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Communications in Theoretical Physics

Geometric Quantization of Anyon Systems

Yue Yu, Zhu Zhongyuan

Published Apr 1, 1991• 1 citation
Publisher

Abstract

Geometric quantization of multiparticle systems in 2 dimensions is studied. The interpretation of the global geometry to the phase factors describing the quantum statistics of the systems is presented.
DOI
10.1088/0253-6102/15/3/339

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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.
The natural role played by coherent states in the geometric quantization program is brought out by studying the mathematical equivalence between two physical interpretations that have recently been proposed for this program. These interpretations are based, respectively, on the modular algebra structure of prequantization, and the reproducing kernel structure of phase space quantization. The arguments are presented in this paper for the particular case where the phase space of the system considered is the cotangent bundle T*M of a homogeneous manifold M, and for didactic reasons, the latter is taken to be a real vector space.