Session W3: Invited Session: Fractional Topological Insulators

Fractional Topological Insulators in 2 and 3 dimensions

I will present a family of exactly solvable models whose low energy physics is that of a 3D topological band insulator of fractionally charged fermions. When time reversal is broken at the surface, these insulators display a fractional magnetoelectric effect, leading to fractional quantum Hall surface states. Further, some -- but not all -- of them can be shown to be genuine topological insulators, whose gapless surface states are protected by time reversal. This gives an explicit construction of fractional topological insulators in 3D. This work has been done in collaboration with Michael Levin (University of Maryland), Maciej Koch-Janusz (Weizmann Institute), and Ady Stern (Weizmann Institute).   

Composite fermions for fractionally filled Chern bands

We consider fractionally filled bands with a non-zero Chern index that exhibit the Fractional Quantum Hall Effect~in zero external field\footnote{R. Roy and S. Sondhi, \textit{Physics }\textbf{4}, 46 (2011) and papers reviewed therein.} a possibility supported by numerical work.\footnote{Ibid.} Analytic treatments are complicated by a non-constant Berry flux and the absence of Composite Fermions (CF), which would not only single out preferred fractions, but also allow us compute numerous response functions at nonzero frequencies, wavelengths and temperature using either Chern-Simons field theory or our Hamiltonian formalism.\footnote{G. Murthy and R. Shankar, Rev. Mod. Phys., \textbf{75}, 1101, (2003)} We describe a way to introduce CF's by embedding the Chern band in an auxiliary problem involving Landau levels. The embedded band can be designed to approximate a prescribed Chern density in k space which determines the commutation relations of the charge densities and hence preserve all dynamical and algebraic aspects of the original problem. We find some states for which the filling fraction and dimensionless Hall conductance are not equal. The approach extends to two-dimensional time-reversal invariant topological insulators and to composite bosons.   

Fractional Topological Insulators

The prediction and experimental discovery of topological band insulators and topological superconductors are recent examples of how topology can characterize phases of matter. In these examples, electronic interactions do not play a fundamental role. In this talk I shall discuss cases where interactions lead to new phases of matter of topological character. Specifically, I shall discuss fractional topological states in lattice models which occur when interacting electrons propagate on flattened Bloch bands with non-zero Chern number. Topologically ordered many-particle states can emerge when these bands are partially filled, including a possible realization of the fractional quantum Hall effect without external magnetic fields. I shall also ponder on the possible practical applications, beyond metrology, that the quantized charge Hall effect might have if it could be realized at high temperatures and without external magnetic fields in strongly correlated materials.   

Correlated topological insulators and the fractional magnetoelectric effect

I will describe the recent theoretical construction of electronic phases in 3d that combine the physics of electron fractionalization with that of topological insulators. Called fractional topological insulators, these states of matter host protected surface states and fractionally charged quasiparticle excitations. I will then discuss the emergent gauge theory description of these phases with an emphasis on the crucial role of deconfinement at low energies. I will also describe a wide variety of experimental signatures of fractional topological insulators as well as suggesting directions for experimental searches.