Opening remarks

Topological Josephson junctions and solitons with non-Abelian statistics

• Eytan Grosfeld (Ben-Gurion University)

Topological Josephson junctions and solitons with non- Abelian statistics Abstract: In recent years, chiral topological superconductors have been the subject of intensive study. These superconductors are characterized by the presence of charge-neutral Majorana edge states. Driven by prospects for applications in fault- tolerant quantum computation, Majorana fermions are ardently sought after. When a Josephson junction is formed between two topological superconductors, Josephson vortices through this junction carry a localized Majorana mode which affects their dynamics. In this talk I will discuss some of the properties of these Josephson vortices.

Transport properties of helical edge states

• Patrik Recher (TU Braunschweig)

Topological Surface States in Topological Insulators, Superconductors and Beyond

• Zahid Hasan (Princeton University)

Bulk Topological Insulators are a new phase of electronic matter which realizes a non-quantum-Hall-like topological state in the bulk matter and unlike the quantum Hall liquids can be turned into superconductors. In this talk, I will first briefly review the basic theory and experimental probes that reveal topological order. I will then discuss experimental results that demonstrate the fundamental properties of topological insulators such as spin-momentum locking, non- trivial Berry’s phases, mirror Chern number, absence of backscattering or no U-turn rule, protection by time- reversal symmetry and the existence of room temperature topological order (at the level of M.Z.H and C.L. Kane, Rev. of Mod. Phys., 82, 3045 (2010)). I will then discuss the possible exotic roles of broken symmetry phases such as superconductivity and magnetism in doped topological insulators and their potential device applications in connection to our recent results as well as briefly outline the emerging research frontiers of the field as a whole. If time permits, I will discuss experimental progress in finding topological insulators beyond Kane-Mele Z2 paradigm.

Pfaffian versus anti-pfaffian at 5/2 and 3/8

• Jainendra Jain (Penn State University)

Dirac fermions in HgTe

• Laurens Molenkamp

Bulk edge correlations and entanglement in a Maxwell-Chern-Simons model

• Israel Klich (University of Virginia)

Mapping of parent Hamiltonians: from Abelian and non-Abelian quantum Hall states to exact models of critical spin chains

• Martin Greiter (Karlsruher Institut für Technologie)

We derive an exact model for a critical spin chain with arbitrary spin S, which includes the Haldane-Shastry model as the special case S=1/2. While spinons in the Haldane--Shastry model obey abelian half-fermi statistics, the spinons in the general model introduced here obey non-abelian statistics. This manifests itself through topological choices for the fractional momentum spacings. The general model is derived by mapping exact models of quantized Hall states onto spin chains. The lecture begins with pedagogical review of the Gutzwiller wave functions for S=1/2.

The two-channel Kondo fixed point in two dimensional topological superconductor

• Ting Pong Choy (University of Leiden)

Competition between d-wave and topological p-wave superconductivity in the doped Kitaev-Heisenberg model

• Timo Hyart (Leipzig University)

The competition between Kitaev and Heisenberg interactions away from half filling is studied for the doped Kitaev-Heisenberg $t$-$J_K$-$J_H$ model on a honeycomb lattice. While the isotropic Heisenberg coupling supports a time-reversal violating d-wave singlet state, we find that the Kitaev interaction favors a time-reversal invariant p-wave superconducting phase, which obeys the rotational symmetries of the microscopic model, and is robust for $J_H<J_K/2$. Within the p-wave superconducting phase, a critical chemical potential $|\mu|=\mu_c \approx t$ separates a topologically trivial phase for $|\mu|< \mu_c$ from a topologically non-trivial $Z_2$ time-reversal invariant spin-triplet phase for $|\mu|>\mu_c$. Because the topological p-wave superconductivity in the doped Kitaev model is considerably more robust to adding a Heisenberg exchange than the Kitaev spin liquid phase itself at zero doping, we expect that this type of physics might be observable in doped honeycomb iridates, in particular in Li$_2$IrO$_3$.

