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SUMMARY:Topological Matter Protected by Momentum Space Crystallographic Gr
 oups\, Prof. Yu-Xin Zhao [趙宇心] University of Hong Kong
DTSTART:20260310T060000Z
DTEND:20260310T080000Z
DTSTAMP:20260420T121800Z
UID:indico-event-176@indico.phys.nthu.edu.tw
DESCRIPTION:While crystallographic groups are typically considered in real
  space\, momentum-space crystallographic groups (MCGs) have recently emerg
 ed as an active research area. This development is largely driven by the f
 ramework of projective crystal symmetry\, where all non-symmorphic crystal
 lographic groups arise from phase factors between real-space translations 
 and point-group elements\, according to Mackey’s representation theory [
 1\,2]. A key implication of non-symmorphic MCGs is that the momentum-space
  unit—traditionally regarded as a torus—can take the form of any compa
 ct flat manifold\, known as the ten platycosms\, which are the orbital spa
 ces of the ten Bieberbach groups [3]. For each platycosm\, the topological
  classification\, specifically the reduced K-group\, is isomorphic to the 
 second integral cohomology group of the corresponding Bieberbach group [3]
 . We will further demonstrate that the cohomology groups of MCGs can exhau
 stively classify all Abelian crystalline topological insulators\, as well 
 as all twistings of point-group actions over the Brillouin torus [4]. By e
 stablishing an isomorphism between the integral cohomology and a one-degre
 e-lower cohomology with U(1)-valued functions over momentum space as coeff
 icients\, we can algebraically formulate a complete set of topological inv
 ariants for classifying Abelian crystalline topological insulators and alg
 ebraically represent all twistings of point-group actions.References:[1] Z
 . Y. Chen\, S. A. Yang\, Y. X. Zhao\, Nat. Commun. 13\, 2215 (2022)[2] C. 
 Zhang\, Z. Y. Chen\, Z. Zhang\, Y. X. Zhao\, Phys. Rev. Lett. 130\, 256601
  (2023)[3] C. Zhang\, P. Wang\, J. Lyu\, Y. X. Zhao\, Phys. Rev. Lett. 135
 \, 136601 (2025)[4] T. R. Liu\, Z. Zhang\, Y. X. Zhao\, arXiv:2512.21844 (
 2025)\n\nhttps://indico.phys.nthu.edu.tw/event/176/
URL:https://indico.phys.nthu.edu.tw/event/176/
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