Gauge-theoretic invariants for topological insulators: A bridge between Berry, Wess-Zumino, and Fu-Kane-Mele

TL;DR

This paper connects two geometric approaches to understanding the Fu-Kane-Mele invariant in topological insulators, linking Berry phase methods with bundle gerbe techniques through explicit formulas and homotopy theory.

Contribution

It establishes a direct link between Berry connection-based and bundle gerbe-based formulas for the Fu-Kane-Mele invariant, providing a new geometric perspective.

Findings

Proves the equality between Wess-Zumino amplitude and Berry phase.

Provides an equivariant Polyakov-Wiegmann formula for $U(N)$-valued fields.

Circumvents bundle gerbe language using basic homotopy theory.

Abstract

We establish a connection between two recently-proposed approaches to the understanding of the geometric origin of the Fu-Kane-Mele invariant $\mathrm{FKM} \in \mathbb{Z}_2$, arising in the context of 2-dimensional time-reversal symmetric topological insulators. On the one hand, the $\mathbb{Z}_2$ invariant can be formulated in terms of the Berry connection and the Berry curvature of the Bloch bundle of occupied states over the Brillouin torus. On the other, using techniques from the theory of bundle gerbes it is possible to provide an expression for $\mathrm{FKM}$ containing the square root of the Wess-Zumino amplitude for a certain $U(N)$-valued field over the Brillouin torus. We link the two formulas by showing directly the equality between the above mentioned Wess-Zumino amplitude and the Berry phase, as well as between their square roots. An essential tool of independent interest is an equivariant version of the adjoint Polyakov-Wiegmann formula for fields $\mathbb{T}^2 \to U(N)$, of which we provide a proof employing only basic homotopy theory and circumventing the language of bundle gerbes.