Instability in magnetic materials with dynamical axion field

• Masaki Oshikawa (University of Tokyo)

In the context of string theory, an instability of axion electrodynamics in the presence of a background electric field was found. We show that this instability leads to a complete screening of an applied electric field above a certain critical value and the excess energy is converted into a magnetic field. We clarify the physical origin of the screening effect and discuss its possible experimental realization in magnetic materials where magnetic fluctuations play the role of the dynamical axion field. Journal-Ref: Phys. Rev. Lett. 108, 161803 (2012)

Signatures of topological order in Coulomb blockaded transport through semiconductor-superconductor nanowire rings

• Mats Horsdal (Leipzig University)

In semiconductor-superconductor hybrid structures a topological phase transition is expected as a function of chemical potential or magnetic field strength. We show that signatures of this transition can be observed in nonlinear Coulomb blockaded transport through a ring shaped structure. In particular, on the scale of the superconducting gap and for a fixed electron parity of the ring, the excitation spectrum is independent of flux in the topologically trivial phase but acquires a characteristic h/e-periodicity in the nontrivial phase. We relate the h/e-periodicity to the recently predicted 4?-periodicity of the Josephson current across a junction formed by two topological superconductors.

Topological liquid nucleation induced by vortex-vortex interactions in Kitaev’s honeycomb model

• Ville Lahtinen (NORDITA)

We provide a comprehensive microscopic understanding of the nucleation of topological quantum liquids, a general mechanism where interactions between non- Abelian anyons cause a transition to another topological phase, which we study in the context of Kitaev’s honeycomb lattice model. For non-Abelian vortex excitations arranged on superlattices, we observe the nucleation of several distinct Abelian topological phases whose character is found to depend on microscopic parameters such as the superlattice spacing or the spin exchange couplings. By reformulating the interacting vortex superlattice in terms of an effective model of Majorana fermion zero modes, we show that the nature of the collective many-anyon state can be fully traced back to the microscopic pairwise vortex interactions. Due to RKKY-type sign oscillations in the interactions, we find that longer-range interactions beyond nearest neighbor can influence the collective state and thus need to be included for a comprehensive picture. Corresponding results should hold for vortices forming an Abrikosov lattice in a p-wave superconductor or quasiholes forming a Wigner crystal in non-Abelian quantum Hall states.

Topological nematic states, twist defects and genons

• Xiaoliang Qi (Stanford University)

In this talk, I will discuss a general class of non-abelian statistics which can be created in topological states by introducing extrinsic defects, such as lattice dislocations or superconductor-ferromagnet domain walls in conventional quantum Hall states or topological insulators. In particular, these defects can be realized in certain lattice fractional Chern insulators named as topological nematic states. In this paper, we begin by placing these defects within the broader conceptual scheme of extrinsic twist defects associated with symmetries of the topological state. We explicitly study several classes of examples, including Z2 and Z3 twist defects, where the topological state with N twist defects can be mapped to a topological state without twist defects on a genus g / N surface. To emphasize this connection we refer to the twist defects as genons. We develop methods to compute the projective non-abelian braiding statistics of the genons, and we find the braiding is given by adiabatic modular transformations, or Dehn twists, of the topological state on the effective genus g surface. As an particularly interesting example, we find situations where the genons have quantum dimension 2 and can be used for universal topological quantum computing (TQC), while the host topological state is by itself non-universal for TQC.

Surface Majorana Fermions in Topological Superconductors

• Masatoshi Sato (Institute for Solid State Physics, University of Tokyo)

Recently, there is much interest in Majorana fermions (MFs) in topological superconductors (TSCs). MFs were originally introduced as elementary particles such neutrinos, but no elementary particles have been identified as MFs yet. Now it has been known that MFs can be realized as topologically protected surface states of TSCs. In this talk, I will report on our works of MFs in TSCs. Although spin-triplet superconductors had been known to host MFs, recently it was found that even s-wave superconducting states may support MFs [1] if the spin- orbit interaction is taken into account [2]. By combing with Zeeman magnetic fields, one finds that the system hosts chiral MFs on its surface and Majorana zero modes in a vortex, which implies the non Abelian topological order. This scheme was first established in Ref [1], and later applied to various systems. Very recently, an experimental realization of MFs based on this scheme was reported in a nanowire system [3]. In this talk, I discuss MFs of the recent discovered superconducting topological insulator (STI), CuxBi2Se3, which is expected to be an odd-parity superconductor. From the Fermi surface criterion [4], surface helical MFs have been predicted. Here we develop a theory of the tunneling spectroscopy for STIs [5]. Based on the symmetry and topological nature of parent topological insulators, we find that the MFs in the STIs have a profound structural transition in the energy dispersions .We clarify that MFs in the vicinity of the transitions give rise to robust zero bias peaks that is consistent with the recent experimental result [6] of the tunneling conductance. [1] M. Sato et al, Phys. Rev. Lett. 103, 020401 (2009); Phys. Rev. B 82, 134521 (2010). [2] M. Sato, S. Fujimoto, Phys. Rev. B 79, 094504 (2009). [3] L. Kouwenhoven, Nature, News 483, 132 (2012). [4] M. Sato, Phys. Rev. B 81, 220504(R) (2010). [5] A. Yamakage, K. Yada, M.Sato, Y. Tanaka, Phys. Rev. B85, 180509(R) (2012) [6] S.Sasaki et al, Phys. Rev. Lett. 107, 217001 (2011).

