Anyons and Lowest Landau Level Anyons

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 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...

Semi-classical limit of the Levy–Lieb functional in Density Functional Theory

“…Since U is C ∞ on D c α where P ε is supported, U √ P ε has the same regularity as √ P ε and satisfies an estimate similar to (1.2), with a worse dependence in N . In dimension d 3, the situation is more complex, due to some well known topological obstructions [13,16,17]. Indeed, for N 2 and d 3, there does not exist any anti-symmetric function U :…”

Many-anyon trial states

“…This is then called the 'magnetic gauge picture' for anyons. We refer to the extensive reviews [7][8][9][10][11][12][13][14][15][16] for a more complete background on the topic.…”

“…[37,38,10,39,16] for review), where the individual anyons are replaced by their average magnetic field, something which is arguably reasonable in a sufficiently dense regime. Other approximations assume either a very strong external magnetic field, thereby reducing to lowest-Landau-level anyons which turn out to be solvable with a connection to Calogero-Sutherland models [40,14], or in the case of the free dilute gas, that it is sufficient to only take two-particle interactions into account [32,41]. It has however been stressed that real progress in understanding the anyon gas cannot be made without knowledge of the true many-body spectrum.…”

Local density approximation for the almost-bosonic anyon gas

“…This is a generalization of the usual types of particles: bosons have α = 0 (symmetric wave functions) and their mean-field description is via models of the form (1.4), fermions have α = 1 (anti-symmetric wave functions) and appropriate models for them are Hartree-Fock functionals (see [Bac92,LS77,Lio87,Lio88,FLS15] and references therein). For general α one speaks of anyons [Kha05,Myr99,Ouv07,Wil90], which are believed to emerge as quasi-particle excitations of certain condensedmatter systems [ASW84,Hal83,Hal84,ZSGJ14,CS15,LR16].…”