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Anyons and Lowest Landau Level Anyons

  • Chapter
The Spin

Part of the book series: Progress in Mathematical Physics ((PMP,volume 55))

Abstract

Intermediate statistics interpolating from Bose statistics to Fermi statistics are allowed in two dimensions. This is due to the particular topology of the two dimensional configuration space of identical particles, leading to non trivial braiding of particles around each other. One arrives at quantum many-body states with a multivalued phase factor, which encodes the any- onic nature of particle windings. Bosons and fermions appear as two limiting cases. Gauging away the phase leads to the so-called anyon model, where the charge of each particle interacts “à la Aharonov-Bohm” with the fluxes carried by the other particles. The multivaluedness of the wave function has been traded off for topological interactions between ordinary particles. An alternative Lagrangian formulation uses a topological Chern-Simons term in 2+1 dimensions. Taking into account the short distance repulsion between particles leads to an Hamiltonian well defined in perturbation theory, where all perturbative divergences have disappeared. Together with numerical and semi-classical studies, perturbation theory is a basic analytical tool at disposal to study the model, since finding the exact N-body spectrum seems out of reach (except in the 2-body case which is solvable, or for particular classes of N-body eigenstates which generalize some 2-body eigenstates). However, a simplification arises when the anyons are coupled to an external homogeneous magnetic field. In the case of a strong field, by projecting the system on its lowest Landau level (LLL, thus the LLL-anyon model), the anyon model becomes solvable, i.e., the classes of exact eigenstates alluded to above provide for a complete interpolation from the LLL-Bose spectrum to the LLL-Fermi spectrum. Being a solvable model allows for an explicit knowledge of the equation of state and of the mean occupation number in the LLL, which do interpolate from the Bose to the Fermi cases. It also provides for a combinatorial interpretation of LLL-anyon braiding statistics in terms of occupation of single particle states. The LLL-anyon model might also be relevant experimentally: a gas of electrons in a strong magnetic field is known to exhibit a quantized Hall conductance, leading to the integer and fractional quantum Hall effects. Haldane/exclusion statistics, introduced to describe FQHE edge excitations, is a priori different from anyon statistics, since it is not defined by braiding considerations, but rather by counting arguments in the space of available states. However, it has been shown to lead to the same kind of thermodynamics as the LLL-anyon thermodynamics (or, in other words, the LLL-anyon model is a microscopic quantum mechanical realization of Hal- dane’s statistics). The one dimensional Calogero model is also shown to have the same kind of thermodynamics as the LLL-anyons thermodynamics. This is not a coincidence: the LLL-anyon model and the Calogero model are intimately related, the latter being a particular limit of the former. Finally, on the purely combinatorial side, the minimal difference partition problem — partition of integers with minimal difference constraints on their parts — can also be mapped on an abstract exclusion statistics model with a constant one-body density of states, which is neither the LLL-anyon model nor the Calogero model.

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  1. Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris-Sud, CNRS, UMR 8626, 91405, Orsay Cedex, France

    Stéphane Ouvry

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  1. Service de Physique Théorique Orme des Merisiers, CEA - Saclay, 91191, Gif-sur-Yvette Cedex, France

    Bertrand Duplantier

  2. Laboratoire Kastler-Brossel Département Physique, ENS Paris, Rue Lhomond 24, 75231, Paris CX 05, France

    Jean-Michel Raimond

  3. Laboratoire de Physique Théorique, Université Paris-Sud, Campus d’Orsay, 91405, Orsay Cedex, France

    Vincent Rivasseau

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Ouvry, S. (2009). Anyons and Lowest Landau Level Anyons. In: Duplantier, B., Raimond, JM., Rivasseau, V. (eds) The Spin. Progress in Mathematical Physics, vol 55. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8799-0_3

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