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Almost Commuting Unitary Matrices Related to Time Reversal

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Abstract

The behavior of fermionic systems depends on the geometry of the system and the symmetry class of the Hamiltonian and observables. Almost commuting matrices arise from band-projected position observables in such systems. One expects the mathematical behavior of almost commuting Hermitian matrices to depend on two factors. One factor will be the approximate polynomial relations satisfied by the matrices. The other factor is what algebra the matrices are in, either \({{\bf M}_n(\mathbb{A})}\) for \({\mathbb{A} = \mathbb{R}}\) , \({\mathbb{A} = \mathbb{C}}\) or \({\mathbb{A} = \mathbb{H}}\) , the algebra of quaternions.

There are potential obstructions keeping k-tuples of almost commuting operators from being close to a commuting k-tuple.We consider two-dimensional geometries and so this obstruction lives in \({KO_{-2}(\mathbb{A})}\) . This obstruction corresponds to either the Chern number or spin Chern number in physics. We show that if this obstruction is the trivial element in K-theory then the approximation by commuting matrices is possible.

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Author notes

  1. Adam P. W. Sørensen

    Present address: School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW, 2522, Australia

Authors and Affiliations

  1. University of New Mexico, Department of Mathematics and Statistics, Albuquerque, New Mexico, 87131, USA

    Terry A. Loring

  2. Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen Ø, Denmark

    Adam P. W. Sørensen

Authors
  1. Terry A. Loring
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  2. Adam P. W. Sørensen
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Correspondence to Terry A. Loring.

Additional information

Communicated by A. Connes

This work was partially supported by a grant from the Simons Foundation (208723) and by the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation.

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Loring, T.A., Sørensen, A.P.W. Almost Commuting Unitary Matrices Related to Time Reversal. Commun. Math. Phys. 323, 859–887 (2013). https://doi.org/10.1007/s00220-013-1799-6

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