On superconformal anyons

Published for SISSA by Springer Received: November 25, 2015 Accepted: January 8, 2016 Published: January 22, 2016 Nima Doroud, David Tong and Carl Turner Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 OWA, U.K. E-mail: , , Abstract: In d = 2 + 1 dimensions, there exist field theories which are non-relativistic and superconformal. These theories describe two species of anyons, whose spins differ by 1/2, interacting in a harmonic trap. We compute the dimensions of chiral primary operators. These operators receive large anomalous dimensions which are related to the unusual angular momentum properties of anyons. Surprisingly, we find that the dimensions of some chiral primary operators violate the unitarity bound and we trace this to the fact that the associated wavefunctions become non-normalisable. We also study BPS nonperturbative states in this theory: these are Jackiw-Pi vortices. We show that these emerge at exactly the point where perturbative operators hit the unitarity bound. To describe the low-energy dynamics of these vortices, we construct a novel type of supersymmetric gauged linear sigma model. Keywords: Chern-Simons Theories, Conformal and W Symmetry, Anyons ArXiv ePrint: 1511.01491 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP01(2016)138 JHEP01(2016)138 On superconformal anyons Contents 1 2 Superconformal field theory 2.1 Symmetries 2.2 Operators, states and chiral primaries 3 5 8 3 The spectrum 3.1 Repulsive interactions 3.2 Attractive interactions: the view from quantum mechanics 13 13 15 4 Jackiw-Pi vortices 4.1 Vortices in the plane 4.2 Vortices in a trap 4.3 Dynamics of vortices 4.4 Open questions 20 20 22 24 27 A Perturbative analysis A.1 Conformal fixed points A.2 Anomalous dimensions 28 30 34 1 Introduction and summary It’s hard to be free when you’re an anyon. Statistical obligations mean that you constantly have to be aware of your standing relative to your neighbours. This has consequences. Even the simplest quantum mechanics problem involving multiple anyons is challenging. For example, the spectrum of anyons in a harmonic trap remains unsolved. The purpose of this paper is to study this simple system, and its supersymmetric extension, from the perspective of non-relativistic conformal field theory. It was realised long ago that the dynamics of anyons has a hidden SO(2, 1) conformal symmetry. This is true whether the problem is expressed in the “first-quantised” formalism of quantum mechanics [1], or in the “second-quantised” formalism of field theory [2]. In the latter, field theoretic framework, anyons are described in terms of non-relativistic matter fields interacting with a Chern-Simons gauge field. More recently, Nishida and Son revisited the quantum dynamics of non-relativistic field theories which exhibit conformal invariance [3]. They showed, among other things, that there is a non-relativistic version of the state-operator map, with the spectrum of the dilatation operator mapped to the spectrum of the Hamiltonian in the presence of a harmonic trap. In this manner, solving the quantum mechanics problem of anyons in a trap is equivalent to finding the scaling dimensions of operators in a non-relativistic conformal field theory. –1– JHEP01(2016)138 1 Introduction and summary where J is the angular momentum of the operator, while R is an R-symmetry, counting (roughly) the difference between the population of the two species of anyons. The spectrum of chiral primary operators has been discussed in both [5] and [6]. However, there are a number of subtleties that arise in this spectrum that were previously overlooked. The first subtlety is associated to the angular momentum of multiple anyons. Suppose that a single anyonic particle has spin j. If n identical anyons are placed in a trap, their combined spin has the unusual property of scaling quadratically with the number of particles [7–9]: J = n2 j. This well known result follows directly from the spin-statistics theorem: exchanging two groups of n anyons requires each anyon from the first group to be exchanged with each from the second, giving the n2 behaviour to the statistical phase. The spin-statistics theorem, which is valid as our theory is the low-energy limit of a relativistic theory, then says that this scaling should also be manifest in the spin. Here, our interest lies in the implications of this result for (1.1) since it means that the dimension of n-particle operators should also scale as n2 . Indeed, we will show that the spectrum of n-particle chiral primary operators is given by ∆ = n − n(n − 1)j (1.2) The first term is the classical dimension of the operator. We will see, following [3], that the second term arises as an anomalous dimension for chiral primary operators. This anomalous dimension is one-loop exact. The exact result (1.2) gives rise to a new puzzle because for j > 0 the dimension of the operator would appear to become arbitrarily negative for a sufficiently large number of particles. This is in conflict with unitarity which requires ∆ ≥ 1. We resolve this puzzle. We will see that the j > 0 spectrum corresponds to anyons interacting through an attractive delta-function potential. This potential means that the quantum mechanical wavefunction diverges as two anyons approach. For a small number of anyons this is not an issue. 1 This is a property of non-relativistic theories which is not shared by the more familiar relativistic supersymmetric theories. The bosonic sector of a relativistic theory knows about the presence of fermions through loop effects. But in a non-relativistic theory there are no anti-particles and, correspondingly, no virtual loops involving anti-particles. –2– JHEP01(2016)138 In this paper we will exploit the fact that one may introduce superconformal symmetry into the problem. Indeed, a non-relativistic superconformal field theory of anyons was constructed in [4]. It consists of two species of anyons whose spin differs by 1/2. If we focus on the sector of the Hilbert space where only a single species of anyon is populated, this reduces to the problem above.1 Embedding the problem in a larger theory with superconformal symmetry will turn out to be useful in exhibiting some of the properties of the anyonic spectrum. The representation theory of the non-relativistic superconformal group was discussed in [5, 6]. There are both long multiplets and short multiplets. The short multiplets are built on (anti)-chiral primary operators whose dimension ∆ is dictated by the algebra   3 ∆=± J− R (1.1) 2 2 Superconformal field theory Starting from Chern-Simons theories coupled to gapped, relativistic matter, one may take a non-relativistic limit in which anti-particles decouple but particles remain. Surprisingly, supersymmetry not only survives this limit but is enhanced to a non-relativistic superconformal (or super-Schrödinger) symmetry. Th (...truncated)