中国物理B ›› 2019, Vol. 28 ›› Issue (11): 117101-117101.doi: 10.1088/1674-1056/ab4d3b

所属专题： TOPICAL REVIEW — Topological semimetals

• TOPICAL REVIEW—Topological semimetals • 上一篇    下一篇

Stiefel-Whitney classes and topological phases in band theory

Junyeong Ahn, Sungjoon Park, Dongwook Kim, Youngkuk Kim, Bohm-Jung Yang   

  1. 1 Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea;
    2 Center for Correlated Electron Systems, Institute for Basic Science(IBS), Seoul 08826, Korea;
    3 Center for Theoretical Physics(CTP), Seoul National University, Seoul 08826, Korea;
    4 Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
  • 收稿日期:2019-03-30 修回日期:2019-09-30 出版日期:2019-11-05 发布日期:2019-11-05
  • 通讯作者: Bohm-Jung Yang E-mail:bjyang@snu.ac.kr

Stiefel-Whitney classes and topological phases in band theory

Junyeong Ahn1,2,3, Sungjoon Park1,2,3, Dongwook Kim4, Youngkuk Kim4, Bohm-Jung Yang1,2,3   

  1. 1 Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea;
    2 Center for Correlated Electron Systems, Institute for Basic Science(IBS), Seoul 08826, Korea;
    3 Center for Theoretical Physics(CTP), Seoul National University, Seoul 08826, Korea;
    4 Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
  • Received:2019-03-30 Revised:2019-09-30 Online:2019-11-05 Published:2019-11-05
  • Contact: Bohm-Jung Yang E-mail:bjyang@snu.ac.kr

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摘要： We review the recent progress in the study of topological phases in systems with space-time inversion symmetry IST. IST is an anti-unitary symmetry which is local in momentum space and satisfies IST2=1 such as PT in two dimensions (2D) and three dimensions (3D) without spin-orbit coupling and C2T in 2D with or without spin-orbit coupling, where P, T, C2 indicate the inversion, time-reversal, and two-fold rotation symmetries, respectively. Under IST, the Hamiltonian and the periodic part of the Bloch wave function can be constrained to be real-valued, which makes the Berry curvature and the Chern number vanish. In this class of systems, gapped band structures of real wave functions can be topologically distinguished by the Stiefel-Whitney numbers instead. The first and second Stiefel-Whitney numbers w1 and w2, respectively, are the corresponding invariants in 1D and 2D, which are equivalent to the quantized Berry phase and the Z2 monopole charge, respectively. We first describe the topological phases characterized by the first Stiefel-Whitney number, including 1D topological insulators with quantized charge polarization, 2D Dirac semimetals, and 3D nodal line semimetals. Next we review how the second Stiefel-Whitney class characterizes the 3D nodal line semimetals carrying a Z2 monopole charge. In particular, we explain how the second Stiefel-Whitney number w2, the Z2 monopole charge, and the linking number between nodal lines are related. Finally, we review the properties of 2D and 3D topological insulators characterized by the nontrivial second Stiefel Whitney class.

关键词: topological, semimetal

Abstract: We review the recent progress in the study of topological phases in systems with space-time inversion symmetry IST. IST is an anti-unitary symmetry which is local in momentum space and satisfies IST2=1 such as PT in two dimensions (2D) and three dimensions (3D) without spin-orbit coupling and C2T in 2D with or without spin-orbit coupling, where P, T, C2 indicate the inversion, time-reversal, and two-fold rotation symmetries, respectively. Under IST, the Hamiltonian and the periodic part of the Bloch wave function can be constrained to be real-valued, which makes the Berry curvature and the Chern number vanish. In this class of systems, gapped band structures of real wave functions can be topologically distinguished by the Stiefel-Whitney numbers instead. The first and second Stiefel-Whitney numbers w1 and w2, respectively, are the corresponding invariants in 1D and 2D, which are equivalent to the quantized Berry phase and the Z2 monopole charge, respectively. We first describe the topological phases characterized by the first Stiefel-Whitney number, including 1D topological insulators with quantized charge polarization, 2D Dirac semimetals, and 3D nodal line semimetals. Next we review how the second Stiefel-Whitney class characterizes the 3D nodal line semimetals carrying a Z2 monopole charge. In particular, we explain how the second Stiefel-Whitney number w2, the Z2 monopole charge, and the linking number between nodal lines are related. Finally, we review the properties of 2D and 3D topological insulators characterized by the nontrivial second Stiefel Whitney class.

Key words: topological, semimetal

中图分类号:  (Electron density of states and band structure of crystalline solids)

  • 71.20.-b
73.20.-r (Electron states at surfaces and interfaces)