The fractional quantum Hall effect (FQHE), now entering it's fourth decade, continues to draw attention from the condensed matter community. New experiments in recent years are raising hopes that it will be possible to observe quasi-particles with non-abelian anyonic statistics. These particles could form the building blocks of a quantum computer. The quantum Hall states have topologically protected energy gaps to the low-lying set of excitations. This topological order is not a locally measurable quantity but rather a non-local object, and it is one of the keys to it's stability. From an early stage understanding of the FQHE has been facilitate by constructing trial wave functions. The topological classification of these wave functions have given further insight to the nature of the FQHE. An early, and successful, wave function construction for filling fractions ν=p/(2p+1) was that of composite fermions on planar and spherical geometries. Recently, new developments using conformal field theory have made it possible to also construct the full Haldane-Halperin hierarchy wave functions on planar and spherical geometries. In this thesis we extend this construction to a toroidal geometry, i.e. a flat surface with periodic boundary conditions. One of the defining features of topological states of matter in two dimensions is that the ground state is not unique on surfaces with non trivial topology, such as a torus. The archetypical example is the fractional quantum Hall effect, where a state at filling fraction ν=p/q, has at least a q-fold degeneracy on a torus. This has been shown explicitly for a few cases, such as the Laughlin states and the the Moore-Read states, by explicit construction of candidate electron wave functions with good overlap with numerically found states. In this thesis, we construct explicit torus wave functions for a large class of experimentally important quantum liquids, namely the chiral hierarchy states in the lowest Landau level. These states, which includes the prominently observed positive Jain sequence at filling fractions ν=p/(2p+1), are characterized by having boundary modes with only one chirality. Our construction relies heavily on previous work that expressed the hierarchy wave functions on a plane or a sphere in terms of correlation functions in a conformal field theory. This construction can be taken over to the torus when care is taken to ensure correct behaviour under the modular transformations that leave the geometry of the torus unchanged. Our construction solves the long standing problem of engineering torus wave functions for multi-component many-body states. Since the resulting expressions are rather complicated, we have carefully compared the simplest example, that of ν=2/5, with numerically found wave functions. We have found an extremely good overlap for arbitrary values of the modular parameter τ, that describes the geometry of the torus. Having explicit torus wave functions allows us to use the methods developed by Read and Read \&amp; Rezayi to numerically compute the quantum Hall viscosity. Hall viscosity is conjectured to be a topologically protected macroscopic transport coefficient characterizing the quantum Hall state. It is related to the shift of the same QH-fluid when it is put on a sphere. The good agreement with the theoretical prediction for the 2/5 state strongly suggests that our wave functions encodes all relevant topologically information. We also consider the Hall viscosity in the limit of a very thin torus. There we find that the viscosity changes as we approach the thin torus limit. Because of this we study the Laughlin state in that limit and see how the change in viscosity arises from a change in the Hamiltonian hopping elements. Finally we conclude that there are both qualitative and quantitative difference between the thin and the square torus. Thus, one has to be careful when interpreting results in the thin torus limit. / <p>At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 4: Manuscript.</p>

No description available.

This thesis scrutinizes the condensed matter physics phenomenon known as the fractional quantum Hall effect (FQHE), in particular fractional quantum Hall effects occurring in multicomponent systems. Broadly speaking, the FQHE can be defined as a many-electron quantum phenomenon, driven by strong interactions, that occurs in two-dimensional electron gasses in the presence of a perpendicular external magnetic field (and it is also predicted to occur for any two-dimensional particles, such as confined cold atoms, in an external gauge field). Multicomponent systems are systems where the constituent particles (such as electrons or cold atoms) possess internal degrees of freedom, for instance a spin or valley index. These internal degrees of freedom are often overlooked when modeling the FQHE. Taking into account the multicomponent degree of freedom yields an abundance of possibilities for the intellection of new types of so-called “topological phases of matter”, which are ubiquitously associated with the FQHE. In this thesis several different cases are considered. The first topic discussed herein is a study of phase transitions that can take place between FQHE phases with different net values of their multicomponent degrees of freedom. Examples are phase transitions between phases of different uniform net spin polarization, tunable as a function of certain system parameters. Some significant technical refinements are made to a previous model and comparisons are made with a variety of different experiments. The results are relevant for multicomponent FQHEs occurring in GaAs,AlAs and SiGe semiconductor systems where the electronic structure is confined to two dimensions, as well as in two-dimensional materials such as graphene. The second topic discussed herein is the introduction of the multiparticle multicomponent pseudopotential formalism. This methodology is oriented towards considerably expanding an existing framework for the construction of exactly solvable FQHE models by parameterizing multicomponent interactions. The final topic is the first example application of this new formalism to the construction of an exactly solvable FQHE model.

