In the quantum Hall effect, a planar electron gas exposed to a perpendicular magnetic field condenses into a macroscopic quantum mechanical liquid characterized by a topological integer or fractional quantization of the transverse conductance. For the fractionally quantized liquid, the low-energy elementary excitations carry fractional charge and obey fractional statistics. At low energies, the degrees of freedom in the quantum Hall states are purely topological. The states are classified into universality classes according to their topological order, and for each universality class, there is a low-energy effective quantum field theory, which determines the properties of the associated quasiparticles. In a finite system with a boundary, the two-dimensional effective theory reduces into a one-dimensional edge theory, characteristic of each universality class. The thesis investigates few-electron quantum Hall droplets and the associated quasiparticles by means of numerical exact diagonalization of the Hamiltonian matrix in a high magnetic field approximation, which constitutes a Landau level projection. The inter-electron interaction depends on the Landau level as well as the details of the experimental sample. In the results, we compare the states obtained with realistic effective interactions to the theoretical models concerning the universality classes of the Laughlin states and the Pfaffian state, by studying overlaps of the state vectors, phase structure of the wave functions, as well as the quasiparticle and edge excitations.
In the quantum Hall effect, a planar electron gas exposed to a perpendicular magnetic field condenses into a macroscopic quantum mechanical liquid characterized by a topological integer or fractional quantization of the transverse conductance. For the fractionally quantized liquid, the low-energy elementary excitations carry fractional charge and obey fractional statistics. At low energies, the degrees of freedom in the quantum Hall states are purely topological. The states are classified into universality classes according to their topological order, and for each universality class, there is a low-energy effective quantum field theory, which determines the properties of the associated quasiparticles. In a finite system with a boundary, the two-dimensional effective theory reduces into a one-dimensional edge theory, characteristic of each universality class. The thesis investigates few-electron quantum Hall droplets and the associated quasiparticles by means of numerical exact diagonalization of the Hamiltonian matrix in a high magnetic field approximation, which constitutes a Landau level projection. The inter-electron interaction depends on the Landau level as well as the details of the experimental sample. In the results, we compare the states obtained with realistic effective interactions to the theoretical models concerning the universality classes of the Laughlin states and the Pfaffian state, by studying overlaps of the state vectors, phase structure of the wave functions, as well as the quasiparticle and edge excitations.
The subject of this work is the application of conformal field theory in the fractional quantum Hall effect. The fractional quantum Hall effect is observed in a two-dimensional electron gas in a strong perpendicular magnetic field and at low temperature. The electron system forms a strongly correlated quantum liquid and exhibits the exact quantization of the Hall conductivity in to a fractional value vet/h, where v is the fractional filling fraction of Landau-levels in the system. In addition, the elementary excitations of the electron fluid have fractional charge and fractional statistics of anyons. First, we review the basic of two-dimensional conformal field theory and show how certain conformal quantum field theories can be solved non-perturbatively by using the representation theory of conformal symmetries. The low-energy effective field theory in the bulk of a fractional quantum Hall state is known to be a 2+1 dimensional topological field theory. The states of the topological quantum field theory are directly connected to the holomorphic correlation functions of an equivalent two-dimensional conformal field theory. It is known that the gapless excitations at the boundary of the quantum Hall state are described by the same conformal field theory. By using this connection, the so-called Laughlin and Moore-Read states are derived in detail using conformal field theory. The fractional charge and statistics of the quasiparticle excitations in these states are obtained from the conformal field theory description. Furthermore, the non-Abelian statistics of the Moore-Read states are verified explicitly from the quasiparticle wave functions. We argue how the different fractional quantum Hall phases of matter are described by the quantum numbers obtained using conformal field theory. Lastly, we study model interactions that have the Laughlin v = 1/3 and the Moore-Read v = 1/2 states as exact ground states. We compute the matrix elements of these interactions and diagonalize the Hamiltonian matrix in the disk geometry for both states. The low-energy edge excitations of these states are studied in the finite geometry and we verify that they are in accordance with the conformal field theory on the boundary. By creating a localized density disturbance in the system, we create the e/4 quasiparticle of the Moore-Read state, and by studying the properties of the edge theory it is seen that quasiparticle has the same properties as the non-Abelian quasihole in the conformal field theory. The obtained results verify the correspondence between the bulk state and the boundary conformal field theory at the edge in the fractional quantum Hall states in question.
