Perturbative three-body spectrum and the third virial coefficient in the anyon model

McCabe, J.F.; Ouvry, Stéphane

doi:10.1016/0370-2693(91)90977-x

Cited by 74 publications

(91 citation statements)

References 21 publications

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“…Here again the experience gained in the study of the anyon model is helpful. A naïve perturbative analysis might make no sense due to the very singular 5 Aharonov-Bohm interaction α 2 ij /r 2 ij . One can circumvent this difficulty by noticing that the singular gauge parameter Ω ′′ = i<j α ij θ ij is the imaginary part of the meromorphic function…”

Section: Some Exact and Perturbative Results I) Exact Resultsmentioning

confidence: 99%

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Topological two-dimensional quantum mechanics

Veigy

Ouvry

1993

Physics Letters B

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Section: Some Exact and Perturbative Results I) Exact Resultsmentioning

confidence: 99%

“…In the Nanyon case with harmonic attraction to the origin, linear eigenstates have been constructed [4], but they are known to be only part of the spectrum. Perturbative approaches [5] have undercovered a very complex structure for the equation of state. Finally, several numerical analysis have been developped [6].…”

Section: Introductionmentioning

confidence: 99%

Topological two-dimensional quantum mechanics

Veigy

Ouvry

1993

Physics Letters B

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“…It is understood that all the results below are obtained for α in this interval, but they can be periodically continued to the whole real axis. Before introducing an external magnetic field, let us come back to the anyon Hamiltonian (17) and take advantage of wavefunctions vanishing at least as r −α ij when r ij → 0 (exclusion of the diagonal of the configuration space in the quantum mechanical formulation) by encoding this short distance behavior in the N -body bosonic wave function [10] …”

Section: The Lll-anyon Modelmentioning

confidence: 99%

“…Coming back to the Hamiltonian formulation (8), one might ask how the exclusion of the diagonal of the configuration space materializes in the Hamiltonian formulation. One way to look at it is perturbation theory [10,11]. Let us simplify the problem by considering the standard A-B problem, i.e.…”

Section: Introductionmentioning

confidence: 99%

Anyons and Lowest Landau Level Anyons

Ouvry

2009

The Spin

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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 anyonic 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 AharonovBohm" 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 1 arXiv:0712.2174v1 [cond-mat.stat-mech] 13 Dec 2007 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 Haldane'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...

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“…However, perturbation theory meets certain difficulties near Bose statistics, as originally noticed in [2]. In order to overcome these difficulties, it was pointed out in [3] [4], that certain modifications of the singular N -anyon Hamiltonian are required.…”

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Magnetic moment and perturbation theory with singular magnetic fields

1995

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: The spectrum of a charged particle coupled to Aharonov-Bohm/anyon gauge fields displays a nonanalytic behavior in the coupling constant. Within perturbation theory, this gives rise to certain singularities which can be handled by adding a repulsive contact term to the Hamiltonian. We discuss the case of smeared flux tubes with an arbitrary profile and show that the contact term can be interpreted as the coupling of a magnetic moment spinlike degree of freedom to the magnetic field inside the flux tube.We also clarify the ansatz for the redefinition of the wave function.

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Perturbative three-body spectrum and the third virial coefficient in the anyon model

Cited by 74 publications

References 21 publications

Topological two-dimensional quantum mechanics

Topological two-dimensional quantum mechanics

Anyons and Lowest Landau Level Anyons

Magnetic moment and perturbation theory with singular magnetic fields

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