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Phys. Rev. B 75, 121306(R) (2007) [4 pages]

Topological invariants of time-reversal-invariant band structures

J. E. Moore1,2 and L. Balents3
1Department of Physics, University of California, Berkeley, California 94720, USA
2Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
3Department of Physics, University of California, Santa Barbara, California 93106, USA

Rapid Received 28 July 2006; revised 1 February 2007; published 26 March 2007

The topological invariants of a time-reversal-invariant band structure in two dimensions are multiple copies of the [openface Z]2 invariant found by Kane and Mele. Such invariants protect the “topological insulator” phase and give rise to a spin Hall effect carried by edge states. Each pair of bands related by time reversal is described by one [openface Z]2 invariant, up to one less than half the dimension of the Bloch Hamiltonians. In three dimensions, there are four such invariants per band pair. The [openface Z]2 invariants of a crystal determine the transitions between ordinary and topological insulators as its bands are occupied by electrons. We derive these invariants using maps from the Brillouin zone to the space of Bloch Hamiltonians and clarify the connections between [openface Z]2 invariants, the integer invariants that underlie the integer quantum Hall effect, and previous invariants of [script T]-invariant Fermi systems.

©2007 The American Physical Society

URL: http://link.aps.org/doi/10.1103/PhysRevB.75.121306
DOI: 10.1103/PhysRevB.75.121306
PACS: 73.43.-f; 73.20.At; 85.75.-d
  • 73.43.-f
    Quantum Hall effects
  • 73.20.At
    Surface states, band structure, electron density of states
  • 85.75.-d
    Magnetoelectronics; spintronics: devices exploiting spin polarized transport or integrated magnetic fields
  • YEAR: 2007
KEYWORDS: band structure, Brillouin zones, topology, quantum Hall effect

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