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arXiv:cond-mat/0502071 · Physical Review B

Realizing non-Abelian statistics in time-reversal-invariant systems

Paul Fendley, Eduardo Fradkin

Published Jul 1, 2005• 86 citations
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Abstract

We construct a series of $(2+1)$-dimensional models whose quasiparticles obey non-Abelian statistics. The adiabatic transport of quasiparticles is described by using a correspondence between the braid matrix of the particles and the scattering matrix of $(1+1)$-dimensional field theories. We discuss in depth lattice and continuum models whose braiding is that of SO(3) Chern-Simons gauge theory, including the simplest type of non-Abelian statistics, involving just one type of quasiparticle. The ground-state wave function of an SO(3) model is related to a loop description of the classical two-dimensional Potts model. We discuss the transition from a topological phase to a conventionally ordered phase, showing in some cases there is a quantum critical point.

DOI
10.1103/physrevb.72.024412

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We construct a series of (2+1)(2+1)(2+1)-dimensional models whose quasiparticles obey non-Abelian statistics. The adiabatic transport of quasiparticles is described by using a correspondence between the braid matrix of the particles and the scattering matrix of (1+1)(1+1)(1+1)-dimensional field theories. We discuss in depth lattice and continuum models whose braiding is that of SO(3) Chern-Simons gauge theory, including the simplest type of non-Abelian statistics, involving just one type of quasiparticle. The ground-state wave function of an SO(3) model is related to a loop description of the classical two-dimensional Potts model. We discuss the transition from a topological phase to a conventionally ordered phase, showing in some cases there is a quantum critical point.
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Topological quantum field theory (TQFT) is a very powerful theoretical tool to study topological phases and phase transitions. In 2+1d2+1d2+1d, it is well known that the Chern-Simons theory captures all the universal topological data of topological phases, e.g., quasiparticle braiding statistics, chiral central charge, and even provides us a deep insight for the nature of topological phase transitions. Recently, topological phases of quantum matter are also intensively studied in 3+1d3+1d3+1d and it has been shown that looplike excitation obeys the so-called three-loop-braiding statistics. In this paper, we will try to establish a TQFT framework to understand the quantum statistics of particle and looplike excitation in 3+1d3+1d3+1d. We will focus on Abelian topological phases for simplicity, however, the general framework developed here is not limited to Abelian topological phases.
Many trial wave functions for fractional quantum Hall states in a single Landau level are given by functions called conformal blocks, taken from some conformal field theory. Also, wave functions for certain paired states of fermions in two dimensions, such as px+ipy{p}_{x}+i{p}_{y}px​+ipy​ states, reduce to such a form at long distances. Here we investigate the adiabatic transport of such many-particle trial wave functions using methods from two-dimensional field theory. One context for this is to calculate the statistics of widely separated quasiholes, which has been predicted to be non-Abelian in a variety of cases. The Berry phase or matrix (holonomy) resulting from adiabatic transport around a closed loop in parameter space is the same as the effect of analytic continuation around the same loop with the particle coordinates held fixed (monodromy), provided the trial functions are orthonormal and holomorphic in the parameters so that the Berry vector potential (or connection) vanishes. We show that this is the case (up to a simple area term) for paired states (including the Moore-Read quantum Hall state) and present general conditions for it to hold for other trial states (such as the Read-Rezayi series). We argue that trial states based on a nonunitary conformal field theory do not describe a gapped topological phase, at least in many cases. By considering adiabatic variation in the aspect ratio of the torus, we calculate the Hall viscosity, a nondissipative viscosity coefficient analogous to Hall conductivity, for paired states, Laughlin states, and more general quantum Hall states. Hall viscosity is an invariant within a topological phase and is generally proportional to the ``conformal spin density'' in the ground state.
