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To construct a crystal you need a lattice and a basis. The lattice represents the translational symmetry of the system. Namely, graphene has a hexagonal lattice, meaning the two lattice vectors are 60 degress apart. Since the brillouin zone is constructed by inverting the lattice vectors, the brillouin zone is shaped based upon the lattice, but not the ...


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There are similar topological invariants for band structures in one dimension, but an important difference is that these invariants always require some symmetry in the band Hamiltonians, for example particle-hole symmetry. In such cases, typically the invariant is given by $$ C=\int\frac{dk}{2\pi} a_k \text{ mod }1 $$



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