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.
While elementary particles obey either bosonic or fermionic exchange statistics, generalized exchange statistics that interpolate between bosons and fermions---applicable to quasiparticles---constitute an intriguing topic from both the fundamental and practical points of view. This work develops a scattering framework for two identical one-dimensional (1D) bosonic anyons and two identical 1D fermionic anyons with zero-range contact interactions. The two-body system with zero-range interactions, in both free space and under external confinement, is used to illustrate the recently proposed bosonic-anyon---fermionic-anyon mapping (R. Hidalgo-Sacoto et al., Phys. Rev. A 112, 063309 (2025)), which connects the eigenstates of bosonic anyons to those of fermionic anyons and vice versa. Performing explicit calculations for two-particle systems, the momentum distributions and the off-diagonal correlations of the single-particle density matrix for bosonic anyons and fermionic anyons are confirmed to be distinct. We also confirm the previously derived asymptotic coefficients of the momentum distribution tail at orders ${k}^{\ensuremath{-}2}$ and ${k}^{\ensuremath{-}3}$ for two harmonically confined anyons. Nonuniversal contributions at order ${k}^{\ensuremath{-}4}$ are discussed.
The momentum distribution of an expanding cloud of one-dimensional hard-core anyons is studied by an exact numerical approach, and shown to become indistinguishable from that of a noninteracting spin-polarized Fermi gas for large enough times (dynamical fermionization). We also consider the expansion of one-dimensional anyons with strongly attractive short-range interactions suddenly released from a parabolic external potential, and find that the momentum distribution approaches that of its dual system, the ideal Bose gas (dynamical bosonization). For both processes the characteristic time scales are identified, and the effect of the initial confinement is analyzed comparing the dynamics associated with both harmonic and hard-wall traps.
We study a one-dimensional system of interacting anyons with short-range interactions under external confinement. This model, which describes anyons with zero-range
s-wave and
p-wave scattering interactions in the strong interaction regime, continuously interpolates between spin-polarized fermions with
p-wave interactions and free bosons. At zero temperature, the correlation functions decay exponentially with distance, with oscillations governed by the statistics parameter. The decay rate is maximal for
p-wave fermions and decreases monotonically as the statistics parameter approaches the bosonic limit, where it vanishes. The momentum distribution is asymmetric, a hallmark of one-dimensional anyons, and takes the form of a shifted Lorentzian with universal power-law tails, ${lim}_{k\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}}n(k)\ensuremath{\sim}C/{k}^{2}$. We prove analytically that, following release from a harmonic trap, the asymptotic momentum distribution converges to that of free bosons in the same trap, a phenomenon known as dynamical bosonization. We also establish the universality of the ground-state
n-particle reduced density matrices: their natural occupations are independent of the confining potential, while the associated natural
n-point functions for different confinements are related through a simple analytical transformation. In particular, for the one-particle reduced density matrix, we derive exact expressions for both the natural occupations and the natural orbitals at arbitrary particle number. These results extend and unify earlier partial findings for
p-wave fermions, and they provide a clear conceptual explanation of the double degeneracy observed in their spectrum.
While elementary particles obey either bosonic or fermionic exchange statistics, generalized exchange statistics that interpolate between bosons and fermions -- applicable to quasi-particles -- constitute an intriguing topic, both from the fundamental and practical points of view. This work develops a scattering framework for two identical 1D bosonic anyons and two identical 1D fermionic anyons with zero-range contact interactions. The two-body system with zero-range interactions, both in free space and under external confinement, is used to illustrate the recently proposed bosonic-anyon -- fermionic-anyon mapping~(R. Hidalgo-Sacoto {\em{et al.}}, arXiv:2505.17669), which connects the eigenstates of bosonic anyons to those of fermionic anyons and vice versa. Performing explicit calculations for two-particle systems, the momentum distributions and the off-diagonal correlations of the single-particle density matrix for bosonic anyons and fermionic anyons are confirmed to be distinct. We also confirm the previously derived asymptotic coefficients of the momentum distribution tail at orders
k−2 and
k−3 for two harmonically confined anyons. Non-universal contributions at order
k−4 are discussed.
Cold atomic gases provide a remarkable testbed to study the physics of interacting many-body quantum systems. They have started to play a major role as quantum simulators, given the high degree of control that is possible. A crucial element is given by the necessarily non-zero temperature. However cooling to the required ultralow temperatures or even simply measuring the temperature directly on the system can prove to be very challenging tasks. Here, we implement thermometry on strongly interacting two- and one-dimensional Bose gases with high sensitivity in the nano-Kelvin temperature range. Our method is aided by the fact that the decay of the first-order correlation function is very sensitive to the temperature when interactions are strong. We find that there may be a significant temperature variation when the three-dimensional quantum gas is cut into two-dimensional slices or into one-dimensional tubes. Strikingly, the temperature for the one-dimensional case can be much lower than the initial temperature. Our findings show that this decrease results from the interplay of dimensional reduction and strong interactions.