We show that the Wannier obstruction and the fragile topology of the nearly flat bands in twisted bilayer graphene at magic angle are manifestations of the nontrivial topology of two-dimensional real wave functions characterized by the Euler class. To prove this, we examine the generic band topology of two dimensional real fermions in systems with space-time inversion I_ST symmetry. The Euler class is an integer topological invariant classifying real two band systems. We show that a two-band system with a nonzero Euler class cannot have I_ST -symmetric Wannier representation. Moreover, a two-band system with the Euler class e₂ has band crossing points whose total winding number is equal to 2e₂. Thus the conventional Nielsen-Ninomiya theorem fails in systems with a nonzero Euler class. We propose that the topological phase transition between two insulators carrying distinct Euler classes can be described in terms of the pair creation and annihilation of vortices accompanied by winding number changes across Dirac strings. When the number of bands is bigger than two, there is a Z₂ topological invariant classifying the band topology, that is, the second Stiefel Whitney class (w₂). Two bands with an even (odd) Euler class turn into a system with w₂=0 (w₂=1) when additional trivial bands are added. Although the nontrivial second Stiefel-Whitney class remains robust against adding trivial bands, it does not impose Wannier obstruction when the number of bands is bigger than two. However, when the resulting multi-band system with the nontrivial second Stiefel-Whitney class is supplemented by additional chiral symmetry, a nontrivial second-order topology and the associated corner charges are guaranteed.

Authors: Junyeong Ahn (서울대, 공동 제1저자), Sungjoon Park(서울대, 공동 제1저자), Bohm-Jung Yang^* (서울대)

Journal: Physical Review X 9, 021013 (2019)

Publication date: 22 April 2019

https://journals.aps.org/prx/abstract/10.1103/PhysRevX.9.021013