Topological Invariants and Ground-State Wave functions of Topological Insulators on a Torus

We define topological invariants in terms of the ground-state wave functions on a torus. This approach leads to precisely defined formulas for the Hall conductance in four dimensions and the topological magnetoelectric $\theta$ term in three dimensions, and their generalizations in higher dimensions. They are valid in the presence of arbitrary many-body interactions and disorder. These topological invariants systematically generalize the two-dimensional Niu-Thouless-Wu formula and will be useful in numerical calculations of disordered topological insulators and strongly correlated topological insulators, especially fractional topological insulators.

Popular Summary

Exotic materials that are insulators in their interior but conductors at their boundary, called “topological insulators,” have been discovered quite recently. The fundamental difference between such a topological insulator and a conventional insulator lies in the different topologies of their respective electronic structures. The precise quantities that capture the topological differences are called topological invariants, not unlike the concept of “genus,” which distinguishes a ball (genus 0) from a donut (genus 1). They are therefore very powerful tools for identifying topological insulators. A major difficulty in this field is, however, that most of these topological invariants are formulated and calculated under the assumption that electrons are freely moving, namely, that the electrons do not interact with each other—an assumption that many materials do not satisfy. In this theoretical paper, we develop a new, broadly applicable approach to formulating and calculating topological invariants that goes back to the basic description of the quantum world: many-body wave functions.

In quantum mechanics, the wave function of an electronic system encodes all physical information about the system. It is then natural to expect that the wave function contains all information about the “topology” of the system’s electronic structure. The question is how to extract this information, namely, how to express topological invariants in terms of the wave function. For a two-dimensional quantum Hall insulator, the simplest example of topological insulators, Niu, Thouless, and Wu developed a formula using a technical approach based on “twisted boundary conditions,” which involves deforming the shape of the wave function in a judiciously chosen way.

By confining the space that the wave function lives in to a higher-dimensional torus (or “donut”), we have found a meaningful way to systematically generalize the earlier simple formula to new precise expressions of topological invariants for a broad class of topological insulators with strong electronic interactions, including three-dimensional topological insulators and also the theoretically interesting four-dimensional quantum Hall insulators. The underlying technical insight was that a slightly different “twisting” (from Niu, Thouless, and Wu’s) of the shape of the wave function could actually enable such generalizations.

One main advantage of our approach is that it applies to materials involving both electronic interaction of arbitrary strength and disorder of arbitrary degree. It should also be very valuable in facilitating numerical calculations of topological invariants for a broad range of topological insulators.