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arXiv:1510.07698

Three Lectures On Topological Phases Of Matter

Edward Witten

Published Oct 26, 2015• 171 citations
arXivHTMLPDF

Abstract

These notes are based on lectures at the PSSCMP/PiTP summer school that was held at Princeton University and the Institute for Advanced Study in July, 2015. They are devoted largely to topological phases of matter that can be understood in terms of free fermions and band theory. They also contain an introduction to the fractional quantum Hall effect from the point of view of effective field theory.

DOI
10.48550/arxiv.1510.07698

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These notes are based on lectures at the PSSCMP/PiTP summer school that was held at Princeton University and the Institute for Advanced Study in July, 2015. They are devoted largely to topological phases of matter that can be understood in terms of free fermions and band theory. They also contain an introduction to the fractional quantum Hall effect from the point of view of effective field theory.
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Abstract These lecture notes start by revisiting the definition of non-interacting topological phases of matter for fermions, placing some of the recurrent concepts that arise in other presentations at the Les Houches school on a time line beginning in 1931. They then focus exclusively on two-dimensional realizations of fractional topological insulators. The tenfold way for non-interacting fermions in the presence of local interactions and static local disorder is rederived. Attention is then turned to determining whether interactions between fermions can produce topological phases of matter with protected boundary states that are not captured by the tenfold way.
Some geometry and toplogy, G. Marmo and G. Morandi gauge symmetries, toplogy and quantisation, A.P. Balachandran the Chern-Simons-Landau-Ginzburg theory of the fractional quantum Hall effect, S.C. Zhang universitality in the fractional quantum hall, E. Fradkin and A. Lopez anyons and anyon superconductivity, A.L. Fetter bosonization - how to make it work for you in condensed matter, R. Shankar methods of conformal field theory in condensed matter physics, A.W.W. Ludwig integrable models in condensed matter physics, N. Andrei quantum antiferromagnets in two dimensions, S. Sachdev.
In the presence of symmetry-protection, topological invariants of topological phases of matter in free fermion systems, e.g., topological band insulators, can be directly computed via the properties of band structure. Nevertheless, it is usually difficult to extract topological invariants in strongly-correlated topological phases of matter in which band structure is not well-defined. One typical example is the fractional quantum Hall effect whose low-energy physics is governed by Chern-Simons topological gauge theory and Hall conductivity plateaus involve extremely fruitful physics of strong correlation. In this article, we focus on intrinsic topological order (iTO), symmetry-protected topological phases (SPT), and symmetry-enriched topological phases (SET) in boson and spin systems. Through gauge field-theoretical approach, we review some research progress on these topological phases of matter from the aspects of projective construction, low-energy effective theory and topological response theory.
Abstract These lecture notes are mainly concerned with how the concept of topological insulators is generalized when interactions are included. More generally, the interplay of symmetry and topology is discussed. Traditionally, phases of matter were distinguished on the basis of symmetry alone. On the other hand, fractional quantum Hall phases are examples of topological states whose essential character does not require a discussion of symmetry. However, topological insulators are an example of a new phase of matter that combines both symmetry and topology.