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.
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 authors explore here the interesting ``chemistry'' of anyons -- collective excitations emerging from a large amount of correlated electrons -- in strongly interacting fractional quantum Hall fluids. Anyons can form clusters (``molecules'') because of the effective anyon-anyon interaction induced by tunable electron-electron interactions. Laughlin anyons form hydrogen molecule like bound states with short-range electron interactions, potentially explaining the mystery of the observed 2
e/3 effective charge at extremely low temperatures. In the non-Abelian Moore--Read phase, fusion channel dependent effective interactions lift the ``1/$\ensuremath{\psi}$'' degeneracy, enabling the possible energetic control of non-Abelian anyons. The authors also propose STM as a useful experimental tool for detecting the fingerprint of various configurations of anyon clusters.
Quantum phases of matter can be classified into two kinds: gapped or gapless, depending on whether the equilibrium ground state is separated from excited states by a finite energy window or not in the thermodynamic limit. Topology has played an important role in the understanding of both gapped and gapless phases of matter. This thesis uncovers several robust physical properties of both kinds of systems which are largely unexplored before. \\ For gapped systems, we study topological orders, which are characterized by the existence of anyonic quasiparticles with fractional braiding statistics. In the first part, we introduce a family of Abelian quantum Hall insulators, at filling factors
ν=p/2q for bosonic systems and
ν=p/(p+2q) for fermionic systems, with
p and
q being two coprime integers. These states are termed the \textit{nondiagonal} quantum Hall states, which are constructed in a coupled wire model that exhibits an intimate relation to the nondiagonal conformal field theory and has a constrained pattern of motion for anyons. We unveil a nontrivial interplay between charge symmetry and translation symmetry in this system, and establish the nondiagonal quantum Hall states as symmetry-enriched topological orders. In the second part, we introduce a related model of interacting electrons, but in the absence of a strong quantizing magnetic field, known as the \textit{toric code insulator}. It is constructed from a two-dimensional array of strongly coupled one-dimensional topological superconductors, which feature electron fractionalization and possesses anyonic excitations described by the
Z2 topological order. The motion of anyons is again constrained, and different types of anyons can be related by the translation symmetry. Nondiagonal quantum Hall states and the toric code insulator are convincing examples that itinerant electronic systems can realize interesting symmetry-enriched topological orders.\\ For gapless quantum matters we focus on a particularly familiar class, the metals, but we uncover their topological aspects which were much less appreciated. Every metal is characterized by a Fermi surface of gapless excitations, and the Fermi surface surrounds a manifold in the momentum space, known as the Fermi sea. The ground state of a non-interacting metal, or a Fermi gas, consists of all single-electron states within the Fermi sea. The shape of the Fermi sea has a natural topological characterization in terms of the Euler characteristic
χF, and a fundamental question that we address in this thesis is: \textbf{does
χF imply any robust physical properties in metals, and how can they be measured?} In the third part of the thesis,
χF is connected to physically measurable quantities, such as equal-time density-density correlation functions, and multipartite entanglement measures, such as the mutual information. Particularly, for three dimensions, we establish that the 4-partite mutual information is a robust probe of the Fermi sea topology even in the presence of Fermi-liquid interactions. In the last part, we focus on two dimensions, and propose an experimentally accessible platform to probe
χF for any two-dimensional electron gas (2DEG) that respects time-reversal symmetry. The proposed experimental setup is a Josephson
π-junction, and we predict that the electrical transport in the direction \textit{along} the junction exhibits a current-rectification effect that is controlled by the underlying Fermi sea topology. Due to the relation to Andreev state transport, we term this effect \textit{topological Andreev rectification}. The size of rectification is quantified by a \textit{nonlocal rectified conductance}, which is established to attain an integer quantized value that equals
χF (in unit of
e2/h). Extensive numerical simulations are performed to support our theoretical proposal, and material platforms such as InAs heterostructures and graphene are identified as promising candidates to demonstrate this 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.
For decades, Landau’s theory of phase transition has provided a successful classification for quantum and classical states of matter based on their symmetry broken patterns, except for certain exotic quantum states such as the fractional quantum Hall (FQH) effect. However, such exotic phenomena are crucial for a complete understanding of the nature. This thesis explores new physical principles emerging from topology and topological states. First, we use an example to demonstrate Landau’s theory by studying a pair-density wave system, where the symmetry-breaking paradigm is applicable. It turns out that such a system exhibits the Kosterlitz-Thouless (KT) transition, a phase transition driven by the proliferation of topological defects, i.e. vortex-antivortex pairs. For topological systems, where symmetry broken pattern cannot be used as a classification tool, we prove a theorem that two gapped systems with non-vanishing ground-state overlap must be adiabatically connected, and thus are necessarily in the same topological phase. This theorem provides a simple and generic approach to classify topological band insulators/superconductors, without the needs to calculate any known or yet-to-known topological indices. Once the overlap is found nonzero, the two systems must be topologically identical. After presenting a generic proof, the theorem is also verified through calculating the overlap for several milestone topological band insulators and certain interacting systems. Such an overlap technique is then generalized to (2+1)-D strongly-interacting topological systems at fixed points, including both symmetry-protected topological (SPT) states and intrinsic topological states like FQH. For interacting topological states, the main challenge of utilizing wave-function overlaps to classify them lies in the famous Anderson orthogonality catastrophe (AOC), which states that two different many-body wave functions must have zero overlap in the thermodynamic limit. In this thesis, we found that wave-function overlaps indeed carry critical information about the topological nature of quantum states and this information can be extracted from the finite-size scaling of the overlaps. In the finite-size scaling analysis, we found a universal topological response term as a sub-leading contribution. This term depends on both the central charge of the corresponding conformal field theory (CFT) and the Euler characteristics of the underlying manifolds on which the system is defined. This term reveals a fundamental connection between ground-state overlaps and CFT’s. In addition, surprisingly, the overlap between an intrinsic topological state and a topologically trivial product state shows a decay faster than the exponential behavior expected via a typical AOC analysis. Such finite-size scaling behaviors could be utilized to theoretically detect the gapless edge modes, and to distinguish the topology of quantum states or serve as a signature of topological phase transitions. Possible generalization to higher dimensions and generic non-fixed-point topological systems is also discussed.