Abstract:
Edge excitations of a fractional quantum Hall system can be derived as surface excitations of an incompressible quantum droplet using one dimensional chiral bosonization. Here we show that an analogous approach can be developed to characterize surface states of three-dimensional time reversal invariant topological insulators. The key ingredient of our theory is the Luther's multidimensional bosonization construction.

Abstract:
Effective field theories that describes the dynamics of a conserved U(1) current in terms of "hydrodynamic" degrees of freedom of topological phases in condensed matter are discussed in general dimension $D=d+1$ using the functional bosonization technique. For non-interacting topological insulators (superconductors) with a conserved U(1) charge and characterized by an integer topological invariant [more specifically, they are topological insulators in the complex symmetry classes (class A and AIII) and in the "primary series" of topological insulators in the eight real symmetry classes], we derive the BF-type topological field theories supplemented with the Chern-Simons (when $D$ is odd) or the $\theta$-term (when $D$ is even). For topological insulators characterized by a $\mathbb{Z}_2$ topological invariant (the first and second descendants of the primary series), their topological field theories are obtained by dimensional reduction. Building on this effective field theory description for non-interacting topological phases, we also discuss, following the spirit of the parton construction of the fractional quantum Hall effect by Block and Wen, the putative "fractional" topological insulators and their possible effective field theories, and use them to determine the physical properties of these non-trivial quantum phases.

Abstract:
We review various features of interacting Abelian topological phases of matter in two spatial dimensions, placing particular emphasis on fractional Chern insulators (FCIs) and fractional topological insulators (FTIs). We highlight aspects of these systems that challenge the intuition developed from quantum Hall physics - for instance, FCIs are stable in the limit where the interaction energy scale is much larger than the band gap, and FTIs can possess fractionalized excitations in the bulk despite the absence of gapless edge modes.

Abstract:
We analyze generalizations of two dimensional topological insulators which can be realized in interacting, time reversal invariant electron systems. These states, which we call fractional topological insulators, contain excitations with fractional charge and statistics in addition to protected edge modes. In the case of s^z conserving toy models, we show that a system is a fractional topological insulator if and only if \sigma_{sH}/e^* is odd, where \sigma_{sH} is the spin-Hall conductance in units of e/2\pi, and e^* is the elementary charge in units of e.

Abstract:
We propose a $\mathbb{Z}_{2}$ classification of Abelian time-reversal fractional topological insulators in terms of the composite fermions picture. We consider the standard toy model where spin up and down electrons are subjected to opposite magnetic fields and only electrons of the same spin interact via a repulsive force. By applying the composite fermions approach to this time-reversal symmetric system, we are able to obtain a hierarchy of topological insulators with spin Hall conductance $\sigma_{s}=\frac{e}{2\pi}\frac{p}{2mp+1} $, being $p,m \in\mathbb{N}$. They show stable edge states only for odd $p$, as a direct consequence of the Kramer's theorem.

Abstract:
Topological insulators can be generally defined by a topological field theory with an axion angle theta of 0 or pi. In this work, we introduce the concept of fractional topological insulator defined by a fractional axion angle and show that it can be consistent with time reversal (T) invariance if ground state degeneracies are present. The fractional axion angle can be measured experimentally by the quantized fractional bulk magnetoelectric polarization P_3, and a `halved' fractional quantum Hall effect on the surface with Hall conductance of the form (p/q)(e^2/2h) with p,q odd. In the simplest of these states the electron behaves as a bound state of three fractionally charged `quarks' coupled to a deconfined non-Abelian SU(3) `color' gauge field, where the fractional charge of the quarks changes the quantization condition of P_3 and allows fractional values consistent with T-invariance.

Abstract:
In this paper we construct fully symmetric wavefunctions for the spin-polarized fractional Chern insulators (FCI) and time-reversal-invariant fractional topological insulators (FTI) in two dimensions using the parton approach. We show that the lattice symmetry gives rise to many different FCI and FTI phases even with the same filling fraction $\nu$ (and the same quantized Hall conductance $\sigma_{xy}$ in FCI case). They have different symmetry-protected topological orders, which are characterized by different projective symmetry groups. We mainly focus on FCI phases which are realized in a partially filled band with Chern number one. The low-energy gauge groups of a generic $\sigma_{xy}=1/m\cdot e^2/h$ FCI wavefunctions can be either $SU(m)$ or the discrete group $Z_m$, and in the latter case the associated low-energy physics are described by Chern-Simons-Higgs theories. We use our construction to compute the ground state degeneracy. Examples of FCI/FTI wavefunctions on honeycomb lattice and checkerboard lattice are explicitly given. Possible non-Abelian FCI phases which may be realized in a partially filled band with Chern number two are discussed. Generic FTI wavefunctions in the absence of spin conservation are also presented whose low-energy gauge groups can be either $SU(m)\times SU(m)$ or $Z_m\times Z_m$. The constructed wavefunctions also set up the framework for future variational Monte Carlo simulations.

Abstract:
We propose a systematical approach to construct generic fractional quantum anomalous Hall (FQAH) states, which are generalizations of the fractional quantum Hall states to lattice models with zero net magnetic field and full lattice translation symmetry. Local and translationally invariant Hamiltonians can also be constructed, for which the proposed states are unique ground states. Our result demonstrates that generic chiral topologically ordered states can be realized in lattice models, without requiring magnetic translation symmetry and Landau level structure. We further generalize our approach to the time-reversal invariant analog of fractional quantum Hall states--fractional topological insulators, and provide the first explicit wavefunction description of fractional topological insulators in the absence of spin conservation.

Abstract:
Fractional topological insulators are electronic systems that carry fractionally charged excitations, conserve charge and are symmetric to reversal of time. In this review we introduce the basic essential concepts of the field, and then survey theoretical understanding of fractional topological insulators in two and three dimensions. In between, we discuss the case of "two and a half dimensions", the fractional topological insulators that may form on the two dimensional surface of an unfractionalized three dimensional topological insulator. We focus on electronic systems and emphasize properties of edges and surfaces, most notably the stability of gapless edge modes to perturbations.

Abstract:
Topological insulators are characterized by the presence of gapless surface modes protected by time-reversal symmetry. In three space dimensions the magnetoelectric response is described in terms of a bulk theta term for the electromagnetic field. Here we construct theoretical examples of such phases that cannot be smoothly connected to any band insulator. Such correlated topological insulators admit the possibility of fractional magnetoelectric response described by fractional theta/pi. We show that fractional theta/pi is only possible in a gapped time reversal invariant system of bosons or fermions if the system also has deconfined fractional excitations and associated degenerate ground states on topologically non-trivial spaces. We illustrate this result with a concrete example of a time reversal symmetric topological insulator of correlated bosons with theta = pi/4. Extensions to electronic fractional topological insulators are briefly described.