Topological phases: Anyonic quantum spin chains and gapped and gapless spin liquid

Topologically ordered phases of matter represent a novel form of order that does not fit the familiar Landau symmetry breaking paradigm. One type of topological phase occurs in two dimensional systems of electrons in a strong magnetic field and is known as the fractional quantum Hall state. Such fractional quantum Hall states are known to have low energy excitations with anyon exchange statistics, generalizing Bosonic and Fermionic statistics. When the ground state of the quantum Hall state populated with well-separated anyons is degenerate, the anyon statistics are non-commutative and such anyons are known as non-abelian anyons.;In particular, some of the simplest such systems are Bosonic quantum Hall states containing anyons which generalize the ordinary SU(2) quantum spin degrees of freedom. These theories are described by so-called SU(2) at level k Chern-Simons theories, well known from Witten's work on the Jones Polynomial and are called Read-Rezayi states. Here we will investigate the collective phase assumed by a chain of interacting spin --1 anyons in the SU(2) at level k=5 state, which turns out to be the simplest anyonic generalization of the ordinary bilinear-biquadratic SU(2) chain of interacting spin --1 quantum spins. Using numerical simulations to obtain the spectrum of the anyonic chain we will show that when such an anyonic spin --1 SU(2) at level 5 chain is staggered, new phase transitions appear in the phase diagram and that these phase transitions are described by conformal field theories in the universality class of the well studied conformal minimal models which we identify. Because the anyons in the SU(2) at level k phase are deformations of ordinary SU(2) spins, it will also be instructive to review the effect of staggering in ordinary SU(2) spin chains of spin 1/2 and spin--1, and to draw parallels with the staggered anyonic chain.;We will also study a model of spin 1/2 quantum spins on the Kagome lattice with an SU(2) invariant chiral three spin interaction on each triangular plaquette which breaks time-reversal and parity symmetry. In the case of uniform chirality, where each triangular plaquette has the same chirality, we demonstrate that this provides a simple local Hamiltonian whose ground state is the long sought-after Kalmeyer-Laughlin state, which is a bosonic quantum Hall state at filling factor 1/2 . We arrive at this result, which we first justify physically through heuristic arguments using a Chalker-Coddington type network model approach, by comparing numerically obtained universal characteristics of the topological phase, including entanglement properties and modular S and T matrices of our model to those known analytically for the Kalmeyer-Laughlin state. The underlying Hamiltonian is shown to arise simply from the half-filled Hubbard model on the Kagome lattice in an applied magnetic field. Therefore, these results set the stage for possible future discovery of the elusive Kalmeyer-Laughlin quantum Hall state in conventional interacting Mott-insulating electronic solid state materials.;When the chiralities of the three-spin interactions on the plaquettes of the Kagome lattice are staggered, a heuristic network model analysis as well as results from numerical work on the quasi-one dimensional system suggest that the physical system is an unusual gapless spin liquid which possesses lines in momentum space at which gapless SU(2) spin excitations, known as "spinons", reside. Many-body systems possessing such lines in momentum space which host gapless Bosonic excitations represent a poorly understood class of non-Fermi-liquids in which the gapless Fermionic excitations occurring at the Fermi surface of an ordinary Landau Fermi-liquid are replaced by a similar "Bose surface" structure of gapless Bosonic excitations in momentum space. We present arguments for the presence of this exotic non-Fermi-liquid state in the staggered Kagome spin model by (i) relating the latter to a "chiral" 2-channel Kondo lattice model which is more amenable to an analytic description than other Kondo lattice models due to the chiral nature of the Kondo interaction, and by (ii) analyzing a similar but different system built from Majorana zero modes as opposed to quantum spins.