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We show that the Wannier obstruction and the fragile topology of the nearlyflat bands in twisted bilayer graphene at magic angle are manifestations of thenontrivial topology of two-dimensional real wave functions characterized by theEuler class. To prove this, we examine the generic band topology of twodimensional real fermions in systems with space-time inversion $I_{ST}$symmetry. The Euler class is an integer topological invariant classifying realtwo band systems. We show that a two-band system with a nonzero Euler classcannot have an $I_{ST}$-symmetric Wannier representation. Moreover, a two-bandsystem with the Euler class $e_{2}$ has band crossing points whose totalwinding number is equal to $-2e_2$. Thus the conventional Nielsen-Ninomiyatheorem fails in systems with a nonzero Euler class. We propose that thetopological phase transition between two insulators carrying distinct Eulerclasses can be described in terms of the pair creation and annihilation ofvortices accompanied by winding number changes across Dirac strings. When thenumber of bands is bigger than two, there is a $Z_{2}$ topological invariantclassifying 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 thenontrivial second Stiefel-Whitney class remains robust against adding trivialbands, it does not impose a Wannier obstruction when the number of bands isbigger than two. However, when the resulting multi-band system with thenontrivial second Stiefel-Whitney class is supplemented by additional chiralsymmetry, a nontrivial second-order topology and the associated corner chargesare guaranteed.

physical review. x

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