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

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.

23 pages, 1 figure. To appear in Letters in Mathematical Physics

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Cited by in corpus (9)

• Tensor Berry connections and their topological invariants
• Spin Conductance and Spin Conductivity in Topological Insulators: Analysis of Kubo-like terms
• Parseval frames of exponentially localized magnetic Wannier functions
• Three dimensional two-band Floquet topological insulator with $Z_2$ index
• Symmetric Fermi projections and Kitaev's table: topological phases of matter in low dimensions
• Eigenvalue crossings in Floquet topological systems
• Differential geometric invariants for time-reversal symmetric Bloch-bundles II: The low dimensional "Quaternionic" case
• Gerbes in Geometry, Field Theory, and Quantisation
• Topology vs localization in synthetic dimensions