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Non-commutative Brillouin Torus

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A Computational Non-commutative Geometry Program for Disordered Topological Insulators

Part of the book series: SpringerBriefs in Mathematical Physics ((BRIEFSMAPHY,volume 23))

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Abstract

In this Chapter, we review the fundamental theoretical tools, starting with the space of disordered configurations and its associated dynamical systems, the \(C^*\)-algebra \({\mathcal A}_d\) of the physical observables, together with its Fourier and differential calculus. The latter is provided by a set of commuting derivations \(\partial \) and a trace \({\mathcal T}\). The triple \(({\mathcal A}_d,\partial ,{\mathcal T})\) defines a non-commutative manifold known as the non-commutative Brillouin torus. We reformulate the topological invariants and other response functions in this new framework. We also introduce the magnetic derivations and investigate the behavior of the correlation functions w.r.t. the magnetic fields. This Chapter also fixes the notation and defines the precise settings for the rest of our calculations.

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  1. Department of Physics, Department of Mathematical Sciences, Yeshiva University, New York, NY, USA

    Emil Prodan

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  1. Emil Prodan
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Prodan, E. (2017). Non-commutative Brillouin Torus. In: A Computational Non-commutative Geometry Program for Disordered Topological Insulators. SpringerBriefs in Mathematical Physics, vol 23. Springer, Cham. https://doi.org/10.1007/978-3-319-55023-7_3

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