From Majorana to Parafermion quantum chains

• Kirill Shtengel (UC Riverside)

Topological lattice models in 3D and domain-wall anyons

• Curt von Keyserlingk (Rudolf Peierls Centre for Theoretical Physics)

Little is known about the kinds of topological phases that exist in 3D, or how to classify them. It therefore makes sense to investigate solvable models exhibiting topological order. In this work we study such a class of exactly solvable spin models, first put forward by Walker and Wang (2011). While these are not models of interacting fermions, they may well capture the topological behaviour of some strongly correlated systems. In this work we give a full treatment of a special case, which we call the 3D semion model: We calculate its ground state degeneracies for a variety of boundary conditions, and classify its low-lying excitations. While point defects in the bulk are confined in meson-like pairs, the surface excitations are more interesting: The model has deconfined point defects pinned to the boundary of the lattice, and these exhibit semionic statistics. The surface physics is reminiscent of a $\nu=1/2$ bosonic fractional quantum hall effect, and these considerations help motivate an effective field theoretic description for the lattice models in their topological limit based on a kind of BF theory. Our special example of the 3D semion model captures much of the behaviour of more general `confined Walker-Wang models'.

CP1 and CP2 skyrmions and stable fractional vortices in superconductors with broken time reversal symmetry

• Egor Babaev (KTH and University of Massachusetts Amherst)

Recently there was much interest in superconducting states which break time reversal symmetry. Examples being p+ip superconductors or three band superconductors with frustrated interband couplings. We will discuss that when this symmetry is broken the system posses stable or, in certain cases metastable skyrmions characterized by CP1, or in case of three band superconductors CP2 symmetry. These Skyrmions can be viewed as stable configuration of spatially separated fractional vorteces.

An unexpected turn for twisted graphenes

• Eugene Mele (University of Pennsylvania)

Twisted graphenes are a family of multilayer graphenes in which the crystallographic axes of neighboring layers are rotated by angles \theta \neq n \pi/3. Experimentally the misalignment is associated with a reduction of the energy scale for coherent interlayer electronic motion relative to its known value for Bernal stacked graphenes and for related multilayer graphenes where \theta \eq n \pi/3. The microscopic origin and nature of this reduction is attracting significant theoretical interest. In this work we focus on the physics for small rotation angles, a regime where a continuum treatment is thought to be valid, and find that rotational anisotropy in the interlayer coupling Hamiltonian plays an unexpected role by selecting one of two geometrically distinct electronic states for the coupled system. This talk will discuss the topological classification of these states and their spectral signatures.

Status of Quantum Hall Interferometry Experiments

• Steven Simon (Oxford University)

Fractional topological phases and broken time reversal symmetry in strained graphene

• Jerome Cayssol (Max Planck Institute for the Physics of Complex Systems)

