Laughlin's quantum Hall wave function is obtained as an exact ground state of an N-particle Hamiltonian, in which the electrons themselves are coupled to the Chern-Simons field. The proof requires quantum field-theoretic methods in the Schr\"odinger picture, and clearly exhibits the relation to conformal field theory since the Knizhnik-Zamolodchikov connection shows up and the Laughlin ground state is recognized as a conformal block. A refined version of this approach can be applied for toroidal boundary conditions, i.e., the electrons are confined to the physical interior of the Hall sample. The result turns out to be a nontrivial modification of the p-fold degenerate Haldane-Rezayi wave function. Furthermore, it is shown that the degeneracy of the ground state is accounted for by the quantum group
Uq(su2) with deformation parameter $q=\mathrm{exp}(2\ensuremath{\pi}i/p)$ and p odd; put differently, the possible quantum numbers p of the quantum Hall effect, determining the filling fractions $\ensuremath{\nu}=1/p,$ can be explained by an underlying quantum symmetry.
The braid group dynamics captures the fractional quantum Hall effect (FQHE) as a manifestation of puncture phase. When the dynamics is generalized for particles on a multi-sheeted surface, we obtain new tools which determine the fractional charges, the quantum statistics, and the filling factors of the multi-layered FQHE. A many-quasi-hole wavefunction is proposed for the bilayered samples. We also predict a
ν=5/7 FQHE for triple-layered samples. The viability of {\em 3-dimensional} FQHE and the application of the concept of generalized duality to anyonic superconductivity are discussed.
Master's thesis in Theoretical Physics. The official university title is "Fractional Quantum Hall Effect: Current State and Perspectives"; this record uses the author's preferred, more descriptive title. This thesis develops a unified description of the fractional quantum Hall effect (FQHE) and topological order through three complementary frameworks: microscopic trial many-body wavefunctions, the algebraic theory of anyons and unitary modular tensor categories, and Chern-Simons effective field theory. The first part covers the physical foundations—Landau quantization, the TKNN invariant, Laughlin's wavefunctions, the hierarchy and composite fermion theories, and some non-Abelian states like the Moore-Read Pfaffian state. The second part develops the algebraic theory of anyons, from the braid group to fusion rules, quantum dimensions, topological spins, and the F- and R-matrices that characterize a unitary modular tensor category. The third part constructs the Chern-Simons effective field theory, deriving fractional charge, statistics, and ground-state degeneracy, unifying Abelian states via the K-matrix formalism, and then generalizing to non-Abelian states through 𝑈(1)ₘ × 𝑆𝑈(𝑁)ₖ Chern-Simons theory. The central result is that Chern–Simons theory provides a field-theoretic realization of the topological order introduced microscopically and algebraically in the earlier chapters, closing the circle between the three descriptions and providing a unified framework for the FQHE and topological order and anyon's physics.
A bstract Fibonacci anyons ε provide the simplest possible model of non-Abelian fusion rules: [1] × [1] = [0] ⊕ [1]. We propose a conformal field theory construction of topological quantum registers based on Fibonacci anyons realized as quasiparticle excitations in the ℤ 3 parafermion fractional quantum Hall state. To this end, the results of Ardonne and Schoutens for the correlation function of four Fibonacci fields are extended to the case of arbitrary number n of quasi-holes and N = 3 r electrons. Special attention is paid to the braiding properties of the obtained correlators. We explain in details the construction of a monodromy representation of the Artin braid group
B B n acting on n -point conformal blocks of Fibonacci anyons. The matrices of braid group generators are displayed explicitly for all n ≤ 8. A simple recursion formula makes it possible to extend without efforts the construction to any n . Finally, we construct
N N qubit computational spaces in terms of conformal blocks of
2N 2 N + 2 Fibonacci anyons.
In this paper we explore the braiding properties of the Moore-Read fractional Hall sequence, which amounts to computing the adiabatic evolution of the Hall liquid when the anyons are moved along various trajectories. In this work, the anyons are pinned to precise spatial configurations by using specific external potentials. Such external potentials break the translational symmetry and it appears that one will be forced to simulate the braidings on the entire many-body Hilbert space, an absolutely prohibitive scenario. We demonstrate how to overcome this difficulty and obtain the exact braidings for fairly large Hall systems. For this, we show that the incompressible state of a general
(k,m) fractional Hall sequence can be viewed as the unique zero mode of a specific Hamiltonian
H(k,m), whose form is explicitly derived by using
k-particle creation operators. The compressible Hall states corresponding to $n\ifmmode\times\else\texttimes\fi{}k$ anyons fixed at
w1,…,wnk are shown to be the zero modes of a pinning Hamiltonian
Hw1,…,wnk(k,m), which is also explicitly derived. The zero modes of
Hw1,…,wnk(k,m) are shown to be contained in the space of the zero modes of
H(k,m). Therefore, the computation of the braidings can be done entirely within this space, which we map out for a number of Hall systems. Using this efficient computational method, we study various properties of the Moore-Read states. In particular, we present direct confirmation of their topological and non-Abelian properties that were previously predicted using the underlying conformal field theory structure of the Moore-Read state.