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arXiv:2605.04538

Second quantization of anyons and spin-anyon duality

Priyanshi Bhasin, Diptiman Sen, Tanmoy Das

Published May 6, 2026• 0 citations
arXivHTMLPDF

Abstract

Anyons exhibit a non-trivial interplay between local exclusion rules and non-local braiding and exchange phases, making a consistent commutation algebra and second-quantized formulation challenging. We develop an algebraic framework for Abelian anyons in one dimension with statistical phase $θ$ = $π$/N that enforces a finite on-site occupancy of N-1 anyons with the exchange phase $θ$ between different sites. Moreover, we introduce an exact Jordan-Wigner duality between $π$/3 anyons and spin-1 operators, allowing us to map a tight-binding anyon model to an XY-like spin-1 model. The model exhibits anyon-density-dependent flux, incompressible or gapless regions, and critical points with level crossings that appear as discontinuities in ground-state currents, momenta, fidelities, and correlation functions. Our second-quantization formalism establishes a novel spin anyon duality, offering a conceptually new route to realize anyons from spin Hamiltonians and to engineer corresponding device architectures.

DOI
10.48550/arxiv.2605.04538

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Anyons exhibit a non-trivial interplay between local exclusion rules and non-local braiding and exchange phases, making a consistent commutation algebra and second-quantized formulation challenging. We develop an algebraic framework for Abelian anyons in one dimension with statistical phase θθθ = πππ/N that enforces a finite on-site occupancy of N-1 anyons with the exchange phase θθθ between different sites. Moreover, we introduce an exact Jordan-Wigner duality between πππ/3 anyons and spin-1 operators, allowing us to map a tight-binding anyon model to an XY-like spin-1 model. The model exhibits anyon-density-dependent flux, incompressible or gapless regions, and critical points with level crossings that appear as discontinuities in ground-state currents, momenta, fidelities, and correlation functions. Our second-quantization formalism establishes a novel spin anyon duality, offering a conceptually new route to realize anyons from spin Hamiltonians and to engineer corresponding device architectures.
Abstract We review and develop the many-body spectral theory of ideal anyons, i.e. identical quantum particles in the plane whose exchange rules are governed by unitary representations of the braid group on N strands. Allowing for arbitrary rank (dependent on N ) and non-abelian representations, and letting N→∞N \rightarrow \infty N→∞ N → ∞ , this defines the ideal non-abelian many-anyon gas. We compute exchange operators and phases for a common and wide class of representations defined by fusion algebras, including the Fibonacci and Ising anyon models. Furthermore, we extend methods of statistical repulsion (Poincaré and Hardy inequalities) and a local exclusion principle (also implying a Lieb–Thirring inequality) developed for abelian anyons to arbitrary geometric anyon models, i.e. arbitrary sequences of unitary representations of the braid group, for which two-anyon exchange is nontrivial.
In one spatial dimension, anyons in the original description of Leinaas and Myrheim are formally equivalent to locally interacting bosons described by the Lieb-Liniger model. This allows an interesting reinterpretation of interacting bosons in the context of anyons. We elaborate on this parallel, particularly including the many-body bound states from the attractive Lieb-Liniger model. In the anyonic context these bound states are created solely by quantum-statistical attraction and coined the quantum-statistical condensate, which is shown to be more robust than the Bose-Einstein condensate. We introduce the second quantization formalism for the present anyons and construct the generalized Jordan-Wigner transformation that connects them to the bosons of the Lieb-Liniger model.
Low-dimensional quantum systems can host anyons, particles with exchange statistics that are neither bosonic nor fermionic. However, the physics of anyons in one dimension remains largely unexplored. In this work, we realize Abelian anyons in one dimension with arbitrary exchange statistics using ultracold atoms in an optical lattice, where we engineer the statistical phase through a density-dependent Peierls phase. We explore the dynamical behavior of two anyons undergoing quantum walks and observe the anyonic Hanbury Brown-Twiss effect as well as the formation of bound states without on-site interactions. Once interactions are introduced, we observe spatially asymmetric transport in contrast to the symmetric dynamics of bosons and fermions. Our work forms the foundation for exploring the many-body behavior of one-dimensional anyons.
