We review the recent progress in the study of topological phases in systems with space–time inversion symmetry IST.ISTis an anti-unitary symmetry which is local in momentum space and satisfies I2 ST= 1 such as PT in ...We review the recent progress in the study of topological phases in systems with space–time inversion symmetry IST.ISTis an anti-unitary symmetry which is local in momentum space and satisfies I2 ST= 1 such as PT in two dimensions(2 D)and three dimensions(3 D) without spin–orbit coupling and C2 T in 2 D 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 1 D and 2 D, 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, including1 D topological insulators with quantized charge polarization, 2 D Dirac semimetals, and 3 D nodal line semimetals. Next we review how the second Stiefel–Whitney class characterizes the 3 D 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 2 D and 3 D topological insulators characterized by the nontrivial second Stiefel Whitney class.展开更多
基金supported by IBS-R009-D1supported by the Institute for Basic Science in Korea (Grant No. IBS-R009-D1)+6 种基金Basic Science Research Program through the National Research Foundation of Korea (NRF) (Grant No. 0426-20180011)the POSCO Science Fellowship of POSCO TJ Park Foundation (No. 042620180002)supported in part by the U.S. Army Research Office under Grant Number W911NF-18-1-0137supported by Institute for Basic Science (IBS-R011D1)NRF grant funded by the Korea government (MSIP) (NRF-2017R1G1B5018169)supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1701-07Basic Science Research Program through NRF funded by the Ministry of Education (NRF2018R1A6A3A11044335)
文摘We review the recent progress in the study of topological phases in systems with space–time inversion symmetry IST.ISTis an anti-unitary symmetry which is local in momentum space and satisfies I2 ST= 1 such as PT in two dimensions(2 D)and three dimensions(3 D) without spin–orbit coupling and C2 T in 2 D 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 1 D and 2 D, 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, including1 D topological insulators with quantized charge polarization, 2 D Dirac semimetals, and 3 D nodal line semimetals. Next we review how the second Stiefel–Whitney class characterizes the 3 D 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 2 D and 3 D topological insulators characterized by the nontrivial second Stiefel Whitney class.
