The quantum Hall effect has been the prototypical example for many emerging fields of Physics, including the study of topological phases of matter, topological order, and topological insulators. The objects of interest in this thesis are anyons, which are elementary excitations in fractional quantum Hall (FQH) fluids. Anyons are quasiparticles with fractional charge and fractional statistics. The latter prop- erty makes it a subject of immense interest due to application in fault-tolerant quantum computing. Here, we employ a perspective that anyons are the appro- priate degree of freedom to describe the relevant physics of FQH systems. This is because when a topological phase is assumed and the energy gap goes to infinity, anyons (“quasiholes”) are the only possible elementary excitations, since they are not energetically punished by the correspondng model Hamiltonian. With respect to a realistic Hamiltonian, however, this choice of degree of freedom depends on the exact effective interaction in the system. For example, while the Laughlin state is the model state at filling factor n + 1/3 (where n is some integer), when the effective electron-electron interaction is more long-ranged, the quasiholes of a Gaffnian state may be excited and become the better degree of freedom to describe physics in the same filling factor. This perspective results in a surprising observa- tion that a Laughlin quasihole may fractionalize under certain conditions, leading to a quasiparticle of charge e/6 that can potentially be detected in experiments. Another important aspect of anyons is their intrinsic spin and the corresponding spin-statistics relation, which must account for the fact that FQH anyons are not point particles, but objects with finite sizes and intricate internal structures. Our investigation leads to a rigorous derivation of the general spin-statistics relation for anyons in Abelian FQH phases. Furthermore, we show it is possible to fully capture the important features of the statistics by a conformal Hilbert space (CHS) description. Our analysis provides the necessary tools to study such geometric effects, which we use to investigate how disorder in a realistic system may affect different experimental schemes for measuring anyon statistics.
This paper reviews some of the recent progresses in understanding the dynamical and statistical properties of anyons in fractional quantum Hall (FQH) systems. These are strongly interacting topological systems for electrons confined to a two-dimensional manifold, and anyons are fundamental particles emerging from the truncation of the Hilbert space as a result of both the kinetic and interaction energies. We introduce the concept of the conformal Hilbert spaces (CHS) from such Hilbert space truncation, and the hierarchy of these Hilbert spaces allows us to understand the internal structures of the anyons that can strongly affect their dynamics. The bulk-edge correspondence and the conformal mapping of the CHS also enable us to derive a rigorous bosonization scheme for anyons in these two-dimensional systems, and to capture the statistical interaction between anyons in the form of microscopic interaction Hamiltonians. Examples about the fractionalization of Laughlin quasiholes and a new family of bosonic FQH phases as the dual descriptions of the well-known fermionic FQH phases are given, as applications to the proposed theoretical constructions in this work.
Quasihole excitations in fractional quantum Hall (FQH) systems exhibit fractional statistics and fractional spin, but how the spin-statistics relation emerges from many-body physics remains poorly understood. Here we prove a spin-statistics relation using only FQH wave functions, on both the sphere and disk geometry. In particular, the proof on the disk generalizes to all quasiholes in realistic systems, which have a finite size and could be deformed into arbitrary shapes. Different components of the quasihole spins are linked to different conformal Hilbert spaces (CHSs), which are null spaces of model Hamiltonians that host the respective FQH ground states and quasihole states. Understanding how the intrinsic spin of the quasiholes is linked to different CHSs is crucial for the generalized spin-statistics relation that takes into account the effect of metric deformation. In terms of the experimental relevance, this enables us to study the effect of deformation and disorder that introduces an additional source of Berry curvature, an aspect of anyon braiding that has been largely neglected in previous literature.
We show the low-lying excitations at filling factor ν=n+1/3 with realistic interactions can be understood as quantum fluids with "Gaffnian quasiholes" as the proper elementary degrees of freedom. Each Laughlin quasihole can thus be understood as a bound state of two Gaffnian quasiholes, which in the lowest Landau level (LLL) behaves like "partons" with "asymptotic freedom" mediated by neutral excitations acting as "gluons." Near the experimentally observed nematic FQH phase in higher LLs, quasiholes become weakly bound and can fractionalize with rich dynamical properties. By studying the effective interactions between quasiholes, we predict a finite temperature phase transition of the Laughlin quasiholes even when the Laughlin ground state remains incompressible, and derive relevant experimental conditions for its possible observations.
