Abstract:
We clarify the relation between two kinds of statistics for particle excitations in planar systems: the braid statistics of anyons and the Haldane exclusion statistics(HES). It is shown non-perturbatively that the HES exists for incompressible anyon liquid in the presence of a Hall response. We also study the statistical properties of a specific quantum anomalous Hall model with Chern-Simons term by perturbation in both compressible and incompressible regimes, where the crucial role of edge states to the HES is shown.

Abstract:
The integer quantum Hall effect is a topological state of quantum matter in two dimensions, and has recently been observed in three-dimensional topological insulator thin films. Here we study the Landau levels and edge states of surface Dirac fermions in topological insulators under strong magnetic field. We examine the formation of the quantum plateaux of the Hall conductance and find two different patterns, in one pattern the filling number covers all integers while only odd integers in the other. We focus on the quantum plateau closest to zero energy and demonstrate the breakdown of the quantum spin Hall effect resulting from structure inversion asymmetry. The phase diagrams of the quantum Hall states are presented as functions of magnetic field, gate voltage and chemical potential. This work establishes an intuitive picture of the edge states to understand the integer quantum Hall effect for Dirac electrons in topological insulator thin films.

Abstract:
We study anisotropic lattice strips in the presence of a magnetic field in the quantum Hall effect regime. At specific magnetic fields, causing resonant Umklapp scattering, the system is gapped in the bulk and supports chiral edge states in close analogy to topological insulators. These gaps result in plateaus for the Hall conductivity exactly at the known fillings n/m (both positive integers and m odd) for the integer and fractional quantum Hall effect. For double strips we find topological phase transitions with phases that support midgap edge states with flat dispersion. The topological effects predicted here could be tested directly in optical lattices.

Abstract:
Edge states in the integral quantum Hall effect on a lattice are reviewed from a topological point of view. For a system with edges which is realized inevitably in an experimental situation, the Hall conductance $\sigma_{xy}$ is given by a winding number of the edge state on a complex energy surface. A relation between two topological invariants (bulk and edge) is also clarified. In a macroscopic system, mixture of the edge states are exponentially small and negligible. Quantum Coherence between the two edge states gives the quantization of $\sigma_{xy}$. However, when the system is mesoscopic, the mixture between the edges states plays a physical role. We focus on this point and show numerical results.

Abstract:
The Hofstadter model is a simple yet powerful Hamiltonian to study quantum Hall physics in a lattice system, manifesting its essential topological states. Lattice dimerization in the Hofstadter model opens an energy gap at half filling. Here we show that even if the ensuing insulator has a Chern number equal to zero, concomitantly a doublet of edge states appear that are pinned at specific momenta. We demonstrate that these states are topologically protected by inversion symmetry in specific one-dimensional cuts in momentum space, define and calculate the corresponding invariants and identify a platform for the experimental detection of these novel topological states.

Abstract:
We study pairing effects in the edge states of paired fractional quantum Hall states by using persistent edge currents as a probe. We give the grand partition functions for edge excitations of paired states (Pfaffian, Haldane-Rezayi, 331) coupling to an Aharanov-Bohm flux and derive the exact formulas of the persistent edge current. We show that the currents are flux periodic with the unit flux $\phi_0=hc/e$. At low temperatures, they exhibit anomalous oscillations in their flux dependence. The shapes of the functions depend on the bulk topological order. They converge to the sawtooth function with period $\phi_0/2$ at zero temperature, which indicates pair condensation. This phenomenon provides an interesting bridge between superconductivity in 2+1 dimensions and superconductivity in 1+1 dimensions. We propose experiments of measuring the persistent current at even denominator plateau in single or double layer systems to test our predictions.

Abstract:
We consider (2+1)-dimensional topological quantum states which possess edge states described by a chiral (1+1)-dimensional Conformal Field Theory (CFT), such as e.g. a general quantum Hall state. We demonstrate that for such states the reduced density matrix of a finite spatial region of the gapped topological state is a thermal density matrix of the chiral edge state CFT which would appear at the spatial boundary of that region. We obtain this result by applying a physical instantaneous cut to the gapped system, and by viewing the cutting process as a sudden "quantum quench" into a CFT, using the tools of boundary conformal field theory. We thus provide a demonstration of the observation made by Li and Haldane about the relationship between the entanglement spectrum and the spectrum of a physical edge state.

Abstract:
We conjecture that the counting of the levels in the orbital entanglement spectra (OES) of finite-sized Laughlin Fractional Quantum Hall (FQH) droplets at filling $\nu=1/m$ is described by the Haldane statistics of particles in a box of finite size. This principle explains the observed deviations of the OES counting from the edge-mode conformal field theory counting and directly provides us with a topological number of the FQH states inaccessible in the thermodynamic limit- the boson compactification radius. It also suggests that the entanglement gap in the Coulomb spectrum in the conformal limit protects a universal quantity- the statistics of the state. We support our conjecture with ample numerical checks.

Abstract:
We introduce the concept of super universality in quantum Hall and spin liquids which has emerged from previous studies. It states that all the fundamental features of the quantum Hall effect are generically displayed as general topological features of the $\theta$ parameter in nonlinear sigma models in two dimensions. To establish super universality in spin liquids we revisit the mapping by Haldane who argued that the anti ferromagnetic Heisenberg spin $s$ chain is effectively described by the O(3) nonlinear sigma model with a $\theta$ term. By combining the path integral representation for the dimerized spin $s=1/2$ chain with renormalization group decimation techniques we generalise the Haldane approach to include a more complicated theory, the fermionic rotor chain, involving four different renormalization group parameters. We show how the renormalization group calculation technique can be used to lay the bridge between the fermionic rotor chain and the sigma model. As an integral and fundamental aspect of the mapping we establish the topological significance of the dangling spin at the edge of the chain which is in all respects identical to the massless chiral edge excitations in quantum Hall liquids. We consider various different geometries of the spin chain and show that for each of the different geometries correspond to a topologically equivalent quantum Hall liquid.

Abstract:
We show that every even-denominator fractional quantum Hall (FQH) state possesses at least two robust, topologically distinct gapless edge phases if charge conservation is broken at the boundary by coupling to a superconductor. The new edge phase allows for the possibility of a direct coupling between electrons and emergent neutral fermions of the FQH state. This can potentially be experimentally probed through geometric resonances in the tunneling density of states at the edge, providing a probe of fractionalized, yet electrically neutral, bulk quasiparticles. Other measurable consequences include a charge $e$ fractional Josephson effect, a charge $e/4q$ quasiparticle blocking effect in filling fraction $p/2q$ FQH states, and modified edge electron tunneling exponents.