We consider a many-body system of 2D anyons, free quantum particles with general statistics parameter \alpha ∈ ]0,2[ α ∈ ] 0 , 2 [ . In the magnetic gauge picture they are described as bosons attached to Aharonov-Bohm fluxes of intensity 2\pi \alpha 2 π α , generating long-range magnetic forces. A dimensional reduction to 1D is obtained by imposing a strongly anisotropic trapping potential. This freezes the motion in the direction of strong trapping, leading to 1D physics along the weak direction. The latter is governed to leading order by the Tonks-Girardeau model of impenetrable bosons, independently of \alpha α .
We consider a many-body system of 2D anyons, free quantum particles with general statistics parameter \alpha ∈ ]0,2[ α ∈ ] 0 , 2 [ . In the magnetic gauge picture they are described as bosons attached to Aharonov-Bohm fluxes of intensity 2\pi \alpha 2 π α , generating long-range magnetic forces. A dimensional reduction to 1D is obtained by imposing a strongly anisotropic trapping potential. This freezes the motion in the direction of strong trapping, leading to 1D physics along the weak direction. The latter is governed to leading order by the Tonks-Girardeau model of impenetrable bosons, independently of \alpha α .
This thesis addresses the dimensional reduction of two-dimensional (2D) Abelian anyons under strong anisotropic confinement, studied both at the microscopic many-body level and at the mean-field/almost-bosonic level.Anyons are quasi-particles with fractional statistics that interpolate between bosons and fermions, characterized by an exchange phase e^{i·π·α} for α in (0, 1). The theoretical concept of anyons dates back to the work of Leinaas and Myrheim in 1977, while convincing experimental evidence for their existence emerged only in 2020, through their realization in fractional quantum Hall systems. Investigating their behavior under confinement not only clarifies the connections between one- and two-dimensional anyons, but also offers new insights into the mathematical structure of anyonic systems.At the microscopic level, we consider N anyons in the magnetic gauge picture with a harmonic trapping potential that freezes the motion in the longitudinal (y) direction. We rigorously prove that, after subtracting the frozen longitudinal contribution, the associated 2D energy Γ-converges to that of the one-dimensional (1D) Tonks-Girardeau (TG) gas. This establishes a precise mathematical connection between 2D anyons and 1D impenetrable bosons, and, via the Bose-Fermi mapping, to a free Fermi gas. Importantly, this 1D limit model is independent of the exchange parameter α at leading order, indicating that the effect of the magnetic gauge field is gauged away up to o(1) corrections. Equivalently, the long-range Aharonov-Bohm interactions collapse into a hard-core exclusion constraint in this dimensional reduction limit.At the mean-field level, in the average-field regime (N → ∞ with α proportional to N^{-1}), the 2D anyon system is described by the Chern-Simons-Schrödinger (CSS) functional, a nonlinear magnetic Schrödinger model with a self-consistent gauge field. Performing the same anisotropic dimensional reduction limit, the resulting 1D effective model is described by a quintic nonlinear Schrödinger (NLS) equation, whose quintic nonlinearity stems from the magnetic self-consistent gauge in the CSS model. This dimensional reduction is established both at the level of ground-state energies and, under suitable well-posedness assumptions on the CSS dynamics, at the level of time evolution. Unlike the TG gas, the effective coupling constant g in the quintic NLS retains a dependence on the exchange parameter α through g = π·β² for β proportional to α·N.Taken together, these results yield two distinct effective 1D descriptions arising from different orders of limits. Crucially, the mean-field/almost-bosonic limit and the dimensional reduction (strong confinement) limit do not commute: the TG gas limit is α-independent, whereas the quintic NLS limit keeps an α-dependence through β proportional to α·N. This non-commutativity highlights the subtle interplay between scaling limits and confinement in anyonic systems. Future work might investigate whether a critical regime separates these two situations.
We report the observation of a one-dimensional (1D) Tonks-Girardeau (TG) gas of bosons moving freely in 1D. Although TG gas bosons are strongly interacting, they behave very much like noninteracting fermions. We enter the TG regime with cold rubidium-87 atoms by trapping them with a combination of two light traps. By changing the trap intensities, and hence the atomic interaction strength, the atoms can be made to act either like a Bose-Einstein condensate or like a TG gas. We measure the total 1D energy and the length of the gas. With no free parameters and over a wide range of coupling strengths, our data fit the exact solution for the ground state of a 1D Bose gas.