This dissertation is devoted to theoretical studies of strongly interacting one-dimensional and quasi one-dimensional electron systems. The properties of one-dimensional electron systems can be studied within the bosonization technique, which presents fermions as collective bosonic density excitations. The power of this approach is the ability to treat electron-electron interaction exactly in the low-energy limit. The approach predicts the failure of Fermi liquid and an absence of long-range order in one-dimensions. The low-energy description of one-dimensional interacting systems is called the Tomonaga-Luttinger liquid theory. For example, the edges of quantum Hall systems are one-dimensional and described by a chiral Tomonaga-Luttinger liquid. Another example is a quantum spin Hall system with helical edge states, which are also described by a Tomonaga-Luttinger liquid. In our first work, a study of magnetized edge states of quantum spin-Hall system is presented. A magnetic field dependent signature of such edges is obtained, which can be verified in a Coulomb drag experiment. The second part of the dissertation is devoted to quasi-one dimensional antiferromagnetic lattices. A spatially anisotropic lattice antiferromagnet can be viewed as an array of one dimensional spin chains coupled in a way to match the lattice symmetry. This allows to use the non-Abelian bosonization technique to describe the low-energy physics of spin chains and study the inter-chain interactions perturbatively. The work presented in the dissertation studies the effect of Dzyaloshinskii-Moriya interaction on the magnetic phase diagram of the spatially anisotropic kagome antiferromagnet. We predict a Dzyaloshinskii-Moriya interaction driven phase transition from Neel to Neel+dimer state. In the third work, a novel model of the fractional quantum Hall effect is given. Wave functions of two-dimensional electrons in strong and quantizing magnetic field are essentially one-dimensional. That invites one to use the one-dimensional phenomenological bosonization to describe the density fluctuations of the two-dimensional interacting electrons in magnetic field. Remarkably, the constructed trial bosonized fermion operator describing the electron states with a fixed Landau gauge momentum is effectively two-dimensional. / text

The discovery of the fractional quantum Hall effect for two-dimensional electron gases immersed in a strong orthogonal magnetic field represents a cornerstone of modern physics. The states responsible for the appearance of the fractional quantum Hall effect have been found to be part of a whole new class of phases of matter, characterized by an internal order with unprecedented properties and known as topological order. This fact opened up a completely new territory for physical studies, paving the way towards many of the current hot topics in physics, such as topological phases of matter, topological order and topological quantum computing. As it happens for most topologically-ordered phases, fractional quantum Hall states are breeding ground for the observation of many exotic physical phenomena. Important examples include the appearance of degenerate ground states when the system in placed on a space with non-trivial topology, the existence of chiral gapless edge excitations which unidirectionally propagate without suffering of back-scattering processes, and the possibility of hosting elementary excitations, known as quasiparticles and quasiholes, carrying fractional charge and anyonic statistics. Even though for years since their discovery fractional quantum Hall states have been studied only in electronic systems, the recent advances made in the domains of quantum simulators and artificial gauge fields opened the possibility to realize bosonic analogs of these states in platforms based on ultracold atoms and photons. Reaching the appropriate conditions for the simulation of the fractional quantum Hall effect with neutral particles (such as atoms and photons) has required decades of both theoretical and experimental efforts and passed through the implementation of many topological models at the single-particle level. However, we strongly believe that the stage is set finally and that bosonic fractional quantum Hall states will be realized soon in different set-ups. Motivated by this fact, we dedicate this Thesis to the study of the edge and quasihole excitations of bosonic fractional quantum Hall states with the goal of guiding near future experiments towards exciting discoveries such as the observation of anyons. In the first part of the Thesis we focus our attention on the behavior of the edge excitations of the bosonic $ u=1/2$ Laughlin state (a paradigmatic wave function for the fractional quantum Hall effect) in the presence of cylindrically symmetric hard-wall confining potentials. With respect to electronic devices, atomic and photonic platforms offers indeed a more precise control on the external potential confining the systems, as confirmed by the recent realization of flat-bottomed traps for ultracold atoms and by the flexibility in designing optical cavities. At the same time, most of the theoretical works in this direction have considered harmonic confinements, for which the edge states have been found to display the standard chiral Luttinger liquid behavior, leaving the field open for our analysis of new physics beyond the Luttinger paradigm. In the second part we propose a novel method to probe the statistical properties of the quasihole excitations on top of a fractional quantum Hall state. As compared to the previous proposals, it does not rely on any form of interference and it has the undeniable advantage of requiring only the measurements of density-related observables. As we have already mentioned, although the existence of anyons have been theoretically predicted long time ago, it still lacks a clear-cut experimental evidence and this motivated people working with ultracold atoms and photons to push their systems into the fractional quantum Hall regime. However, while there exist plenty of proposals for the detection of anyons in solid-state systems (mostly based on interferometric schemes in which currents are injected into the system and anyons travel along its edges), in the present literature the number of detection schemes applicable in ultracold atomic and/or photonic set-ups is much smaller and they are typically as demanding as those proposed in the electronic context. Finally, in the last part of the Thesis we move to the lattice counterparts of the fractional quantum Hall states, the so-called fractional Chern insulators. Still with the purpose of paving the way for future experimental studies with quantum simulators, we focus our attention of the simplest bosonic version of these states and, in particular, on the properties of its quasihole excitations. Although this topic has already been the subject of intense studies, most of the previous works were limited either to system sizes which are too small to host anyonic excitations, or to unphysical conditions, such as periodic geometries and non-local Hamiltonians. Our study investigates for the first time the properties of genuine quasihole excitations in experimentally relevant situations.

