Statistical aspects of the anyon model

Comtet, Alain; Georgelin, Y. P.; Ouvry, Stéphane

doi:10.1088/0305-4470/22/18/026

Cited by 86 publications

(112 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The structure of the Hamiltonian H N (α ij ) given in (8) allows for the following general comment. Suppose one has an eigenstate ψ(α ij ) of energy E(α ij ).…”

Section: Some Exact and Perturbative Results I) Exact Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Topological two-dimensional quantum mechanics

Veigy

Ouvry

1993

Physics Letters B

View full text Add to dashboard Cite

show abstract

“…The structure of the Hamiltonian H N (α ij ) given in (8) allows for the following general comment. Suppose one has an eigenstate ψ(α ij ) of energy E(α ij ).…”

Section: Some Exact and Perturbative Results I) Exact Resultsmentioning

confidence: 99%

“…Let us confine the particles by a harmonic attraction to the origin. This procedure [1,8] is commonly used in the anyon context since it yields a discrete spectrum. The N -particle Hamiltonian with a harmonic interaction reads…”

Section: Some Exact and Perturbative Results I) Exact Resultsmentioning

confidence: 99%

Topological two-dimensional quantum mechanics

Veigy

Ouvry

1993

Physics Letters B

View full text Add to dashboard Cite

show abstract

“…Therefore, the equations derived here also apply in the case of inverse square potentials in arbitrary spatial dimension and other SO(2, 1) invariant systems such as anyons [37][38][39][40][41][42].…”

Section: Comments and Conclusionmentioning

confidence: 99%

Path-integral Fujikawa’s approach to anomalous virial theorems and equations of state for systems with SO(2,1) symmetry

Ordóñez

2016

Physica A: Statistical Mechanics and its Applications

View full text Add to dashboard Cite

We derive anomalous equations of state for nonrelativistic 2D complex bosonic fields with contact interactions, using Fujikawa's path-integral approach to anomalies and scaling arguments. In the process, we derive an anomalous virial theorem for such systems. The methods used are easily generalizable for other 2D systems, including fermionic ones, and of different spatial dimensionality, all of which share a classical SO(2, 1) Schrödinger symmetry. The discussion is of a more formal nature and is intended mainly to shed light on the structure of anomalies in 2D many-body systems. The anomaly corrections to the virial theorem and equation of state -pressure relationship -may be identified as the Tan contact term. The practicality of these ideas rests upon being able to compute in detail the Fujikawa Jacobian that contains the anomaly. This and other conceptual issues, as well as some recent developments, are discussed at the end of the paper.

show abstract

“…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%

“…These studies were followed by the 3-body [14] and then the N -body problem [15]. Statistical mechanics was also considered (second virial coefficient [16,17], third virial coefficient [18]). However, it soon became apparent that a complete N -body spectrum was out of reach, to the exception of particular classes of exact eigenstates generalizing the 2-body eigenstates.…”

Section: Introductionmentioning

confidence: 99%

Anyons and Lowest Landau Level Anyons

Ouvry

2009

The Spin

View full text Add to dashboard Cite

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...

show abstract

Statistical aspects of the anyon model

Cited by 86 publications

References 30 publications

Topological two-dimensional quantum mechanics

Topological two-dimensional quantum mechanics

Path-integral Fujikawa’s approach to anomalous virial theorems and equations of state for systems with SO(2,1) symmetry

Anyons and Lowest Landau Level Anyons

Contact Info

Product

Resources

About