Nicholas Read

Entanglement subspaces, trial wave functions, and special Hamiltonians in the fractional quantum Hall effect

We consider spaces of trial wavefunctions for ground states and edge excitations in the fractional quantum Hall effect that can be obtained in various ways. In one way, functions are obtained by analyzing the entanglement of the ground state wavefunction, partitioned into two parts. In another, functions are defined by the way in which they vanish as several coordinates approach the same value, or by a projection-operator Hamiltonian that enforces those conditions. In a third way, functions are given by conformal blocks from a conformal field theory (CFT). These different spaces of functions are closely related. The use of CFT methods permits an algebraic formulation to be given for all of them. In some cases, we can prove that there is a ground state, a Hamiltonian, and a CFT such that, for any number of particles, all of these spaces are the same. For such cases, this resolves several questions and conjectures: it gives a finite-size bulk-edge correspondence, and we can use the analysis of functions to construct a projection-operator Hamiltonian that produces those functions as zero-energy states. For a model related to the N = 1 superconformal algebra, the corresponding Hamiltonian imposes vanishing properties involving only three particles; for this we determine all the wavefunctions explicitly. We do the same for a sequence of models involving the M(3,p) Virasoro minimal models that has been considered previously, using results from the literature. We exhibit the Hamiltonians for the first few cases of these. The techniques we introduce can be applied in numerous examples other than those considered here.

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.