Parallel Transport and Band Theory in Crystals
  We show that different conventions for Bloch Hamiltonians on non-Bravais lattices correspond to different natural definitions of parallel transport of Bloch eigenstates. Generically the Berry curvatures associated with these parallel transports differ, while physical quantities are naturally related to a canonical choice of the parallel transport. 
Michel Fruchart, David Carpentier, Krzysztof Gaw\k{e}dzki
cond-mat.quant-gas cond-mat.mes-hall math-ph math.MP