Simplified Topological Invariants for Interacting Insulators

We propose general topological order parameters for interacting insulators in terms of the Green’s function at zero frequency. They provide a unified description of various interacting topological insulators including the quantum anomalous Hall insulators and the time-reversal-invariant insulators in four, three, and two dimensions. Since only the Green’s function at zero frequency is used, these topological order parameters can be evaluated efficiently by most numerical and analytical algorithms for strongly interacting systems.

Popular Summary

In the world of geometry topological invariants are elegant, powerful concepts that define “parallel universes” of geometrical shapes that never intersect. A sphere whose genus (a topological invariant) is zero can expand or shrink, or deform into an ellipse, for example, but it can never take on the shape of a teapot—whose genus is one—through smooth deformations. It turns out that topological invariants are equally powerful when it comes to classifying and understanding topological insulators—a class of materials that have, with their many exotic properties, captured the interest and imagination of a large swath of the condensed matter physics community. Many topological invariants for insulators which contain effectively noninteracting charged particles have already been identified. For insulators in which charged particles interact, however, a general framework that allows for both effective identification and efficient computation of topological invariants is still lacking. In this paper, we provide such a framework along with several topological invariants identified.

Our framework is actually built upon a mature theoretical tool used in a number of fields of physics, the Green’s function method. For any topological insulator that has a periodic lattice structure, it becomes more convenient to speak of electronic states and their energies (i.e., the electronic structure) as functions of momentum reciprocal to real-space position. The topology that we speak of is, therefore, the topology of relevant electronic properties (functions) in the momentum space. The Green’s function we use contains essential information on how an electron propagates in the presence of other electrons interacting with it. In principle, the function encodes in it the electronic structure, and thus the basic topology of the structure. In practice, extracting from it topological properties of insulators with interacting electrons is actually a very difficult task: The function, being a description of electron dynamics, depends naturally on both momentum and frequency and is difficult to obtain in its most general form when electron-electron interactions are present. We have been able to circumvent this difficulty, however, by realizing, and utilizing, a number of special properties of the Green’s function. The result is a number of simple topological invariants that can be defined solely in terms of the Green’s function at zero frequency only, which is much more easily obtained than the general form. They include the topological invariants for two-dimensional quantum Hall insulators, and for topological insulators with time-reversal symmetry in two and three dimensions.

This work has great potential to facilitate discoveries and identifications of new topological insulators with strong many-particle interactions, one of the major goals in the field of topological insulators.