Topological invariants of time-reversal-invariant band structures

By Joel E. Moore, L. Balents. 2006

1,616 citations recorded by the source corpus.

The topological invariants of a time-reversal-invariant band structure in two dimensions are multiple copies of the $\mathbb{Z}_2$ invariant found by Kane and Mele. Such invariants protect the topological insulator and give rise to a spin Hall effect carried by edge states. Each pair of bands related by time reversal is described by a single $\mathbb{Z}_2$ invariant, up to one less than half the dimension of the Bloch Hamiltonians. In three dimensions, there are four such invariants per band. The $\mathbb{Z}_2$ 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 $\mathbb{Z}_2$ invariants, the integer invariants that underlie the integer quantum Hall effect, and previous invariants of ${\cal T}$-invariant Fermi systems.

Subfields: Atomic and Molecular Physics, and Optics, Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/8504df047c13cf400895d1e912f5e4d25a7c6249

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In Atomic and Molecular Physics, and Optics

In Materials Chemistry

Citations in the corpus

Weyl and Dirac semimetals in three-dimensional solids (2017, background): “…dimensions (Bernevig et al., 2006; Chen et al., 2009; Fu and Kane, 2007; Fu et al., 2007; Haldane, 1988; Hsieh et al., 2008; Kane and Mele, 2005; Moore and Balents, 2007; Roy, 2009; Xia et al., 2009b; Zhang et al., 2009), has led to an explosion of activity in the study of topological aspects…”
Photonic topological insulators. (2012, background): “The recent discovery of two [ 1, 2] and three [ 3, 4, 5, 6, 7] dimensional topological insulators has stimulated interest in nontrivial topological phases.”
A topological Dirac insulator in a quantum spin Hall phase (2008, background): “Materials with bulk band structures with Z2 = −1 on the other hand, which are expected to exist in rare systems with strong spin-orbit coupling acting as an internal quantizing magnetic field on the electron system [6], and inverted bands at an odd number of high symmetry points in their bulk 3D BZs”
The physics of quantum materials (2017, background): “However, there does exist a three-dimensional topological insulator phase in bulk materials 74–76 .”