Journal of Statistical Physics

, Volume 161, Issue 5, pp 1236–1267 | Cite as

The Average Field Approximation for Almost Bosonic Extended Anyons

  • Douglas Lundholm
  • Nicolas RougerieEmail author


Anyons are 2D or 1D quantum particles with intermediate statistics, interpolating between bosons and fermions. We study the ground state of a large number N of 2D anyons, in a scaling limit where the statistics parameter \(\alpha \) is proportional to \(N ^{-1}\) when \(N\rightarrow \infty \). This means that the statistics is seen as a “perturbation from the bosonic end”. We model this situation in the magnetic gauge picture by bosons interacting through long-range magnetic potentials. We assume that these effective statistical gauge potentials are generated by magnetic charges carried by each particle, smeared over discs of radius R (extended anyons). Our method allows to take \(R\rightarrow 0\) not too fast at the same time as \(N\rightarrow \infty \). In this limit we rigorously justify the so-called “average field approximation”: the particles behave like independent, identically distributed bosons interacting via a self-consistent magnetic field.


Anyons Fractional statistics Magnetic interaction Mean-field theory Quantum de Finetti theorem 



We thank Michele Correggi for discussions. Part of this work has been carried out during visits at the Institut Henri Poincaré (Paris) and the Institut Mittag-Leffler (Stockholm). D. L. would also like to thank LPMMC Grenoble for kind hospitality. We acknowledge financial support from the French ANR (Project Mathosaq ANR-13-JS01-0005-01), as well as the grant KAW 2010.0063 from the Knut and Alice Wallenberg Foundation and the Swedish Research Council Grant No. 2013-4734.


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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsKTH Royal Institute of TechnologyStockholmSweden
  2. 2.CNRS & Université Grenoble Alpes, LPMMC (UMR 5493)GrenobleFrance

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