Various Fractional Quantum Hall (FQH) states were recently reported in graphene under a strong magnetic field. Those FQH phases result from the effect of enhanced interactions between electrons moving in partially filled flat Landau levels, and exhibit remarkable properties (fractional quasiparticles with richer statistical properties than usual fermions or bosons, ground state degeneracy, ...). I will talk about electronic phases in strained graphene in the absence of magnetic field. The strain generates nearly uniform pseudo magnetic fields which are opposite for the two valleys of graphene. The non-interacting part of our model describes the zero magnetic field pseudo Landau level (PLL) structure recently proposed [1] and experimentally reported [2] in strained graphene. Since the reported effective magnetic fields [2] range from 60 T up to 300 T, the interaction-driven phases might conceivably be realized with larger energy gaps than in FQH states under a real magnetic field. Besides strained graphene, our results also pertain for artificial graphenes such as patterned electron gases and cold atoms in hexagonal lattices. More specifically, we have investigated the zero energy PLL at 2/3 filling [3]. In presence of the unscreened Coulomb interaction, electrons realize a 2/3 Hall state breaking time- reversal symmetry. Upon tuning the local part of the interaction, this 2/3 state can be destabilized towards a time-reversal symmetric state realizing a 1/3 Laughlin state in each valley. This state has a 9-fold ground state degeneracy and can be seen as a valley fractional topological insulator (FTI). For local attractive interactions, the 1/3+1/3 FTI has a transition towards a superconducting state. On raising the filling to the neutrality point, namely for the half-filled zero energy PLL, we find either a ferromagnet or a valley polarized state depending on the strength of the on-site interactions. [1] F. Guinea, M.I. Katsnelson, and A.K. Geim, Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering, Nat. Phys. 6, 30 (2010). [2] L. Levy et al., Strain-induced pseudomagnetic fields greater than 300 tesla in graphene nanobubbles, Science 329, 544 (2010). [3] P. Ghaemi, J. Cayssol, D. N. Sheng and A. Vishwanath, Fractional topological phases and broken time reversal symmetry in strained graphene, Phys. Rev. Lett. 108, 266801 (2012).

Heat equation approach to geometric changes in the Laughlin state

• Alexander Seidel (Washington University in St. Louis)

Local representations of the loop braid group

• Joost Slingerland (National University of Ireland Maynooth)

I will give an introduction to the "loop braid group". This group governs the topological exchange properties of ring shaped "particles" in three dimensional space. It plays a role similar to that of the braid group in 2 spatial dimensions. Loops can perform a number of nontrivial exchange motions including simple exchanges like those of point particles and "leapfrogging" like smoke rings. I will introduce the concept of a local representation of the loop braid group; in such a representation, each ring has an internal Hilbert space and exchange motions act only on the internal Hilbert spaces of the rings that are involved in the motion. Examples of such local representations come from gauge theories and are closely related to the anyons that arise in toric code models, or discrete gauge theories. I will argue that, subject to an additional condition, all local representations of the loop braid group are of this type.

Phases and phase transitions in U(1)xU(1) systems with theta-statistical interactions

• Olexei Motrunich (California Institute of Technology)

We study a $U(1)\times U(1)$ system with short-range interactions and mutual $\theta$ statistics in (2+1) dimensions, using $\theta=\pi$ and $\theta=2\pi/3$ as two examples. We are able to reformulate the model to eliminate the sign problem, and perform a Monte Carlo study. We find a phase diagram containing a phase with only small loops and two phases with one species of proliferated loop. In the intermediate coupling regime in the $\theta=2\pi/3$ case, we also find a phase where both species of loop condense, but without any gapless modes and with a quantized cross-transverse response. On the other hand, in the $\theta=\pi$ case this intermediate coupling region exhibits a first order (coexistence) segment along the self-dual line. Lastly, for $\theta=2\pi/n$ and when the energy cost of loops becomes small, we find a phase which is a condensate of bound states, each made up of $n$ particles of one species and a vortex of the other. We define several exact reformulations of the model, which allow us to precisely describe each phase in terms of gapped excitations. We propose field-theoretic descriptions of the phases and phase transitions, which are particularly interesting on the "self-dual" line where both species have identical interactions. This talk is based on two papers: 1) Scott D. Geraedts and Olexei I. Motrunich, "Monte Carlo Study of a U(1)xU(1) system with $\pi$- statistical Interaction", Phys. Rev. B. 85, 045114 (2012) (arXiv:1110.6561). 2) Scott D. Geraedts and Olexei I. Motrunich, "Phases and phase transitions in a U(1)xU(1) system with $\theta=2\pi/3$ mutual statistics", arXiv:1205.1790.

Adiabatic continuation of Fractional Chern Insulators to Fractional Quantum Hall States

• Gunnar Möller (Cavendish Laboratory)

We show how the phases of interacting particles in topological flat bands, known as fractional Chern insulators, can be adiabatically connected to incompressible fractional quantum Hall liquids in the lowest Landau-level of an externally applied magnetic field. Our approach enables a formal proof of the equality of their topological orders and furthermore, this proof robustly extends to the thermodynamic limit. For the adiabatic continuation we use the hybrid Wannier orbital basis proposed by Qi [Phys. Rev. Lett. 107}, 126803 (2011)] in order to construct interpolation Hamiltonians that provide continuous deformations between the two models. We illustrate the validity of our approach for the groundstate of bosons in the half filled Chern band of the Haldane model (C=1), showing that it is adiabatically connected to the \nu=1/2 Laughlin state of bosons in the continuum fractional quantum Hall problem. We comment on applications of this formalism to bands with C>1.

