We reserve the first part of this thesis to a brief (and by far incomplete, but hopefully self-contained) introduction to the vast subject of quantum Hall physics. We dedicate the first chapter to a discursive broad introduction. The second one is instead used to introduce the integer and fractional quantum Hall effects, with an eye to the synthetic quantum matter platforms for their realization. In the third chapter we present famous Laughlin's wavefunction and discuss its basic features, such as the gapless edge modes and the gapped quasiparticle excitations in the bulk. We close this introductory part with a fourth chapter which presents a brief overview on the chiral Luttinger liquid theory. In the second part of this thesis we instead proceed to present our original results. In the fifth chapter we numerically study the linear and non-linear dynamics of the chiral gapless edge modes of fractional quantum Hall Laughlin droplets -- both fermionic and bosonic -- when confined by anharmonic trapping potentials with model short range interactions; anharmonic traps allow us to study the physics beyond Wen's low-energy/long-wavelength chiral Luttinger liquid paradigm in a regime which we believe is important for synthetic quantum matter systems; indeed, even though very successful, corrections to Wen's theory are expected to occur at higher excitation energies/shorter wavelengths. Theoretical works pointed to a modified hydrodynamic description of the edge modes, with a quadratic correction to Wen's linear dispersion $\omega_k=vk$ of linear waves; even though further works based on conformal field theory techniques casted some doubt on the validity of the theoretical description, the consequences of the modified dispersion are very intriguing. For example, in conjunction with non-linearities in the dynamics, it allowed for the presence of fractionally quantized solitons propagating ballistically along the edge. The strongly correlated nature of fractional quantum Hall liquids poses technical challenges to the theoretical description of its dynamics beyond the chiral Luttinger liquid model; for this reason we developed a numerical approach which allowed us to follow the dynamics of macroscopic fractional quantum Hall clouds, focusing on the neutral edge modes that are excited by applying an external weak time-dependent potential to an incompressible fractional quantum Hall cloud prepared in a Laughlin ground state. By analysing the dynamic structure factor of the edge modes and the semi-classical dynamics we show that the edge density evolves according to a Korteweg-de Vries equation; building on this insight, we quantize the model obtaining an effective chiral Luttinger liquid-like Hamiltonian, with two additional terms, which we believe captures the essential low-energy physics of the edge beyond Wen's highly successful theory. We then move forward by studying -- even though only partially -- some of the physics of this effective model and analyse some of its consequences. In the sixth chapter we look at the spin properties of bulk abelian fractional quantum Hall quasiparticles, which are closely related to their anyonic statistics due to a generalized spin-statistics relation - which we prove on a planar geometry exploiting the fact that when the gauge-invariant generator of rotations is projected onto a Landau level, it fractionalizes among the quasiparticles and the edge. We then show that the spin of Jain's composite fermion quasielectron satisfies the spin-statistics relation and is in agreement with the theory of anyons, so that it is a good anti-anyon for the Laughlin's quasihole. On the other hand, even though we find that the Laughlin’s quasielectron satisfies the spin-statistics relation, it carries the wrong spin to be the anti-anyon of Laughlin’s quasihole. Leveraging on this observation, we show how Laughlin's quasielectron is a non-local object which affects the system's edge and thus affecting the fractionalization of the spin. Finally, in the seventh chapter we draw our conclusions.

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