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

• URL: http://arxiv.org/abs/2507.10807v1
• Date: Mon, 14 Jul 2025 21:02:14 GMT
• Title: The index of a pair of pure states and the interacting integer quantum Hall effect
• Authors: Sven Bachmann, Jacob Shapiro, Clément Tauber,
• Abstract summary: We show that the Hall conductance associated with an invertible state $omega$ of a two-dimensional interacting electronic system may be written as the index $mathcalN(omega,omega_D)$.<n>This exhibits the integrality and continuity properties of the Hall conductance in the context of general topological features of $mathcalN$.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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