Thesis advisor: Ziqiang Wang / Exotic phases and associated phase transitions in low dimensions have been a fascinating frontier and a driving force in modern condensed matter physics since the 80s. Due to strong correlation effect, they are beyond the description of mean-field theory based on a single-particle picture and Landau's symmetry-breaking theory of phase transitions. These new phases of matter require new physical quantities to characterize them and new languages to describe them. This thesis is devoted to the study on exotic phases of correlated electrons in two spatial dimensions. We present the following efforts in understanding two-dimensional exotic phases: (1) Using Zn vertex algebra, we give a complete classification and characterization of different one-component fractional quantum Hall (FQH) states, including their ground state properties and quasiparticles. (2) In terms of a non-unitary transformation, we obtain the exact form of statistical interactions between composite fermions in the lowest Landau level (LLL) with v=1/(2m), m=1,2... By studying the pairing instability of composite fermions we theoretically explains recently observed FQHE in LLL with v=1/2,1/4. (3) We classify different Z2 spin liquids (SLs) on kagome lattice in Schwinger-fermion representation using projective symmetry group (PSG). We propose one most promising candidate for the numerically discovered SL state in nearest-neighbor Heisenberg model on kagome lattice}. (4) By analyzing different Z2 spin liquids on honeycomb lattice within PSG classification, we find out the nature of the gapped SL phase in honeycomb lattice Hubbard model, labeled sublattice pairing state (SPS) in Schwinger-fermion representation. We also identify the neighboring magnetic phase of SPS as a chiral-antiferromagnetic (CAF) phase and analyze the continuous phase transition between SPS and CAF phase. For the first time we identify a SL called 0-flux state in Schwinger-boson representation with one (SPS) in Schwinger-fermion representation by a duality transformation. (5) We show that when certain non-collinear magnetic order coexists in a singlet nodal superconductor, there will be Majorana bound states in vortex cores/on the edges of the superconductor. This proposal opens a window for discovering Majorana fermions in strongly correlated electrons. (6) Motivated by recent numerical discovery of fractionalized phases in topological flat bands, we construct wavefunctions for spin-polarized fractional Chern insulators (FCI) and time reversal symmetric fractional topological insulators (FTI) by parton approach. We show that lattice symmetries give rise to different FCI/FTI states even with the same filling fraction. For the first time we construct FTI wavefunctions in the absence of spin conservation which preserve all lattice symmetries. The constructed wavefunctions also set up the framework for future variational Monte Carlo simulations. / Thesis (PhD) — Boston College, 2011. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

The integer and fractional quantum Hall effects arise in two-dimensional electron systems subject to low temperature and high perpendicular magnetic field. The phenomenology of these two effects is rich and provides interesting insight into quantum physics. We present two experimental studies of phenomena in the fractional quantum Hall regime. The first examines the tunneling conductance of quasiparticles at filling factor 5/2. This state is of significant interest because it lies outside the traditional Jain hierarchy of fractional quantum Hall states and because it may be the first physical system found to exhibit non-abelian particle statistics. A quantum point contact is used to bring edge states on opposite sides of the system in proximity to each other, allowing quasiparticles to tunnel between the edge states. By annealing the gates forming the quantum point contact at different voltages we control the tunneling strength for fixed temperature and bias. We demonstrate a transition from strong to weak tunneling controlled in this manner. In the weak tunneling regime, the DC bias and temperature dependence of the tunneling conductance is fit to a theoretical form, resulting in values for the quasiparticle charge \(e*\) and the interaction parameter \(g\). The values of these parameters are used to help distinguish between proposed candidate states for the 5/2 wave function. Quantitative and qualitative results are most consistent with the abelian 331 state. Our second main focus is the breakdown of the fractional quantum Hall states at filling factors 4/3 and 5/3. Breakdown of integer and fractional quantum Hall states is known to occur when the Hall and longitudinal resistances deviate from their ideal values at nonzero critical currents. Although multiple studies of breakdown in the integer quantum Hall regime have been reported, corresponding results for the fractional regime are scarce. We observe breakdown over a range of integer states that is consistent with previous results. However, breakdown in the fractional regime is found to exhibit markedly different behavior. In particular, the magnitude of the critical current decreases with increased sample width. This behavior is opposite that observed for integer filling factors and does not seem to be explicable based on current theories of breakdown. / Physics