D-Algebra Structure of Topological Insulators

• Benoit Estienne (Princeton University)

In the quantum Hall effect, the density operators at different wave-vectors generally do not commute and give rise to the Girvin MacDonald Plazmann (GMP) algebra with important consequences such as ground-state center of mass degeneracy at fractional filling fraction, and W_{1 + \infty} symmetry of the filled Landau levels. We show that the natural generalization of the GMP algebra to higher dimensional topological insulators involves the concept of a D-algebra formed by using the fully anti-symmetric tensor in D-dimensions. For insulators in even dimensional space, the D-algebra is isotropic and closes for the case of constant non-Abelian F(k) ^ F(k) ... ^ F(k) connection (D-Berry curvature), and its structure factors are proportional to the D/2-Chern number. In odd dimensions, the algebra is not isotropic, contains the weak topological insulator index (layers of the topological insulator in one less dimension) and does not contain the Chern-Simons \theta form (F ^ A - 1/3 A ^ A ^ A in 3 dimensions). The possible relation to D- dimensional volume preserving diffeomorphisms and parallel transport of extended objects is also discussed.

Screening properties and scaling exponents of quantum Hall states

• Parsa Bonderson (Microsoft Station Q)

The Role of Spin in Some Fractional Quantum Hall Ground States

• Simon Davenport (University of Oxford)

In this talk we will discuss fractional quantum Hall physics occurring when the spin of the electrons is not fully polarised. This physics has been the focus of much recent theoretical work aiming to expanding the possibilities of realising non-abelian quantum Hall states in situations where the spin is non-polarised, such as the non-abelian spin singlet (NASS) state. Spin non-polarised FQHE physics has also spawned much recent experimental work on GaAs quantum well structures, with particular interest for fractions in the 2nd LL. To illustrate the ideology we shall recall some earlier experiments done by Kukushkin's group which investigates ``spin-transitions"---transitions between FQHE states having different net spin polarisations as a function of the ratio of Zeeman to Coulomb energy---in GaAs heterojunctions and for fractions in the LLL. We shall then describe some numerical work that we have conducted in order study the spin transitions from a theoretical perspective, based on composite fermion (CF) theory. Our work builds on some earlier work by Park and Jain, which we extend to encompass the negative effective field case describing filling fractions 2/3, 3/5 and 4/7. We will also present some predictions of CF theory for spin transitions in the 2nd LL at filling fractions 8/3, 13/5 and 18/7, the 8/3 case being the subject of recent experiments by Pan et al.. Time permitting, we shall discuss an ongoing project which is to investigate the theoretical possibility of engineering the NASS state at filling factor 4/7 in the LLL or 18/7 in the 2nd LL.

Fractional Quantum Hall Effect of Lattice Bosons Near Commensurate Flux

• Layla Hormozi (Penn State University)

We study interacting bosons on a lattice in a magnetic field. When the number of flux quanta per plaquette is close to a rational fraction, the low energy physics is mapped to a multi-species continuum model: bosons in the lowest Landau level where each boson is given an internal degree of freedom, or pseudospin. We find that the interaction potential between the bosons involves terms that do not conserve pseudospin, corresponding to umklapp processes, which in some cases can also be seen as BCS-type pairing terms. We argue that in experimentally realistic regimes for bosonic atoms in optical lattices with synthetic magnetic fields, these terms are crucial for determining the nature of allowed ground states. In particular, we show numerically that certain paired wavefunctions related to the Moore-Read Pfaffian state are stabilized by these terms, whereas certain other wavefunctions can be destabilized when umklapp processes become strong.

