For C2T/PT-symmetric Bloch bundles, rank two with nonzero Euler class has no exponentially localized symmetric Wannier basis, while every rank not equal to two does.
Degenerate flat bands in twisted bilayer graphene
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abstract
We prove that in the chiral limit of the Bistritzer--MacDonald Hamiltonian, there exist magic angles at which the Hamiltonian exhibits flat bands of multiplicity four instead of two. We analyse the structure of Bloch functions associated with the bands of arbitrary multiplicity, compute the corresponding Chern number to be $ -1 $, and show that there exist infinitely many degenerate magic angles for a generic choice of tunnelling potential, including the Bistritzer--MacDonald potential. Moreover, we demonstrate for generic tunnelling potentials flat bands have only twofold or fourfold multiplicities.
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Fragile topology on solid grounds: a mathematical perspective
For C2T/PT-symmetric Bloch bundles, rank two with nonzero Euler class has no exponentially localized symmetric Wannier basis, while every rank not equal to two does.