Abstract
We show how general principles of symmetry in quantum mechanics lead to twisted notions of a group representation. This framework generalizes both the classical threefold way of real/complex/ quaternionic representations as well as a corresponding tenfold way which has appeared in condensed matter and nuclear physics. We establish a foundation for discussing continuous families of quantum systems. Having done so, topological phases of quantum systems can be defined as deformation classes of continuous families of gapped Hamiltonians. For free particles, there is an additional algebraic structure on the deformation classes leading naturally to notions of twisted equivariant K-theory. In systems with a lattice of translational symmetries, we show that there is a canonical twisting of the equivariant K-theory of the Brillouin torus. We give precise mathematical definitions of two invariants of the topological phases which have played an important role in the study of topological insulators. Twisted equivariant K-theory provides a finer classification of topological insulators than has been previously available.
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Communicated by Krzysztof Gawedzki.
The work of D.S.F. is supported by the National Science Foundation under grants DMS-0603964 and DMS-1207817.
The work of G.W.M. is supported by the DOE under grant DE-FG02-96ER40959. GM also gratefully acknowledges partial support from the Institute for Advanced Study and the Ambrose Monell Foundation.
This material is also based upon work supported in part by the National Science Foundation under Grant No. 1066293 and the hospitality of the Aspen Center for Physics. We also thank the Institute for Advanced Study and the Simons Center for Geometry and Physics for providing support and stimulating environments for discussions related to this paper.
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Freed, D.S., Moore, G.W. Twisted Equivariant Matter. Ann. Henri Poincaré 14, 1927–2023 (2013). https://doi.org/10.1007/s00023-013-0236-x
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DOI: https://doi.org/10.1007/s00023-013-0236-x