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arXiv:1005.4936 · Journal of Physics A Mathematical and Theoretical

Dualities for anyons

Brando Bellazzini

Published Dec 15, 2010• 3 citations
arXivHTMLPDF

Abstract

We show that the low-energy dynamics of anyons in (1+1) dimensions with the smallest number of derivatives and , and symmetric interactions are dual to the sine-Gordon model for bosonic fields. We discuss in particular the Tomonaga–Luttinger, Thirring and Schwinger models, as well as their deformation by relevant and marginal operators. In the presence of electromagnetic interactions, the mass of the meson from anyon confinement and the chiral anomaly get corrected by the statistical parameter.

DOI
10.1088/1751-8113/44/3/035403

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We show that the low-energy dynamics of anyons in (1+1) dimensions with the smallest number of derivatives and , and symmetric interactions are dual to the sine-Gordon model for bosonic fields. We discuss in particular the Tomonaga–Luttinger, Thirring and Schwinger models, as well as their deformation by relevant and marginal operators. In the presence of electromagnetic interactions, the mass of the meson from anyon confinement and the chiral anomaly get corrected by the statistical parameter.
Chapter 31 discussed the generic O(N) non-linear σ-model. We have noticed that the abelian case N = 2 is special because the RG β-function vanishes in two dimensions. The corresponding O(2) invariant spin model is specially interesting: it provides an example of the celebrated Kosterlitz–Thouless phase transition and will be examined in Chapter 33. However, a thorough discussion of this model requires a new technique: we have to establish relations, special to two dimensions, between fermion and boson local field theories, by a method called bosonization. This chapter illustrates the derivation involves several steps with the help of various other two-dimensional models which are physically interesting in their own right. It first studies the free massless boson and fermion fields. It evaluates the determinant of the covariant fermion derivative in the presence of an external gauge field. The result is at the basis of the bosonization technique. The chapter then discusses the sine–Gordon (SG) model. It solves the Schwinger model, QED with massless fermions. Finally, it demonstrates the equivalence between the SG model and two fermion models with current-current interaction: the Thirring model and another model with two species of fermions.
I consider general interacting systems of quantum particles in one spatial dimension. These consist of bosons or fermions, which can have any number of components, arbitrary spin, or a combination thereof, featuring low-energy two- and multiparticle interactions. The single-particle dispersion can be Galilean (nonrelativistic), relativistic, or have any other form that may be relevant for the continuum limit of lattice theories. Using an algebra of generalized functions, statistical transmutation operators that are genuinely unitary are obtained, putting bosons and fermions in a one-to-one correspondence without the need for a short-distance hard core. In the nonrelativistic case, low-energy interactions for bosons yield, order by order, fermionic dual interactions that correspond to the standard low-energy expansion for fermions. In this way, interacting fermions and bosons are fully equivalent to each other at low energies. While the Bose-Fermi mappings do not depend on microscopic details, the resulting statistical interactions heavily depend on the kinetic energy structure of the respective Hamiltonians. These statistical interactions are obtained explicitly for a variety of models, and regularized and renormalized in the momentum representation, which allows for theoretically and computationally feasible implementations of the dual theories. The mapping is rewritten as a gauge interaction, and one-dimensional anyons are also considered.
We discuss the connection between anyons (particles with fractional statistics) and deformed Lie algebras (quantum groups). After a brief review of the main properties of anyons, we present the details of the anyonic realization of all deformed classical Lie algebras in terms of anyonic oscillators. The deformation parameter of the quantum groups is directly related to the statistics parameter of the anyons. Such a realization is a direct generalization of the Schwinger construction in terms of fermions and is based on a sort of bosonization formula which yields the generators of the deformed algebra in terms of the undeformed ones. The entire procedure is well defined on two-dimensional lattices, but it can be consistently reduced also to one-dimensional chains.
We present an exact scheme of bosonization for anyons (including fermions) in the two-dimensional manifold of the quantum Hall fluid. This gives every fractional quantum Hall phase of the electrons one or more dual bosonic descriptions. For interacting electrons, the statistical transmutation from anyons to bosons allows us to explicitly derive the microscopic statistical interaction between the anyons, in the form of the effective two-body and few-body interactions. This also leads to a number of unexpected topological phases of the single component bosonic fractional quantum Hall effect that may be experimentally accessible. Numerical analysis of the energy spectrum and ground state entanglement properties are carried out for simple examples.
This book provides a detailed description of the duality between two integrable systems: the 1+1-dimensional Sine-Gordon model and the 1+1-dimensional Thirring model. While of great importance per se, this duality is only part of the target of the book. In order to reach an understanding of the subtleties involved in the duality, one has to take a journey through the properties of quantum integrable systems, building from the ground up the theory of exact S-matrices and familiarising oneself with the mathematical concept of a quantum group. The book therefore becomes an opportunity for a focussed study of integrability in its wider breadth of interest, always maintaining a clear ultimate purpose in mind: understanding the duality between bosons and fermions in 1+1 dimensions. This should make going through the book from the point of view of the reader/early-career researcher a live enterprise, as opposed to a more passive learning exercise. Key features • The book fills a gap in the existing literature, collating a vast amount scattered literature, with uniformised notation and conventions • The book’s format is ideal for use in PhD Schools and advanced graduate level mini courses, or for beginner PhD students as an introductory reference • Originates for an invited set of lectures • Contains a significant number of worked examples and problems, with solutions. It also presents several open-ended suggestions for literature reviews and guided learning through references
Quantum statistics in low-dimensional systems predicts anyonic particles with fractional exchange statistics which are neither that of bosons nor fermions. While anyons are typically found in two dimensions as excitations of topologically-ordered states of matter, anyon-like exchange statistics has also been discussed in one dimension, for instance, in the context of the anyon-Hubbard model (AHM), the physics of which has recently been observed in experiment [Kwan et al., arXiv:2306.01737; Dhar et al., arXiv:2412.21131; and Bakkali-Hassani et al., arXiv:2602.20421]. The AHM can be formulated in terms of bosons featuring density-dependent Peierls phases, described by a statistical phase angle θθθ, which controls asymmetric transport and the formation of dynamically bound pairs at finite momentum. Here, we show theoretically that the AHM also hosts exact two-body bound states in the continuum (BICs) for arbitrary θ≠0θ\neq 0θ=0, and genuine three- and four-body bound states. Unlike conventional bound states stabilized by attractive (or repulsive) interactions, which are energetically localized with a large effective mass, these clusters here are bound by a purely kinematic mechanism endowing them with fast chiral transport properties. We provide a simple variational approximation to the three-body bound states and explain their binding mechanism. Moreover, we show that the signatures of three-body bound states in the AHM can be directly probed experimentally from the expansion dynamics starting from three localized particles.