Cécile Repellin

Series of Abelian and Non-Abelian States in C>1 Fractional Chern Insulators

We report the observation of a new series of Abelian and non-Abelian topological states in fractional Chern insulators (FCI). The states appear at bosonic filling nu= k/(C+1) (k, C integers) in several lattice models, in fractionally filled bands of Chern numbers C>=1 subject to on-site Hubbard interactions. We show strong evidence that the k=1 series is Abelian while the k>1 series is non-Abelian. The energy spectrum at both groundstate filling and upon the addition of quasiholes shows a low-lying manifold of states whose total degeneracy and counting matches, at the appropriate size, that of the Fractional Quantum Hall (FQH) SU(C) (color) singlet k-clustered states (including Halperin, non-Abelian spin singlet states and their generalizations). The groundstate momenta are correctly predicted by the FQH to FCI lattice folding. However, the counting of FCI states also matches that of a spinless FQH series, preventing a clear identification just from the energy spectrum. The entanglement spectrum lends support to the identification of our states as SU(C) color-singlets but offers new anomalies in the counting for C>1, possibly related to dislocations that call for the development of new counting rules of these topological states.

Fractional Chern Insulators beyond Laughlin states

We report the first numerical observation of composite fermion (CF) states in fractional Chern insulators (FCI) using exact diagonalization. The ruby lattice Chern insulator model for both fermions and bosons exhibits a clear signature of CF states at filling factors 2/5 and 3/7 (2/3 and 3/4 for bosons). The topological properties of these states are studied through several approaches. Quasihole and quasielectron excitations in FCI display similar features as their fractional quantum hall (FQH) counterparts. The entanglement spectrum of FCI ground states shows an identical fingerprint to its FQH partner. We show that the correspondence between FCI and FQH obeys the emergent symmetry already established, proving the validity of this approach beyond the clustered states. We investigate other Chern insulator models and find similar signatures of CF states. However, some of these systems exhibit strong finite size effects.

Z 2 fractional topological insulators in two dimensions

We propose a simple microscopic model to numerically investigate the stability of a two dimensional fractional topological insulator (FTI). The simplest example of a FTI consists of two decoupled copies of a Laughlin state with opposite chiralities. We focus on bosons at half filling. We study the stability of the FTI phase upon addition of two coupling terms of different nature: an interspin interaction term, and an inversion symmetry breaking term that couples the copies at the single particle level. Using exact diagonalization and entanglement spectra, we numerically show that the FTI phase is stable against both perturbations. We compare our system to a similar bilayer fractional Chern insulator. We show evidence that the time reversal invariant system survives the introduction of interaction coupling on a larger scale than the time reversal symmetry breaking one, stressing the importance of time reversal symmetry in the FTI phase stability. We also discuss possible fractional phases beyond

Single-mode approximation for fractional Chern insulators and the fractional quantum Hall effect on the torus

\nu = 1/2

Numerical investigation of gapped edge states in fractional quantum Hall-superconductor heterostructures

.

Lattices for fractional Chern insulators

We analyze the collective magneto-roton excitations of bosonic Laughlin

Dynamics and level statistics of interacting fermions in the lowest Landau level

\nu=1/2

Phase diagram of ν=1/2+1/2 bilayer bosons with interlayer couplings with interlayer couplings

fractional quantum Hall (FQH) states on the torus and of their analog on the lattice, the fractional Chern insulators (FCIs). We show that, by applying the appropriate mapping of momentum quantum numbers between the two systems, the magneto-roton mode can be identified in FCIs and that it contains the same number of states as in the FQH case. Further, we numerically test the single mode approximation to the magneto-roton mode for both the FQH and FCI case. This proves particularly challenging for the FCI, because its eigenstates have a lower translational symmetry than the FQH states. In spite of this, we construct the FCI single-mode approximation such that it carries the same momenta as the FQH states, allowing for a direct comparison between the two systems. We show that the single-mode approximation captures well a dispersive subset of the magneto-roton excitations both for the FQH and the FCI case. We find remarkable quantitative agreement between the two systems. For example, the many-body excitation gap extrapolates to almost the same value in the thermodynamic limit.

Phase diagram ofν=12+12bilayer bosons with interlayer couplings

Fractional quantum Hall-superconductor heterostructures may provide a platform towards non-abelian topological modes beyond Majoranas. However their quantitative theoretical study remains extremely challenging. We propose and implement a numerical setup for studying edge states of fractional quantum Hall droplets with a superconducting instability. The fully gapped edges carry a topological degree of freedom that can encode quantum information protected against local perturbations. We simulate such a system numerically using exact diagonalization by restricting the calculation to the quasihole-subspace of a (time-reversal symmetric) bilayer fractional quantum Hall system of Laughlin ν = 1/3 states. We show that the edge ground states are permuted by spin-dependent flux insertion and demonstrate their fractional 6π Josephson effect, evidencing their topological nature and the Cooper pairing of fractionalized quasiparticles. The versatility and efficiency of our setup make it a well suited method to tackle wider questions of edge phases and phase transitions in fractional quantum Hall systems.Fractional quantum Hall effect: new numerical setupA numerical setup provides a full microscopic model to describe a fractional quantum Hall system coupled to superconducting leads. The fractional quantum Hall effect is a phenomenon in which the Hall conductance of 2D electrons shows characteristic quantised plateaus. Systems exhibiting this effect can host exotic topological states, some of which have potential for universal quantum computation. However, their experimental investigation is challenging. An international team comprising Cécile Repellin, Ashley Cook, Titus Neupert and Nicolas Regnault demonstrated for the first time a fully microscopic model that allows the quantitative study of such systems, and showed that it identifies the expected key signatures. These results will both enable further numerical work relying on the presented setup and provide guidance for experiments by indicating the parameter regime in which each signature can be observed.

Phase diagram of ν = 1 2 + 1 2 bilayer bosons with interlayer couplings

Topological order beyond Landau levels is seen in encapsulated bilayer graphene Individual electrons are elementary particles, but in some solid-state systems, electrons can act collectively as though they had a fraction of an electrons charge. This emergent behavior is spectacularly observed in two-dimensional (2D) electron gases as the fractional quantum Hall (FQH) effect in the form of a fractional quantized transverse (or Hall) conductivity and in shot-noise experiments. These experiments require low temperatures and very large magnetic fields in order to create strong electron interactions. This latter condition now appears not to be as essential as originally thought. On page 62 of this issue, Spanton et al. (1) report on an experimental platform based on bilayer graphene that forms a moiré pattern with an encapsulating hexagonal boron nitride layer. They observed incompressible phases with a fractional filling of the band structure with a nonzero Chern number (it has quantized properties robust to local perturbations, or topologically invariant). Some of which have no analog in traditional FQH systems (see the figure).