For a charged particle on a torus with constant metric and gauge field, nonzero ground-state entanglement entropy occurs only under special degeneracy constraints, and the fourfold degenerate case fixes the metric to the canonical torus metric.
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Exploring Entanglement Entropy for a Particle on a Torus with constant Metric and constant $U(1)$-Gauge Field
For a charged particle on a torus with constant metric and gauge field, nonzero ground-state entanglement entropy occurs only under special degeneracy constraints, and the fourfold degenerate case fixes the metric to the canonical torus metric.