Fenner Harper

Abelian Floquet symmetry-protected topological phases in one dimension

Time-dependent systems have recently been shown to support novel types of topological order that cannot be realised in static systems. In this paper, we consider a range of time-dependent, interacting systems in one dimension that are protected by an Abelian symmetry group. We classify the distinct topological phases that can exist in this setting and find that they may be described by a bulk invariant associated with the unitary evolution of the closed system. In the open system, nontrivial phases correspond to the appearance of edge modes in the many-body quasienergy spectrum, which relate to the bulk invariant through a form of bulk-edge correspondence. We introduce simple models which realise nontrivial dynamical phases in a number of cases, and outline a loop construction that can be used to generate such phases more generally.

Periodic table for Floquet topological insulators

Periodically driven systems have recently been shown to host topological phases that are inherently dynamical in character, opening up a new arena in which to explore topological physics. One important group of such phases, known as Floquet topological insulators, arise in systems of free fermions and exhibit protected topological edge modes analogous to the edge modes of static topological insulators. In this work, the authors use methods from K theory to provide a complete topological classification of Floquet topological phases of this kind. The main result is a periodic table for Floquet topological insulators, which may be viewed as a time-dependent extension of the periodic table of topological insulators and superconductors originally introduced by Alexei Kitaev.

Floquet topological order in interacting systems of bosons and fermions

Periodically driven noninteracting systems may exhibit anomalous chiral edge modes, despite hosting bands with trivial topology. We find that these drives have surprising many-body analogs, corresponding to class A, which exhibit anomalous charge and information transport at the boundary. Drives of this form are applicable to generic systems of bosons, fermions, and spins, and may be characterized by the anomalous unitary operator that acts at the edge of an open system. We find that these operators are robust to all local perturbations and may be classified by a pair of coprime integers. This defines a notion of dynamical topological order that may be applied to general time-dependent systems, including many-body localized phases or time crystals.

Perturbative Approach to Flat Chern Bands in the Hofstadter Model

per plaquette is close to a rational fraction. Within this approximation, certain eigenstates of the system are shown to be multi-component wavefunctions that connect smoothly to the Landau levels of the continuum. The perturbative corrections to these are higher Landau level contributions that break rotational invariance and allow the perturbed states to adopt the symmetry of the lattice. In the presence of interactions, this approach allows for the calculation of generalised Haldane pseudopotentials, and in turn, the many-body properties of the system. The method is suciently general that it can apply to a wide variety of lattices, interactions, and magnetic eld strengths.

Fractional Chern insulators in bands with zero Berry curvature

Even if a noninteracting system has zero Berry curvature everywhere in the Brillouin zone, it is possible to introduce interactions that stabilize a fractional Chern insulator. These interactions necessarily break time-reversal symmetry (either spontaneously or explicitly) and have the effect of altering the underlying band structure. We outline a number of ways in which this may be achieved and show how similar interactions may also be used to create a (time-reversal-symmetric) fractional topological insulator. While our approach is rigorous in the limit of long-range interactions, we show numerically that even for short-range interactions a fractional Chern insulator can be stabilized in a band with zero Berry curvature.

Comment on "Elementary formula for the Hall conductivity of interacting systems"

In a recent paper by Neupert, Santos, Chamon, and Mudry [Phys. Rev. B 86, 165133 (2012)] it is claimed that there is an elementary formula for the Hall conductivity of fractional Chern insulators. We show that the proposed formula cannot generally be correct, and we suggest one possible source of the error. Our reasoning can be generalized to show no quantity (such as Hall conductivity) expected to be constant throughout an entire phase of matter can possibly be given as the expectation of any time independent short ranged operator.