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

Gauge-theoretic invariants for topological insulators: a bridge between Berry, Wess–Zumino, and... We establish a connection between two recently proposed approaches to the understanding of the geometric origin of the Fu–Kane–Mele invariant $$\mathrm {FKM}\in \mathbb {Z}_2$$ FKM ∈ Z 2 , arising in the context of two-dimensional time-reversal symmetric topological insulators. On the one hand, the $$\mathbb {Z}_2$$ 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 $$\mathrm {FKM}$$ 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 $$\mathbb {T}^2 \rightarrow U(N)$$ T 2 → U ( N ) , of which we provide a proof employing only basic homotopy theory and circumventing the language of bundle gerbes. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Letters in Mathematical Physics Springer Journals

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

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Publisher
Springer Netherlands
Copyright
Copyright © 2017 by Springer Science+Business Media Dordrecht
Subject
Physics; Theoretical, Mathematical and Computational Physics; Complex Systems; Geometry; Group Theory and Generalizations
ISSN
0377-9017
eISSN
1573-0530
D.O.I.
10.1007/s11005-017-0946-y
Publisher site
See Article on Publisher Site

Abstract

We establish a connection between two recently proposed approaches to the understanding of the geometric origin of the Fu–Kane–Mele invariant $$\mathrm {FKM}\in \mathbb {Z}_2$$ FKM ∈ Z 2 , arising in the context of two-dimensional time-reversal symmetric topological insulators. On the one hand, the $$\mathbb {Z}_2$$ 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 $$\mathrm {FKM}$$ 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 $$\mathbb {T}^2 \rightarrow U(N)$$ T 2 → U ( N ) , of which we provide a proof employing only basic homotopy theory and circumventing the language of bundle gerbes.

Journal

Letters in Mathematical PhysicsSpringer Journals

Published: Feb 20, 2017

References

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