Quantum spin liquids have fascinated condensed matter physicists for decades because of their unusual properties such as spin fractionalization and long-range entanglement. Unlike conventional symmetry breaking the topological order underlying quantum spin liquids is hard to detect experimentally. Even theoretical models are scarce for which the ground state is established to be a quantum spin liquid. The Kitaev honeycomb model and its generalizations to other tri-coordinated lattices are chief counterexamples -they are exactly solvable, harbor a variety of quantum spin liquid phases, and are also relevant for certain transition metal compounds including the polymorphs of (Na,Li)2IrO3 Iridates and RuCl3. In this review, we give an overview of the rich physics of the Kitaev model, including 2D and 3D fractionalization as well as dynamical correlations and behavior at finite temperatures. We discuss the different materials, and argue how the Kitaev model physics can be relevant even though most materials show magnetic ordering at low temperatures.

Frustrated quantum magnets can harbor unconventional spin-liquid ground states in which the elementary magnetic moments fractionalize into new emergent degrees of freedom. While the fractionalization of quantum numbers is one of the recurring themes in modern condensed matter physics, it often remains a challenge to devise a controlled analytical framework tracking this phenomenon. A notable exception is the exactly solvable Kitaev model, in which spin degrees of freedom fractionalize into Majorana fermions and a Z2 gauge field. Here we discuss the physics of fractionalization in three-dimensional Kitaev models and demonstrate that the itinerant Majorana fermions generically form a (semi)metal which, depending on the underlying lattice structure, exhibits Majorana Fermi surfaces, nodal lines, or topologically protected Weyl nodes. We show that the nature of these Majorana metals can be deduced from an elementary symmetry analysis of the projective time-reversal and inversion symmetries for a given lattice. This allows us to comprehensively classify the gapless spin liquids of Kitaev models for the most elementary tricoordinated lattices in three dimensions. We further expand this classification by addressing the effects of time-reversal symmetry breaking and additional interactions.

Li and Haldane conjectured and numerically substantiated that the entanglement spectrum of the reduced density matrix of ground-states of time-reversal breaking topological phases (fractional quantum Hall states) contains information about the counting of their edge modes when the groundstate is cut in two spatially distinct regions and one of the regions is traced out. We analytically substantiate this conjecture for a series of FQH states defined as unique zero modes of pseudopotential Hamiltonians by finding a one to one map between the thermodynamic limit counting of two different entanglement spectra: the particle entanglement spectrum (PES), whose counting of eigenvalues for each good quantum number is identical to the counting of bulk quasiholes (up to accidental zero eigenvalues of the reduced density matrix), and the orbital entanglement spectrum (OES), considered by Li and Haldane. By using a set of clustering operators which have their origin in conformal field theory (CFT) operator expansions, we show that the counting of the OES eigenvalues in the thermodynamic limit must be identical to the counting of quasiholes in the bulk . The latter equals the counting of edge modes at a hard-wall boundary placed on the sample. Our results can be interpreted as a bulk-edge correspondence in entanglement spectra. Moreover, we show that the counting of the PES and OES is identical even for CFT states which are likely bulk gapless, such as the Gaffnian wavefunction.

The physics of spin-orbital entanglement in effective j = 1/2 Mott insulators, which have been experimentally observed for various 5d transition metal oxides, has sparked an interest in Heisenberg-Kitaev (HK) models thought to capture their essential microscopic interactions. Here we argue that the recently synthesized Ba3IrTi2O9 is a prime candidate for a microscopic realization of the triangular HK model -a conceptually interesting model for its interplay of geometric and exchange frustration. We establish that an infinitesimal Kitaev exchange destabilizes the 120 • order of the quantum Heisenberg model. This results in the formation of an extended Z2-vortex crystal phase in the parameter regime most likely relevant to the real material, which can be experimentally identified with spherical neutron polarimetry. Moreover, using a combination of analytical and numerical techniques we map out the entire phase diagram of the model, which further includes various ordered phases as well as an extended nematic phase around the antiferromagnetic Kitaev point.

The fractional quantum Hall effect, being one of the most studied phenomena in condensed matter physics during the past thirty years, has generated many groundbreaking new ideas and concepts. Very early on it was realized that the zoo of emerging states of matter would need to be understood in a systematic manner. The first attempts to do this, by Haldane and Halperin, set an agenda for further work which has continued to this day. Since that time the idea of hierarchies of quasiparticles condensing to form new states has been a pillar of our understanding of fractional quantum Hall physics. In the thirty years that have passed since then, a number of new directions of thought have advanced our understanding of fractional quantum Hall states, and have extended it in new and unexpected ways. Among these directions is the extensive use of topological quantum field theories and conformal field theories, the application of the ideas of composite bosons and fermions, and the study of nonabelian quantum Hall liquids. This article aims to present a comprehensive overview of this field, including the most recent developments.arXiv:1601.01697v2 [cond-mat.str-el]