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.
The fractional quantum Hall effect was experimentally discovered in 1982. It was observed that the Hall conductivity σyx of a two-dimensional electron system is quantized, σyx=e2/3h, in the vicinity of the Landau level filling factor ν=1/3. In 1983, Laughlin proposed a trial many-body wave function, which he claimed described a “new state of matter”—a homogeneous incompressible liquid with fractionally charged quasiparticles. Here, I develop an exact diagonalization theory that allows one to calculate the energy and other physical properties of the ground and excited states of a system of N two-dimensional Coulomb interacting electrons in a strong magnetic field. I analyze the energies, electron densities, and other physical properties of the systems with N≤7 electrons continuously as a function of magnetic field in the range 1/4≲ν<1. The results show that both the ground and excited states of the system resemble a sliding Wigner crystal whose parameters are influenced by the magnetic field. Energy gaps in the many-particle spectra appear and disappear as the magnetic field changes. I also calculate the physical properties of the ν=1/3 Laughlin state for N≤8 and compare the results with the exact ones. This comparison, as well as an analysis of some other statements published in the literature, show that the Laughlin state and its fractionally charged excitations do not describe the physical reality, neither at small N nor in the thermodynamic limit. The results obtained shed new light on the nature of the ground and excited states in the fractional quantum Hall effect.
A core tenet of condensed matter physics has been that different phases of matter can be classified according to Landau's symmetry breaking paradigm. It has become clear, however, that phases of matter exist that are not distinguished by symmetry, but rather by topology. A paradigmatic example of this topological order are the fractional quantum Hall phases, which are the topic of this dissertation. Such phases exhibit the fractional quantum Hall effect, which occurs when electrons confined to two dimensions are subjected to a strong perpendicular magnetic field at very low temperatures. Characterized by a precise quantization of the Hall resistance and a concomitant vanishing of the longitudinal resistance, the fractional quantum Hall effect results from the formation of a strongly correlated quantum liquid of electrons. This quantum liquid supports remarkable quasiparticle excitations, which carry a fractional charge and are thought to obey fractional statistics beyond the familiar Bose-Einstein and Fermi-Dirac statistics. The theoretical understanding of the topological orders realized by the fractional quantum Hall states has progressed by the proposal of explicit trial wave functions as well as various types of effective field theories. This dissertation focuses on two series of trial wave functions, abelian and non-abelian hierarchy wave functions. We study the non-abelian hierarchy wave functions using conformal field theory techniques, by means of which the associated topological properties are studied. These include the fractional charges of the quasiparticles and their non-abelian fractional statistics. In addition, we study abelian hierarchy wave functions using effective Ginzburg-Landau theories in a way that connects to their known conformal field theory description.
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.
The subject of this work is to study the topological phases of matter in (2+1)-dimensions theoretically and numerically. As the theory is not yet complete, we use various gapped model systems to probe the signature behaviour that arise in the topological phase. Most notably, we study the fractional quantum Hall systems, since they are well-established to exist in nature. The main motivation is that the topological phases can be used for topological quantum computation. The first part of the Thesis studies the topological phase from theoretical principles. This consists of defining modular tensor categories to describe consistent quasiparticle systems with arbitrary number of quasiparticles, and topological quantum field theories to describe the low energy, long wavelength effective field theories for the topological phases. There are also other viewpoints that are covered, namely ground state degeneracy that is due to topological properties and not by any symmetry-breaking, and also topological phases forming universality classes of different longranged entanglements. We solve a fractional quantum Hall system on a sphere with Coulomb interaction numerically and evaluate the topological entanglement entropy. This amounts to finding the ground states with various system parameters by an exact diagonalization technique, evaluating the entanglement entropies based on bipartite splits, and extrapolating the entropies to the thermodynamic limit. The subleading term in the entropy then matches the logarithm of the total quantum dimension, which characterizes the quasiparticles. The results match the theoretical predictions.