| Summary: | We study quantum dynamics on noncommutative spaces of negative curvature,
focusing on the hyperbolic plane with spatial noncommutativity in the presence
of a constant magnetic field. We show that the synergy of noncommutativity and
the magnetic field tames the exponential divergence of operator growth caused
by the negative curvature of the hyperbolic space. Their combined effect
results in a first-order transition at a critical value of the magnetic field
in which strong quantum effects subdue the exponential divergence for {\it all}
energies, in stark contrast to the commutative case, where for high enough
energies operator growth always diverge exponentially. This transition
manifests in the entanglement entropy between the `left' and `right' Hilbert
spaces of spatial degrees of freedom. In particular, the entanglement entropy
in the lowest Landau level vanishes beyond the critical point. We further
present a non-linear solvable bosonic model that realizes the underlying
algebraic structure of the noncommutative hyperbolic plane with a magnetic
field.
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