## Abstract

We review a class of translational-invariant insulators without spin-orbit coupling, as may be realized in intrinsically spinless systems, e.g., photonic crystals and ultra-cold atoms. Some of these insulators have no time-reversal symmetry as well, i.e., the relevant symmetries are purely crystalline. Nevertheless, topological phases exist which are distinguished by their robust surface modes. To describe these phases, we introduce the notions of (a) a halved-mirror chirality: an integer invariant which characterizes half-mirror planes in the 3D Brillouin zone, and (b) a bent Chern number: the traditional Thouless-Kohmoto-Nightingale-Nijs invariant generalized to bent 2D manifolds. Like other well-known topological phases, their band topology is unveiled by the crystalline analog of Berry phases, i.e., parallel transport across certain non-contractible loops in the Brillouin zone. We also identify certain topological phases without any robust surface modes - they are uniquely distinguished by parallel transport along bent loops, whose shapes are determined by the symmetry group. Finally, we describe the Weyl semimetallic phase that intermediates two distinct, gapped phases.

Original language | English (US) |
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Article number | 014013 |

Journal | Physica Scripta |

Volume | 2015 |

Issue number | T164 |

DOIs | |

State | Published - Dec 1 2015 |

Event | 156th Nobel Symposium on New Forms of Matter: Topological Insulators and Superconductors - Lidingo, Sweden Duration: Jun 12 2014 → Jun 15 2014 |

## All Science Journal Classification (ASJC) codes

- Atomic and Molecular Physics, and Optics
- Mathematical Physics
- Condensed Matter Physics

## Keywords

- cold atom
- photonic crystal
- topological insulator