Failure of NielsenNinomiya Theorem and Fragile Topology in TwoDimensional Systems with SpaceTime Inversion Symmetry: Application to Twisted Bilayer Graphene at Magic Angle
Abstract
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 twodimensional real wave functions characterized by the Euler class. To prove this, we examine the generic band topology of twodimensional real fermions in systems with spacetime inversion I_{ST} symmetry. The Euler class is an integer topological invariant classifying real twoband systems. We show that a twoband system with a nonzero Euler class cannot have an I_{ST}symmetric Wannier representation. Moreover, a twoband system with the Euler class e_{2} has band crossing points whose total winding number is equal to 2 e_{2}. Thus the conventional NielsenNinomiya 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 Z_{2} topological invariant classifying the band topology, that is, the second Stiefel Whitney class (w_{2}). Two bands with an even (odd) Euler class turn into a system with w_{2}=0 (w_{2}=1 ) when additional trivial bands are added. Although the nontrivial second StiefelWhitney class remains robust against adding trivial bands, it does not impose a Wannier obstruction when the number of bands is bigger than two. However, when the resulting multiband system with the nontrivial second StiefelWhitney class is supplemented by additional chiral symmetry, a nontrivial secondorder topology and the associated corner charges are guaranteed.
 Publication:

Physical Review X
 Pub Date:
 April 2019
 DOI:
 10.1103/PhysRevX.9.021013
 arXiv:
 arXiv:1808.05375
 Bibcode:
 2019PhRvX...9b1013A
 Keywords:

 Condensed Matter  Mesoscale and Nanoscale Physics;
 Condensed Matter  Strongly Correlated Electrons
 EPrint:
 23 pages, 13 figures