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arXiv · 2507.10807

The index of a pair of pure states and the interacting integer quantum Hall effect

Sven Bachmann·Jacob Shapiro·Clément Tauber
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Abstract

We introduce the index $\mathcal{N}(\omega_1,\omega_2)$ of a pair of pure states on a unital C*-algebra, which is a generalization of the notion of the index of a pair of projections on a Hilbert space. We then show that the Hall conductance associated with an invertible state $\omega$ of a two-dimensional interacting electronic system which is symmetric under $U(1)$ charge transformation may be written as the index $\mathcal{N}(\omega,\omega^D)$, where $\omega^D$ is obtained from $\omega$ by inserting a unit of magnetic flux. This exhibits the integrality and continuity properties of the Hall conductance in the context of general topological features of $\mathcal{N}$.

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Sven Bachmann, Jacob Shapiro, Clément Tauber. 2025-07-14. The index of a pair of pure states and the interacting integer quantum Hall effect. https://doi.org/10.1017/fms.2025.10160

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