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.
Fractional quantum Hall (FQH) states host fractionally charged anyons with exotic exchange statistics. Of particular interest are FQH phases supporting non-Abelian anyons, which can encode topologically protected quantum information. In this work, we construct quasihole bases for a broad family of non-Abelian FQH states using parton wave functions, which reproduces the fusion-space dimensionality expected from their underlying conformal field theory, consistent with level-rank duality across the parton family. As an application, we numerically compute braiding matrices for representative parton states for large systems, providing a general framework for diagnosing non-Abelian characteristics in candidate FQH states.
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.
Fractional quantum Hall (FQH) fluids host quasiparticle excitations that carry a fraction of the electronic charge. Moreover, in contrast to bosons and fermions that carry exchange statistics of 0 and $\ensuremath{\pi}$ respectively, these quasiparticles of FQH fluids, when braided around one another, can accumulate a Berry phase, which is a fractional multiple of $\ensuremath{\pi}$. Deploying the spherical geometry, we numerically demonstrate that composite fermion particle (CFP) excitations in the Jain FQH states carry Abelian fractional statistics. Previously, the exchange statistics of CFPs were studied in the disk geometry, where the statistics get obscured due to a shift in the phase arising from the addition of another CFP, making its determination cumbersome without prior knowledge of the shift. We show that on the sphere this technical issue can be circumvented and the statistics of CFPs can be obtained more transparently. The ideas we present can be extended to determine the statistics of quasiparticles arising in certain non-Abelian partonic FQH states.
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
Representative wave functions, which encode the topological properties of the spin polarized fractional quantum Hall states in the lowest Landau level, can be expressed in terms of correlation functions in conformal field theories. Until now, the constructions have been restricted to flat geometries, but in this paper we generalize to the simplest curved geometry, namely that of a sphere. Except for being of interest for numerical studies, that usually are performed on a sphere, the response of the fractional quantum Hall (FQH) liquids to curvature can be used to detect a topological quantity, the shift
S which is the average orbital spin of the constituent electrons. We give explicit expressions for representative wave functions on the sphere, for the full Abelian FQH hierarchy, and calculate the corresponding shifts. These microscopic results, based on wave functions, agree with the predictions from the effective Chern-Simons field theory. The methods we develop can also be applied to the planar case. It gives simpler expressions for states with both quasiparticle and quasihole condensates, and allows us to give closed form expressions for a general state in the hierarchy, rather than finding the wave function on a case by case basis.