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Recently, fractional Chern insulators (FCIs), also called fractional quantum anomalous Hall (FQAH) states, have been theoretically established in lattice systems with topological flat bands. These systems exhibit similar fractionalization…

Using the infinite density matrix renormalization group method on an infinite cylinder geometry, we characterize the $1/3$ fractional Chern insulator state in the Haldane honeycomb lattice model at $\nu=1/3$ filling of the lowest band and…

Topological insulators and their intriguing edge states can be understood in a single-particle picture and can as such be exhaustively classified. Interactions significantly complicate this picture and can lead to entirely new insulating…

We consider interacting fermions in a magnetic field on a two-dimensional lattice with the periodic boundary conditions. In order to measure the Hall current, we apply an electric potential with a compact support. Then, due to the Lorentz…

The realization of fractional Chern insulators opens up the possibility of exploring fractionally charged excitations and anyonic statistics in the absence of a magnetic field. One of the central questions is whether lattice-based systems…

We review the fermionic Chern-Simons field theory for the Fractional Quantum Hall Effect (FQHE). We show that in this field theoretic approach to the problem of interacting electrons moving in a plane in the presence of an external magnetic…

We construct a many-body quantized invariant that sharply distinguishes among two dimensional non-equilibrium driven phases of interacting fermions. This is an interacting generalization of a band-structure Floquet quasi-energy winding…

The understanding of fractional Chern insulators (FCIs) has been deeply guided by band topology and quantum geometry. Here, we introduce a real-space theoretical framework in which FCIs are understood in terms of composite bosons, local…

Fractional Chern insulators realize the remarkable physics of the fractional quantum Hall effect (FQHE) in crystalline systems with Chern bands. The lowest Landau level (LLL) is known to host the FQHE, but not all Chern bands are suitable…

We consider Mott insulators driven by periodic coherent laser radiation, using both single orbital and multi-orbital models, noting that the latter is of more interest in solid state systems. We derive general expressions for the resulting…

We describe a method for engineering local $k+1$-body interactions ($k=1,2,3$) from two-body couplings in spin-${1}{2}$ systems. When implemented in certain systems with a flat single-particle band with a unit Chern number, the resulting…

We discuss the low-energy limit of three-orbital Kondo-lattice and Hubbard models describing $t_{2g}$ orbitals on a triangular lattice near half-filling. We analyze how very flat bands with non-trivial topological character, a Chern number…

Here, we analyse two Dirac fermion species in two spatial dimensions in the presence of general quartic contact interactions. By employing functional bosonisation techniques, we demonstrate that depending on the couplings of the fermion…

Floquet engineering, the control of a quantum system by means of time-periodic driving, allows to modify the properties of the system so that it becomes described by an approximate effective time-independent Hamiltonian. However, in the…

Fractional Chern insulators (FCI) were proposed theoretically about a decade ago. These exotic states of matter are fractional quantum Hall states realized when a nearly flat Chern band is partially filled, even in the absence of an…

Recently it was shown that the topological properties of $2D$ and $3D$ topological insulators are captured by a $\mathbb{Z}_2$ chiral anomaly in the boundary field theory. It remained, however, unclear whether the anomaly survives…

The Harper-Hofstadter model provides a fractal spectrum containing topological bands of any integer Chern number, $C$. We study the many-body physics that is realized by interacting particles occupying Harper-Hofstadter bands with $|C|>1$.…

The recent progress in engineering topological band structures in optical-lattice systems makes it promising to study fractional Chern insulator states in these systems. Here we consider a realistic finite system of a few repulsively…

We predict twisted double bilayer graphene to be a versatile platform for the realization of fractional Chern insulators readily targeted by tuning the gate potential and the twist angle. Remarkably, these topologically ordered states of…

We study a generic model of a Chern insulator supplemented by a Hubbard interaction in arbitrary even dimension $D$ and demonstrate that the model remains well-defined and nontrivial in the $D \to \infty$ limit. Dynamical mean-field theory…