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Frractional Chern Insulators

Nicolas Regnault

Published Mar 1, 2013• 0 citations
OpenAlex

Abstract

The interest in the field of topological insulators has exploded in recent years. Topological insulators are mainly one body problems, neglecting interactions. As such, the type of interacting topologically ordered state with fractional statistic quasiparticles such as the fractional quantum Hall (FQH) has not been found or proposed in an weakly interact ing topological insulator. We will show that recent numerical studies, based on exact diagonalizations, convincingly indicates that FQH phases may emerge in strongly interacting topological insulators. We will see that the fractional Chern insulators might also lead to phases that are different from those one can find in usual FQH systems.

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The interest in the field of topological insulators has exploded in recent years. Topological insulators are mainly one body problems, neglecting interactions. As such, the type of interacting topologically ordered state with fractional statistic quasiparticles such as the fractional quantum Hall (FQH) has not been found or proposed in an weakly interact ing topological insulator. We will show that recent numerical studies, based on exact diagonalizations, convincingly indicates that FQH phases may emerge in strongly interacting topological insulators. We will see that the fractional Chern insulators might also lead to phases that are different from those one can find in usual FQH systems.
A rich variety of phases can exist in quantum systems. For example, the fractional quantum Hall states have persistent topological characteristics that derive from strong interaction. This thesis uses the exact diagonalization method to investigate quantum lattice models with strong interaction. Our research topics revolve around quantum phase transitions between novel phases. The goal is to find the best schemes for realizing these novel phases in experiments. We studied the fractional Chern insulator and its transition to uni-directional stripes of particles. In addition, we studied topological Mott insulators with spontaneous time-reversal symmetry breaking induced by interaction. We also studied emergent kinetics in one-dimensional lattices with spin-orbital coupling. The exact diagonalization method and its implementation for studying these systems can easily be applied to study other strongly correlated systems.
By numerical exact diagonalization techniques, we obtain the quantum phase diagram of the lattice fractional quantum Hall (FQH) systems in the presence of quenched disorder. By implementing an array of local potential traps representing the disorder, we show that the system undergoes a series of quantum phase transitions as the disorder and/or the interaction is tuned. As the strength of potential traps is increased, the FQH state turns into a compressible liquid and then into a topologically trivial insulator. We use the numerically calculated energy gap, quantum degeneracy, the Chern number, the entanglement spectrum, and the fidelity metric to identify various quantum phases. The connection to continuum FQH effects is also discussed.
Abstract We demonstrate the emergence of a topological ordered phase for non-Hermitian systems. Specifically, we elucidate that systems with non-Hermitian two-body interactions show a fractional quantum Hall (FQH) state. The non-Hermitian Hamiltonian is considered to be relevant to cold atoms with dissipation. We conclude the emergence of the non-Hermitian FQH state by the presence of the topological degeneracy and by the many-body Chern number for the ground state multiplet showing Ctot = 1. The robust topological degeneracy against non-Hermiticity arises from the manybody translational symmetry. Furthermore, we discover that the FQH state emerges without any repulsive interactions, which is attributed to a phenomenon reminiscent of the continuous quantum Zeno effect.
We analytically and numerically analyze the one-dimensional "thin-torus" limit of Fractional Topological Insulators in a series of simple models exhibiting exactly flat bands with local hopping. These models are the one-dimensional limit of two dimensional Chern Insulators, and the Hubbard-type interactions projected into their lowest band take particularly simple forms. By exactly solving the many-body interacting spectrum of these models, we show that, just like in the Fractional Quantum Hall effect, the zero modes of the thin-torus limit are CDW states of occupation numbers satisfying generalized Pauli principles. As opposed to the FQH where the thin-torus CDW appear in orbital space, in the thin-torus FCI states, the CDW states are in real-space. We show the counting of the quasihole excitations in the energy spectrum cannot distinguish between a CDW state and a FQH state. However, by exactly computing the entanglement spectrum for the thin-torus states, we show that it can qualitatively and quantitatively distinguish between a CDW and a fractional topological state such as the FCI. We then discover a previously unknown separation of energy scales of the full FQH energy spectrum in the thin torus limit and find that Chern insulator models exhibiting strong isotropic FCI states have a similar structure in their thin-torus limit spectrum. We close by numerically computing the evolution of energy and entanglement spectra from the thin-torus to the isotropic limit. Our results can also be interpreted as an analysis of one-body, 1-dimensional topological insulators stabilized by inversion symmetry in the presence of interactions.
The fractional quantum Hall (FQH) effect illustrates the range of novel phenomena which can arise in a topologically ordered state in the presence of strong interactions. The possibility of realizing FQH-like phases in models with strong lattice effects has attracted intense interest as a more experimentally accessible venue for FQH phenomena which calls for more theoretical attention. Here we investigate the physical relevance of previously derived geometric conditions which quantify deviations from the Landau level physics of the FQHE. We conduct extensive numerical many-body simulations on several lattice models, obtaining new theoretical results in the process, and find remarkable correlation between these conditions and the many-body gap. These results indicate which physical factors are most relevant for the stability of FQH-like phases, a paradigm we refer to as the geometric stability hypothesis, and provide easily implementable guidelines for obtaining robust FQH-like phases in numerical or real-world experiments.
Recent theoretical papers have demonstrated the realization of fractional quantum anomalous Hall states (also called fractional Chern insulators) in topological flat band lattice models without an external magnetic field. Such newly proposed lattice systems play a vital role in obtaining a large class of fractional topological phases. Here we report the exact numerical studies of edge excitations for such systems in a disk geometry loaded with hard-core bosons, which will serve as a more viable experimental probe for such topologically ordered states. We find convincing numerical evidence of a series of edge excitations characterized by the chiral Luttinger liquid theory for the bosonic fractional Chern insulators in both the honeycomb disk Haldane model and the kagome-lattice disk model. We further verify these current-carrying chiral edge states by inserting a central flux to test their compressibility.