Abstract
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 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.
Highlights
The study of dynamics on negative curvature surfaces has yielded some important results in the subject of classical and quantum chaos [1,2,3,4,5,6]
One of the main motivations of this work was to explore possible deformations of the Schwarzian theory, which emerges in the low energy sector of the Sachdev-Ye-Kitaev model
The Schwarzian action is closely related to the Landau level problem on the hyperbolic plane
Summary
The study of dynamics on negative curvature surfaces has yielded some important results in the subject of classical and quantum chaos [1,2,3,4,5,6]. 2, and the energy spectrum and dynamics are uniquely determined by the S L(2, R) symmetry This algebraic structure is closely related to the Landau level problem on the hyperbolic plane which was solved by Comtet and Houston [12, 13]. Our work in this paper is motivated by the question of what variants of the Landau level problem on the hyperbolic plane can subdue the exponential divergence of operator growth in addition to the magnetic field To this end we consider the quantum dynamics on a noncommutative hyperbolic plane with a constant magnetic field [15]. For θ B > 1 the spectrum consists entirely of Landau levels for all energies with no continuum [15] This transition can be tracked by studying the non-analyticity in the entanglement entropy between the intrinsic “left" and “right" components of the noncommutative Hilbert space.
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