Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: application to twisted bilayer graphene at magic angle*

We examine the band topology of two dimensional real fermions in systems with space-time inversion (STI) symmetry. We show that a two-band system with a nonzero Euler class cannot have STI 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 insulators carrying different Euler classes can be described by pair creation and annihilation of vortices across Dirac strings.

For a multiband system, the Z₂ invariant called the second Stiefel-Whitney class (w₂) can be defined, which is equal to e₂ mod 2 for a 2-band system. Although w₂ remains robust against adding trivial bands, it does not impose Wannier obstruction when the total number of bands is greater than two. However, when the resulting multi-band system with the nontrivial second Stiefel-Whitney class is supplemented by additional mirror and chiral symmetries, a nontrivial second order topology and the associated corner charges are guaranteed. We discuss the implications to the nearly flat bands of twisted bilayer graphene at magic angle.