) dimensions outside of the single-body mean-field band theory description. They carry TR and charge U (1) symmetries, which enrich its topological order (TO) in the sense that a symmetric surface must be anomalous and cannot be realized non-holographically by a true (2 + 1)-D system.…”

Magnon thermal Hall effect in kagome antiferromagnets with Dzyaloshinskii-Moriya interactions

“…[1][2][3] When both electron-electron interactions and SOC are present novel phases may be found, including spin liquids, 4,5 unconventional magnetic orders, [6][7][8] and fractional topological states. 9,10 One important class of such materials is Mott insulators at half-filling, which can be described using local moment (spin) models. When SOC is present it often gives rise to Dzyaloshinskii-Moriya interactions (DMI), 11,12 which provides one route towards topologically nontrivial magnetic excitations in both ordered, [13][14][15][16][17] and disordered 13,18,19 systems.…”

Flat Chern Band from Twisted Bilayer MnBi2Te4

“…The discovery of topological insulator (TI) [1][2][3][4][5][6][7][8][9][10][11], quantum anomalous Hall (QAH) effect [12][13][14][15][16][17][18] and other topological states have significantly enriched the variety of quantum matter, and may lead to potential applications in electronics and quantum computation [19][20][21][22][23]. Electron-electron interaction plays an essential role in fractional quantum Hall effect, and there have been proposals of strongly correlated topological states such as fractional TI and fractional Chern insulator (FCI) without magnetic field [24][25][26][27][28][29][30][31]. Experimentally realizing such states is, however, challenging because flat topological electronic bands are generally required for electron-electron interactions to manifest.Recently, it is shown that Moiré superlattices in twisted or lattice mismatched two-dimensional (2D) materials can give rise to flat topological bands.…”