We introduce a novel gauge-invariant, quantized interband index in two-dimensional (2D) multiband systems. It provides a bulk topological classification of a submanifold of parameter space (e.g., an electron valley in a Brillouin zone), and therefore overcomes difficulties in characterizing topology of submanifolds. We confirm its topological nature by numerically demonstrating a one-to-one correspondence to the valley Chern number in $k·p$ models (e.g., gapped Dirac fermion model) and the first Chern number in lattice models (e.g., Haldane model). Furthermore, we derive a band-resolved topological charge and demonstrate that it can be used to investigate the nature of edge states due to band inversion in valley systems like multilayer graphene.

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• Edge states
• Geometric & topological phases
• Valley degrees of freedom
• Valleytronics
• 2-dimensional systems
• Lattice models in condensed matter
• Numerical techniques
• k dot p method