Topological phases and phase transitions in a two-dimensional fermionic lattice: Magnetic field versus spin-orbit coupling

• Wouter BEUGELING (Utrecht University)

Topological states of matter are characterised by a topological invariant, which is protected against disorder effects. In two-dimensional systems, a perpendicular magnetic field can induce the quantum Hall effect, where the Hall conductivity is quantised and protected because it is carried by chiral edge currents. Systems with large spin- orbit coupling in absence of a magnetic field exhibit the quantum spin Hall effect, where the protected quantity is the spin Hall conductivity, carried by helical edge states. In this talk, I address the combined effect of a magnetic field and spin-orbit coupling within a tight-binding model of electrons in a honeycomb lattice. First, I will explore the effects of the intrinsic and Rashba spin-orbit and Zeeman terms individually. Secondly, I will discuss the subtle competitions that arise if these terms are combined. In particular, I will discuss the topological phase transitions induced by variation of the intrinsic spin-orbit coupling. I will conclude by exploring the candidates for experimental realisation of this model: Graphene, artificial honeycomb lattices, and cold atoms in optical lattices. References: * N. Goldman, W. Beugeling, and C. Morais Smith, EPL 97, 23003 (2012). * W. Beugeling, N. Goldman, and C. Morais Smith, arXiv:1204.2212.

Conformal Field Theory of Composite Fermions in the Quantum Hall Effect

• Andrea Cappelli (INFN)

The Jain construction of so-called hierarchical Hall states and the associated picture of composite fermion are reconsidered in the light of recent analyses that have found an exact relation between Jain wavefunctions and conformal field theory correlators. We show that the underlying conformal theory is actually given by the W-infinity minimal models introduced earlier and argue that the composite fermion excitations may obey non- Abelian fractional statistics.

Fractional Topogical Insulators

• Nicolas Regnault (Princeton University)

The Renyi Entropy and the Multifractal Spectrum of Systems Near the Localization Transition

• Eduardo Fradkin (University of Illinois, Urbana-Champaign)

We show that the Renyi entropies of single particle, critical wave functions for disordered systems contain information about the multifractal spectrum. It is shown for moments of the Renyi entropy, S_n, where |n|<1, it is possible to extract universal information about the multifractility of such systems. This is shown through a generic calculation and then illustrated through two example models. We nd good agreement between our analytic formula and numerical simulations of the two test models. Our formalism is easily extendable to generic non-interacting fermion models. It is also suggested that recent experimental advances in measuring the multifractal spectrum might allow some moments of the Renyi entropy to be measured.

Quantum Hall Transitions and Quantum Number Fractionalization in Trapped Cold Atom Systems

• Kun Yang (Florida State University)

Recently there have been experimental attempts to realize quantum Hall physics in trapped cold atom systems, either through rotation or synthetic gauge fields. In this talk I will discuss possible quantum phase transitions between integer and fractional quantum Hall states, driven by attractive interactions between fermionic atoms. Such transitions have no counterparts in electronic quantum Hall liquids, but are related to fractionalization transitions studied in other strongly correlated systems. In one of these examples charge fractionalization is associated with the confinement- deconfinement transition of the (2+1D) Z2 gauge theory, which is in the Ising universality class.

Trial states for the quantum Hall effect: entanglement spectrum and field theory

• Jerome Dubail (Yale University)

A celebrated theoretical approach to the Fractional Quantum Hall Effect consists in cooking up trial wavefunctions for various filling fractions. Some specific trial states (the ones given by conformal blocks) have particularly nice properties. Physically, they describe exotic phases of matter, for instance with non- abelian excitations. On the technical side, they bear some striking resemblance with Tensor Product States. The purpose of this talk is to explain how the edge excitations and the entanglement spectrum of these states both arise naturally from their underlying CFT structure.

Hall viscosity of quantum fluids

• Nicholas Read (Yale University)

Fractional Chern Insulators -- aspects beyond conventional Landau level physics

• Emil Bergholtz (FU Berlin)

A large number of recent work points to the emergence of intriguing analogues of fractional quantum Hall states in two-dimensional lattice models due to effective interactions in nearly flat bands with Chern number C=1. In principle, this opens up a number of intriguing possibilities including lattice realizations of quantum Hall phenomena and anyons at room temperature. Here, I will discuss three related topics. First, I will discuss for which filling fractions there is numerical evidence for fractional Chern insulators (incompressible states), including new evidence for several abelian hierarchy of states, and also point out some important differences compared to conventional Landau level physics and the emergence of competing compressible phases. Second, I will introduce a minimal solvable model describing a “thin-torus” version of a fractional Chern insulator. Third, I will discuss the physics in flat bands with higher Chern number and possible realizations thereof in quasi- two-dimensional slabs of pyrochlore based transition metal oxides as well as correlated states within these bands.