The generalized exclusion statistics (GES) parameter
g characterizes the fractionalization of anyonic excitations in fractional quantum Hall (FQH) states. While
g is phenomenologically linked to the filling factor
ν in Laughlin states, a universal derivation from the underlying topological field theory for arbitrary Abelian and non-Abelian states remains an open theoretical target. Furthermore, whether this topologically derived
g can quantitatively predict the shot-noise Fano factor
F measured in quantum point contact (QPC) tunneling experiments has not been rigorously established across general FQH phases. Here, we derive
g directly from the topological spin
θ=e2πih, where
h is the conformal dimension of the corresponding conformal field theory (CFT) primary field. For Abelian anyons,
g=2h, while non-Abelian anyons yield a matrix form. Using the Keldysh formalism for the QPC tunneling Hamiltonian, we show this topologically derived
g dictates the tunneling density of states exponent and predicts the Fano factor
F=gν in the weak backscattering regime. For Laughlin states at
ν=1/m, this yields
F=ν, matching fractional charge
e∗=νe. For non-Abelian states, the derived
g matrix predicts a modified Fano factor revealing topological spin contributions to nonequilibrium noise. ## 1. Introduction The fractional quantum Hall (FQH) effect provides a physical realization of topological phases of matter hosting anyonic excitations. These excitations obey generalized exclusion statistics (GES), a framework introduced to generalize the Pauli exclusion principle to particles with fractional braiding statistics. In a quantum point contact (QPC) geometry, the tunneling of these anyons generates shot noise, characterized by the Fano factor
F (the ratio of noise power to twice the Poissonian current). While the connection between GES and the filling factor
ν is well-established for Laughlin states, a universal deriv
This thesis investigates interaction-driven phenomena of two-dimensional electrons in strong magnetic fields, with a focus on how the electron-electron (e-e) interaction projected in one Landau level affects the physical properties of the topological bands. The Landau-level-projected interaction favours qualitatively different ground states in different situations, such as topologically ordered fractional quantum Hall liquids that host anyonic quasiparticles, and symmetry-broken electron solids or charge-ordered phases. In the fractional quantum Hall liquid, we study quasiholes as interacting particles rather than featureless point excitations in a dilute and ideal anyon gas. Going beyond model Hamiltonians, we show that realistic e–e interactions generate nontrivial effective anyon–anyon interactions that can endow few-anyon clusters with internal structure and binding, producing “anyon molecules”. Because the same effective interaction also controls how anyons couple to local probes, we compute local density-of-states signatures and show that local tunneling spectra can encode both universal and non-universal information. The analysis is carried out for the Abelian Laughlin states and for the non-Abelian Moore–Read state, the latter one showing characteristic odd–even features tied to its fusion structure. In the high-Landau-level regime, the same interaction-driven physics favors symmetry-broken charge order. We construct a microscopic-to-continuum framework that connects Landau-level-projected Hartree--Fock energetics to elastic constants and, through Kosterlitz--Thouless--Halperin--Nelson--Young defect physics and renormalization-group flow, to a quantitative finite-temperature melting scale. This approach highlights the role of dislocation proliferation-induced thermal renormalization in quantitatively understanding the 2D electron solids' melting, and enables direct comparison with transport signatures of re-entrant insulating phases. Taken together, we provide an interaction-based perspective on both topological quantum Hall liquids and quantum Hall electron solids in Landau levels. This thesis closes by outlining several directions that follow naturally from this work.
We show that even with arbitrarily large cyclotron gap, Landau level (LL) mixing can be dominant with scale-free interaction in a fractional quantum Hall system, as long as the filling factor exceeds certain critical values. Such scale-free interaction with kinetic energy can serve as exact model Hamiltonians for certain composite Fermion or parton states (unlike the well-known TK Hamiltonians where the number of LLs needs to be fixed by hand), and they are natural physical Hamiltonians for 2D systems embedded in higher-dimensional space time. Even with LL mixing the null spaces of such Hamiltonians (spanned by the ground state and the quasiholes) can be analytically obtained, and we show these are the generalization of the conformal Hilbert spaces (CHS) to more than one LLs. The effective interaction between the anyons for these topological phases emerges from the kinetic energy of the ``elementary particles,'' leading to an interesting duality between strongly and weakly interacting systems that can be easily understood via the tuning of parameters in the scale-free interaction. We also propose an experimental platform for approximately realizing such model Hamiltonians with trionlike particles that can potentially lead to very robust (non-Abelian) FQH phases from two-body Coulomb-based interaction.