The quantum Hall effect occurs when a conductor carrying a current is placed in a perpendicular magnetic field. If certain conditions are met, such as strong magnetic field and low temperature, the resistivity becomes quantised, taking values of integer or fractional multiples of h/e2. By analysing the movement of electrons in a magnetic field classically and quantum mechanically information about the integer quantum Hall effect and the fractional quantum Hall effect can be gathered, using the two different gauge potentials of Landau gauge and Symmetric gauge. Resistance Metrology is one field of study that the quantum Hall effect has greatly impacted by providing a way to universally maintain the ohm, with significantly less uncertainty than previously. / Den kvantmekaniska hall-effekten uppstår när en strömbärande ledare placeras i ett vinkelrätt magnetfält. Om vissa villkor är uppfyllda, såsom starkt magnetfält och låg temperatur, blir resistiviteten kvantiserad. Given av heltal (integer) eller fraktions-(fractional) multiplar av h/e2. Genom att analysera elektroners rörelse i ett magnetfält klassiskt och kvantmekaniskt fås information om Hall-effekterna; integer quantum Hall effect och fractional quantum Hall effect, med hjälp av de två gauge potentialerna Landau gauge och Symmetrisk gauge. Resistansmetrologi är ett forskningsområde som kvant Hall-effekten har starkt påverkat genom att tillhandahålla ett sätt att universellt upprätthålla ohm-enheten med betydligt mindre osäkerhet än tidigare.

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.

This Thesis is devoted to the study of topological phases of matter in optical platforms, focusing on non-Hermitian systems with gain and losses involving nonreciprocal elements, and fractional quantum Hall liquids where strong interactions play a central role.In the first part we investigated nonlinear Taiji micro-ring resonators in passive and active silicon photonics setups. Such resonators establish a unidirectional coupling between the two whispering-gallery modes circulating in their perimeter. We started by demonstrating that a single nonlinear Taiji resonator coupled to a bus waveguide breaks Lorentz reciprocity. When a saturable gain is added to a single Taiji resonator, a sufficiently strong unidirectional coupling rules out the possibility of lasing in one of the whispering-gallery modes with independence of the type of optical nonlinearity and gain saturation displayed by the material. This can be regarded as a dynamical time-reversal symmetry breaking. This effect is further enhanced by an optical Kerr nonlinearity. We showed that both ring and Taiji resonators can work as optical isolators over a broad frequency band in realistic operating conditions. Our proposal relies on the presence of a strong pump in a single direction: as a consequence four-wave mixing can only couple the pump with small intensity signals propagating in the same direction. The resulting nonreciprocal devices circumvent the restrictions imposed by dynamic reciprocity. We then studied two-dimensional arrays of ring and Taiji resonators realizing quantum spin-Hall topological insulator lasers. The strong unidirectional coupling present in Taiji resonator lattices promotes lasing with a well-defined chirality while considerably improving the slope efficiency and reducing the lasing threshold. Finally, we demonstrated that lasing in a single helical mode can be obtained in quantum spin-Hall lasers of Taiji resonators by exploiting the optical nonlinearity of the material. In the second part of this Thesis we dived into more speculative waters and explored fractional quantum Hall liquids of cold atoms and photons. We proposed strategies to experimentally access the fractional charge and anyonic statistics of the quasihole excitations arising in the bulk of such systems. Heavy impurities introduced inside a fractional quantum Hall droplet will bind quasiholes, forming composite objects that we label as anyonic molecules. Restricting ourselves to molecules formed by one quasihole and a single impurity, we find that the bound quasihole gives a finite contribution to the impurity mass, that we are able to ascertain by considering the first-order correction to the Born-Oppenheimer approximation. The effective charge and statistical parameter of the molecule are given by the sum of those of the impurity and the quasihole, respectively. While the mass and charge of such objects can be directly assessed by imaging the cyclotron orbit described by a single molecule, the anyonic statistics manifest as a rigid shift of the interference fringes in the differential scattering cross section describing a collision between two molecules.