Topological field theory of time-reversal invariant insulators

TLDR

Topological properties generalize the time‑reversal‑breaking quantum Hall insulator from 2+1 dimensions. The study aims to present a general classification of all topological insulators across dimensions using a unified topological Chern‑Simons field theory in phase space. The authors derive lower‑dimensional TRI insulators via dimensional reduction from a 4+1‑dimensional fundamental TRI insulator, employing a unified topological Chern‑Simons field theory in phase space. They show that a 4+1‑dimensional fundamental TRI insulator is described by a Chern‑Simons field theory classified by the second Chern number, and that dimensional reduction yields the 2+1‑ and 3+1‑dimensional TRI insulators with Z2 classification, fully accounting for measurable topological responses such as the quantized topological magnetoelectric effect.

Abstract

We show that the fundamental time-reversal invariant (TRI) insulator exists in $4+1$ dimensions, where the effective-field theory is described by the $(4+1)$-dimensional Chern-Simons theory and the topological properties of the electronic structure are classified by the second Chern number. These topological properties are the natural generalizations of the time reversal-breaking quantum Hall insulator in $2+1$ dimensions. The TRI quantum spin Hall insulator in $2+1$ dimensions and the topological insulator in $3+1$ dimensions can be obtained as descendants from the fundamental TRI insulator in $4+1$ dimensions through a dimensional reduction procedure. The effective topological field theory and the ${Z}_{2}$ topological classification for the TRI insulators in $2+1$ and $3+1$ dimensions are naturally obtained from this procedure. All physically measurable topological response functions of the TRI insulators are completely described by the effective topological field theory. Our effective topological field theory predicts a number of measurable phenomena, the most striking of which is the topological magnetoelectric effect, where an electric field generates a topological contribution to the magnetization in the same direction, with a universal constant of proportionality quantized in odd multiples of the fine-structure constant $\ensuremath{\alpha}={e}^{2}∕\ensuremath{\hbar}c$. Finally, we present a general classification of all topological insulators in various dimensions and describe them in terms of a unified topological Chern-Simons field theory in phase space.