Quantum Hall physics: Hierarchies and conformal field theory techniques

Abstract

The fractional quantum Hall effect, being one of the most studied phenomena in condensed matter physics during the past thirty years, has generated many groundbreaking new ideas and concepts. Very early on it was realized that the zoo of emerging states of matter would need to be understood in a systematic manner. The first attempts to do this, by Haldane and Halperin, set an agenda for further work which has continued to this day. Since that time the idea of hierarchies of quasiparticles condensing to form new states has been a pillar of our understanding of fractional quantum Hall physics. In the thirty years that have passed since then, a number of new directions of thought have advanced our understanding of fractional quantum Hall states, and have extended it in new and unexpected ways. Among these directions is the extensive use of topological quantum field theories and conformal field theories, the application of the ideas of composite bosons and fermions, and the study of nonabelian quantum Hall liquids. This article aims to present a comprehensive overview of this field, including the most recent developments.arXiv:1601.01697v2 [cond-mat.str-el]

Floquet many-body engineering: topology and many-body physics in phase space lattices

“…Since the concept of topological order was first introduced into condensed matter physics in 1973 [1], topological phenomena have been intensively investigated in the past decades. Today, topology lies at the heart of many research fields, e.g., quantum Hall physics [2], topological insulators/superconductors [3,4], and many more. The origin of topology in physics comes from the geometric phase factor of a quantum state when it moves along an enclosed path.…”

Competing orders in the Hofstadter t–J model

“…With the increasing accuracy of experiments, e.g., in laser-manipulated cold atom systems in a two-dimensional square lattice [2][3][4][5][6][7], it becomes possible to investigate minute details of this non-interacting model. In addition, cold atom systems have proven to be able to emulate interacting fermionic or bosonic systems [4,[8][9][10], which may lead to the realization of exotic material phases such as a cold-atom analogue of the fractional quantum Hall (FQH) effect [11], as suggested by promising results from exact diagonalization (ED) of small clusters [12][13][14][15].…”

Three-dimensional topological insulators and bosonization

“…The topological phases of matter [1][2][3][4] have been described by several approaches, such as wavefunction modeling [5], band theory [6][7][8][9][10] and effective field theory of boundary excitations [11,12], whose interplay has been extremely rich and fruitful.…”

“…The bosonic approach can provide exact results for interacting systems and well as the methods for discussing bulk wavefunctions and braiding statistics [1,5].…”