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Abstract

We study families of Hamiltonians on the lattice arising from the insertion of N flux tubes through lattice cells. In particular, it is shown such systems, under an appropriate notion of equivalence, are in bijective correspondence with the first K-group of the N-torus. A version of spectral flow is defined in the context of systems arising from flux tubes, which serves as an invariant of these systems, and it is shown that this provides a classification for N = 1, 2, and fails to do so for N ≥ 3. We also show that for any path connected locally compact Hausdorff space X, spectral flow can be used to generate a natural homomorphism from the first K-group of X to the first cohomology group of X. Finally, we show that if X is a connected finite CW complex with abelain fundamental group, then this map is surjective.