Réduction dimensionnelle : des anyons 2D à un modèle effectif 1D sous confinement anisotrope Cette thèse traite de la réduction dimensionnelle d'anyons abéliens bidimensionnels soumis à un confinement fortement anisotrope, étudiée à la fois au niveau microscopique (multi-corps) et au niveau champ moyen/presque-bosonique.Les anyons sont des quasi-particules à statistique fractionnaire qui interpolent entre les bosons et les fermions, caractérisées par une phase d'échange e^{i·π·α} pour α compris entre 0 et 1. Le concept théorique des anyons remonte aux travaux de Leinaas et Myrheim en 1977, tandis que des preuves expérimentales convaincantes de leur existence n'ont émergé qu'en 2020, à travers leur réalisation dans les systèmes de Hall quantique fractionnaire. L'étude de leur comportement sous confinement permet non seulement d'éclaircir les liens entre anyons unidimensionnels et bidimensionnels, mais aussi d'apporter de nouvelles perspectives sur la structure mathématique des systèmes anyoniques.Au niveau microscopique, nous considérons N anyons dans le formalisme de jauge magnétique, soumis à un potentiel harmonique qui fige le mouvement dans la direction longitudinale (y). Nous démontrons rigoureusement qu'après avoir soustrait la contribution longitudinale gelée, l'énergie bidimensionnelle associée Γ-converge vers celle du gaz de Tonks-Girardeau (TG) unidimensionnel. Cela établit un lien mathématique précis entre les anyons 2D et les bosons impénétrables 1D, et, via la correspondance Bose-Fermi, avec un gaz de fermions libres. Fait important, ce modèle limite 1D est indépendant du paramètre d'échange α à l'ordre dominant, indiquant que l'effet du champ de jauge magnétique est éliminé jusqu'à des corrections en o(1). Autrement dit, les interactions d'Aharonov-Bohm à longue portée se réduisent à une contrainte d'exclusion à cœur dur dans cette limite de réduction dimensionnelle.Au niveau macroscopique, dans le régime de champ moyen (N tend vers l'infini avec α proportionnel à N^{-1}), le système anyonique 2D est décrit par la fonctionnelle de Chern-Simons-Schrödinger (CSS), un modèle de Schrödinger magnétique non linéaire avec un champ de jauge auto-consistant. En effectuant la même limite de réduction dimensionnelle, le modèle effectif unidimensionnel obtenu est décrit par une équation de Schrödinger non linéaire quintique (NLS), dont la non-linéarité quintique provient du champ de jauge auto-consistant du modèle CSS. Cette réduction dimensionnelle est établie à la fois au niveau des énergies fondamentales et, sous des hypothèses appropriées de bien-positude sur la dynamique CSS, au niveau de l'évolution temporelle. Contrairement au gaz TG, la constante de couplage effective g dans la NLS quintique conserve une dépendance au paramètre d'échange α via g = π·β² pour β proportionnel à α·N.Pris ensemble, ces résultats donnent deux descriptions effectives unidimensionnelles distinctes, issues de différents ordres de limites. De manière cruciale, les limites champ moyen/presque-bosonique et de réduction dimensionnelle (confinement fort) ne commutent pas : la limite TG est indépendante de α, tandis que la limite NLS quintique garde une dépendance en α via β proportionnel à α·N. Cette non-commutativité met en évidence l'interaction subtile entre la limite d'échelle et le confinement dans les systèmes anyoniques. Des travaux futurs pourraient chercher à déterminer s'il existe un régime critique séparant ces deux situations.
We investigate the properties of natural orbitals and their occupation numbers of the ground state of two noninteracting anyons characterized by the fractional statistics parameter α and confined in a harmonic trap. We work in the boson magnetic gauge where the anyons are modeled as composite bosons with magnetic flux quanta attached to their positions. We derive an asymptotic form of the weakly occupied natural orbitals, and show that their corresponding (ordered descendingly) occupation numbers decay according to the power law n−(4+2α) , where n is the index of the natural orbital. We find remarkable numerical agreement of the theory with the natural orbitals and their occupation numbers computed from the spectral decomposition of the system's wave function. We explain that the same results apply to the fermion magnetic gauge. Published by the American Physical Society 2024
The low-energy properties of the one-dimensional anyon gas with a delta-function interaction are discussed in the context of its Bethe ansatz solution. It is found that the anyonic statistical parameter and the dynamical coupling constant induce Haldane exclusion statistics interpolating between bosons and fermions. Moreover, the anyonic parameter may trigger statistics beyond Fermi statistics for which the exclusion parameter alpha is greater than one. The Tonks-Girardeau and the weak coupling limits are discussed in detail. The results support the universal role of alpha in the dispersion relations.
We reformulate 1D boson-fermion duality in path-integral terms. The result is a 1D counterpart of the boson-fermion duality in the 2D Chern-Simons gauge theory. The theory is consistent and enables, using standard resummation techniques, to obtain the long-wave-length asymptotics of the collective mode in 1D boson systems at the Tonks-Girardeau regime. The collective mode has the dispersion of Bogoliubov phonons: ω(q)=qρˉU(q)/m, where ρˉ is the bosons density and U(q) is a Fourier component of the two-body potential.