We study a class of anomalies associated with time-reversal and spatial-reflection symmetry in (2+1)-dimensional bosonic topological phases of matter. In these systems, the topological quantum numbers of the quasiparticles, such as the fusion rules and braiding statistics, possess a Z2{\mathbb{Z}}_{2}Z2​ symmetry which can be associated with either time reversal (denoted Z2T){\mathbb{Z}}_{2}^{\mathbf{T}})Z2T​) or spatial reflections. Under this symmetry, correlation functions of all Wilson loop operators in the low-energy topological quantum field theory (TQFT) are invariant. However, the theories that we study possess a severe anomaly associated with the failure to consistently localize the symmetry action to the quasiparticles, precluding even defining a consistent notion of symmetry fractionalization in such systems. We present simple sufficient conditions which determine when Z2T{\mathbb{Z}}_{2}^{\mathbf{T}}Z2T​ symmetry localization anomalies exist in general. We present an infinite series of TQFTs with such anomalies, some examples of which include USp(4)2\mathrm{USp}{(4)}_{2}USp(4)2​ Chern-Simons (CS) theory and SO(4)4\mathrm{SO}{(4)}_{4}SO(4)4​ CS theory. The theories that we find with these Z2T{\mathbb{Z}}_{2}^{\mathbf{T}}Z2T​ anomalies can all be obtained by gauging the unitary Z2{\mathbb{Z}}_{2}Z2​ subgroup of a different TQFT with a Z4T{\mathbb{Z}}_{4}^{\mathbf{T}}Z4T​ symmetry. We further show that the anomaly can be resolved in several distinct ways: (1) the true symmetry of the theory is Z4T{\mathbb{Z}}_{4}^{\mathbf{T}}Z4T​, or (2) the theory can be considered to be a theory of fermions, with ${\mathbf{T}}^{2}={(\ensuremath{-}1)}^{{N}_{f}}$ corresponding to fermion parity. Finally, we demonstrate that theories with the Z2T{\mathbb{Z}}_{2}^{\mathbf{T}}Z2T​ localization anomaly can be compatible with Z2T{\mathbb{Z}}_{2}^{\mathbf{T}}Z2T​ if they are ``pseudorealized'' at the surface of a (3+1)D symmetry-enriched topological phase. The ``pseudorealization'' refers to the fact that the bulk (3+1)D system is described by a dynamical Z2{\mathbb{Z}}_{2}Z2​ gauge theory and thus only a subset of the quasiparticles are truly confined to the surface.
The fractional quantum Hall effect is the paradigmatic example of topologically ordered phases. One of its most fascinating aspects is the large variety of different topological orders that may be realized, in particular non-Abelian ones. Here we analyze a class of non-Abelian fractional quantum Hall model states which are generalizations of the Abelian Haldane-Halperin hierarchy. We derive their topological properties and show that the quasiparticles obey non-Abelian fusion rules of type su(q)k\mathrm{su}{\left(q\right)}_{k}su(q)k​. For a subset of these states we are able to derive the conformal field theory description that makes the topological properties---in particular braiding---of the state manifest. The model states we study provide explicit wave functions for a large variety of interesting topological orders, which may be relevant for certain fractional quantum Hall states observed in the first excited Landau level.
I propose that non-Abelian topological order can emerge from the organization of quantum particles into identical indistinguishable copies of the same quantum many-body state. Quantum indistinguishability (symmetrization) of the collectivities leads to topological degeneracy in the subspace of elementary excitations, giving rise to non-Abelian braiding statistics. The non-Abelian hidden order of a symmetrized structure is manifested in its entanglement properties, and the corresponding non-Abelian fusion and braiding rules can be derived by analyzing the set of symmetrized states on a surface with non-trivial topology like a torus. To illustrate the emergence of non-Abelian statistics from symmetrization, I consider the case of two identical copies of the toric code model. The resulting model is shown to be non-Abelian, exhibiting two types (charge and flux) of quasiparticles with non trivial fusion channels. The symmetrization construction I present here constitutes a framework for the generation of non-Abelian models from known Abelian ones.