Second quantization of anyons and spin-anyon duality

TL;DR

This paper develops an algebraic framework for Abelian anyons in one dimension, introduces a Jordan-Wigner duality to spin-1 models, and explores their physical properties and potential for device engineering.

Contribution

It presents a novel second-quantization formalism and a spin-anyon duality, enabling realization of anyons from spin Hamiltonians and advancing theoretical understanding.

Findings

Model exhibits anyon-density-dependent flux and critical points with level crossings.

Ground-state properties show discontinuities in currents, momenta, fidelities, and correlations.

Duality maps a tight-binding anyon model to an XY-like spin-1 model.

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 $\theta$ = $\pi$/N that enforces a finite on-site occupancy of N-1 anyons with the exchange phase $\theta$ between different sites. Moreover, we introduce an exact Jordan-Wigner duality between $\pi$/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.