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Fractional Abelian topological phases of matter for fermions in two-dimensional space

Christopher Mudry

Published Jan 26, 2017• 0 citations
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Abstract

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.

DOI
10.1093/acprof:oso/9780198785781.003.0007

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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.
These lecture notes explain the classification of some simple fermionic topological phases of matter in a pedestrian manner, with an aim to be maximally pedagogical = doing things in excruciating detail. We focus on a many-body perspective, even if many of the models we work with are non-interacting. We start out with symmetry protected topological (SPT) phases of free fermions that are protected by U(1) symmetry = topological insulators. We then look at fermion topological phases that don’t even need a symmetry = topological superconductors, and explain how their classification changes in presence of spinless time-reversal symmetry. We close by perturbatively checking which of the 1D topological phases we had found are stable to interactions.
A scheme is proposed to construct integer and fractional topological quantum states of fermions in two spatial dimensions. We devise models for such states by coupling wires of non-chiral Luttinger liquids of electrons, that are arranged in a periodic array. Which inter-wire couplings are allowed is dictated by symmetry and the compatibility criterion that they can simultaneously acquire a finite expectation value, opening a spectral gap between the ground state(s) and all excited states in the bulk. First, with these criteria at hand, we reproduce the tenfold classification table of integer topological insulators, where their stability against interactions becomes immediately transparent in the Luttinger liquid description. Second, we construct an example of a strongly interacting fermionic topological phase of matter with short-range entanglement that lies outside of the tenfold classification. Third, we expand the table to long-range entangled topological phases with intrinsic topological order and fractional excitations.
Starting from an array of interacting fermionic quantum wires, we construct a family of non-Abelian topologically ordered states of matter in three spatial dimensions (3D). These states of matter inherit their non-Abelian topological properties from the su(2)k su(2)^{\,}_{k}su(2)k​ conformal field theories that characterize the constituent interacting quantum wires in the decoupled limit. Thus, the resulting topological phases can be viewed as 3D generalizations of the (bosonic) su(2)k su(2)^{\,}_{k}su(2)k​ Read-Rezayi sequence of fractional quantum Hall states. Focusing in detail on the su(2)2 su(2)^{\,}_{2}su(2)2​ case, we first review how to determine the nature of the non-Abelian topological order (in particular, the topological degeneracy on the torus) in the two-dimensional (2D) case, before generalizing this approach to the 3D case. We also investigate the 2D boundary of the 3D phases, and show for the su(2)2 su(2)^{\,}_{2}su(2)2​ case that there are anomalous gapless surface states protected by an analog of time-reversal symmetry, similar to the massless Dirac surface states of the noninteracting 3D topological insulator.
Abstract Topological condensed matter physics is a recent arrival among the disciplines of modern physics of a distinctive and substantive nature. Its roots reach far back, but much of its current importance derives from exciting developments in the last half-century. The field is advancing rapidly, growing explosively, and diversifying greatly. There is now a zoo of topological phenomena–the quantum spin Hall effect, topological insulators, Coulomb spin liquids, non-Abelian anyonic statistics and their potential application in topological quantum computing, to name but a few–as well as an increasingly sophisticated set of concepts and methods underpinning their understanding. The aim of this Les Houches Summer School was to present an overview of this field, along with a sense of its origins and its place on the map of advances in fundamental physics. The school comprised a set of basic lectures (Part I) aimed at a pedagogical introduction to the fundamental concepts, which was accompanied by more advanced lectures (Part II) covering individual topics at the forefront of today’s research in condensed matter physics.
A scheme is proposed to construct integer and fractional topological quantum states of fermions in two spatial dimensions. We devise models for such states by coupling wires of non-chiral Luttinger liquids of electrons, that are arranged in a periodic array. Which inter-wire couplings are allowed is dictated by symmetry and the compatibility criterion that they can simultaneously acquire a finite expectation value, opening a spectral gap between the ground state(s) and all excited states in the bulk. First, with these criteria at hand, we reproduce the tenfold classification table of integer topological insulators, where their stability against interactions becomes immediately transparent in the Luttinger liquid description. Second, we construct an example of a strongly interacting fermionic topological phase of matter with short-range entanglement that lies outside of the tenfold classification. Third, we expand the table to long-range entangled topological phases with intrinsic topological order and fractional excitations.