Session W2: Invited Session: Theory of Interacting Topological Insulators

Simplified topological invariants for interacting insulators and superconductors

Topological invariants are precise mathematical tools characterizing the topological properties of topological insulators and superconductors. While many simple and powerful topological invariants for noninteracting insulators and superconductors have been well established, the topological invariants for interacting systems are much less investigated, despite of their great importance in studies of topological states in interacting systems. In this talk I will report some recent progress in the search of topological invariants for interacting systems. I will show that topological invariants defined in terms of zero frequency Green's function are precise and convenient tools for interacting topological insulators and superconductors. They have much simpler forms compared to earlier interacting topological invariants, and have the potential to facilitate discoveries of new topological insulators with strong electron-electron interaction.   

Interaction effects on 3D topological insulators and semi-metals

We discuss the effects of interactions on 3D Z2 topological insulators and related phases such as axion insulators, Weyl semi-metals and topological Mott insulators. Our analysis is motivated by the pyrochlore iridates but is of general scope. We begin by studying the effects of interactions on topological phases adiabatically connected to non-interacting Hamiltonians using both regular and dynamical mean field theories. Both the bulk and boundary topological signatures are analyzed. We then move to stronger interactions where a Mott transition from a topological insulator to a fractionalized topological Mott insulator can occur. We discuss the effects of gauge fluctuations on the transition and the resulting spin liquid.   

Topological Insulator Materials with Strong Interaction

All kinds of topological insulator materials have recently been discovered in two-and three-dimensional systems with strong spin-orbit coupling (SOC) hosting helical gapless edge or surface states consisting of odd number of Dirac fermion states inside the bulk band gap. Most of these discovered topological insulators have negligible interaction. Here we theoretically predict a new class of topological insulators with strong interaction. The typical examples are PuTe and AmN, with a simple rocksalt structure, which lie on the boundary between metals and insulators. We show that the interaction can effectively enhance SOC and drives a quantum phase transition to the topological insulator phase with a single Dirac cone on the surface (001). In addition, this kind of compounds has fully or partly filled f states, which could exhibit all kinds of magnetic phases, potentially leads to the discovery of intrinsic quantum anomalous Hall effect (QAHE) and topological magnetic insulators with dynamic axion field.   

Interacting topological phases and quantum anomalies

Since the quantum Hall effect, the notion of topological phases of matter has been extended to those that are well-defined (or: ``protected'') in the presence of a certain set of symmetries, and that exist in dimensions higher than two. In the (fractional) quantum Hall effects (and in ``chiral'' topological phases in general), Laughlin's thought experiment provides a key insight into their topological characterization; it shows a close connection between topological phases and {\it quantum anomalies}. Compared to genuine topological phases, symmetry protected topological phases are more fragile and less entangled states of matter, and hence for their characterization we need to sharpen our understanding on how topological properties of the systems manifest themselves in the form of a quantum anomaly. By taking various kinds of symmetry protected topological phases as an example, I will demonstrate that quantum anomalies serve as a useful tool to diagnose (and even define) topological properties of the systems. I will also discuss quantum anomalies play an essential role when developing descriptions of these topological phases in terms bulk and boundary (effective) theories.   

Braiding statistics approach to symmetry-protected topological phases

Symmetry-protected topological (SPT) phases can be thought of as generalizations of topological insulators. Just as topological insulators have robust gapless boundary modes protected by time reversal and charge conservation symmetry, SPT phases have boundary modes protected by more general symmetries. In this talk, I will describe a method for analyzing 2D SPT phases using braiding statistics. I will present this approach in the context of a simple example: a 2D Ising paramagnet with gapless edge modes protected by Ising symmetry. First, I will show that if the paramagnet is coupled to a $Z_2$ gauge field, the resulting $\pi$-flux excitations have different braiding statistics from that of a usual Ising paramagnet. This result provides a simple proof that the spin model belongs to a distinct quantum phase from a conventional paramagnet. Second, I will show that the $\pi$-flux braiding statistics directly imply the existence of protected edge modes. I will argue that this analysis can be generalized to any 2D SPT phase with unitary symmetries.\\[4pt] [1] M. Levin and Z.-C. Gu, Phys. Rev. B 86, 115109 (2012)