Anyons obeying fractional exchange statistics arise naturally in two dimensions: Hard-core two-body constraints make the configuration space of particles not simply-connected. The braid group describes how topologically-inequivalent exchange paths can be associated to non-trivial geometric phases for abelian anyons. Braid-anyon exchange statistics can also be found in one dimension (1D), but this requires broken Galilean invariance to distinguish different ways for two anyons to exchange. However, recently it was shown that an alternative form of exchange statistics can occur in 1D because hard-core three-body constraints also make the configuration space not simply-connected. Instead of the braid group, the topology of exchange paths and their associated non-trivial geometric phases are described by the traid group. In this article we propose a first concrete model realizing this alternative form of anyonic exchange statistics. Starting from a bosonic lattice model that implements the desired geometric phases with number-dependent Peierls phases, we then define anyonic operators so that the kinetic energy term in the Hamiltonian becomes local and quadratic with respect to them. The ground-state of this traid-anyon-Hubbard model exhibits several indications of exchange statistics intermediate between bosons and fermions, as well as signs of emergent approximate Haldane exclusion statistics. The continuum limit results in a Galilean invariant Hamiltonian with eigenstates that correspond to previously constructed continuum wave functions for traid anyons. This provides not only an a-posteriori justification of our lattice model, but also shows that our construction serves as an intuitive approach to traid anyons, i.e. anyons intrinsic to 1D.
When leaving classical physics and entering the realm of quantum physics, there are many new concepts being introduced. One of the most fundamental ideas in quantum mechanics is that particles no longer have exact known positions, but instead expected values and prob- abilities. This leads to the phenomena of truly identical particles, since they no longer can be distinguished simply by their positions. An important property differentiating different kinds of particles is how a system behaves when two such identical particles are exchanged. Historically, this divided particles into bosons and fermions, corresponding to symmetry and antisymmetry under an exchange. However, in two dimensions a new type of particle appears. These particles are called anyons, and behave differently when particles are exchanged. Anyons can be further divided into abelian and non-abelian anyons, of which this thesis will focus on the latter. The ex- changes can then be represented by the fundamental group of the configuration space of the particles, and in two dimensions this fundamental group is the braid group. Using rotors from a Clifford algebra and studying excitations of Majorana fermions, this thesis will show a way to calculate the exchange matrices of non-abelian anyons, and their corresponding eigenvalues. Furthermore, suggestions on a generalization of this framework along with areas where it can be applied are given.
Abstract Anyon models are algebraic structures that model universal topological properties in topological phases of matter and can be regarded as mathematical characterization of topological order in two spacial dimensions. It is conjectured that every anyon model, or mathematically unitary modular tensor category, can be realized as the representation category of some chiral conformal field theory, or mathematically vertex operator algebra/local conformal net. This conjecture is known to be true for abelian anyon models providing support for the conjecture. We reexamine abelian anyon models from several different angles. First anyon models are algebraic data for both topological quantum field theories and chiral conformal field theories. While it is known that each abelian anyon model can be realized by a quantum abelian Chern–Simons (CS) theory and chiral conformal field theory, the construction is not algorithmic. Our goal is to provide such an explicit algorithm for a K -matrix in CS theory and a positive definite even one for a lattice conformal field theory. Secondly anyon models and chiral conformal field theories underlie the bulk-edge correspondence for topological phases of matter. But there are interesting subtleties in this correspondence when stability of the edge theory and topological symmetry are taken into consideration. Therefore, our focus is on the algorithmic reconstruction of extremal chiral conformal field theories with small central charges. Finally we conjecture that a much stronger reconstruction holds for abelian anyon models: every abelian anyon model can be realized as the representation category of some non-lattice extremal vertex operator algebra generalizing the moonshine realization of the trivial anyon model.