Absence of Weak Localization on Negative Curvature Surfaces.

The interplay between disorder and quantum interference leads to a wide variety of physical phenomena including celebrated Anderson localization-the complete absence of diffusive transport due to quantum interference between different particle trajectories. In two dimensions, any amount of disorder is thought to induce localization of all states at long enough length scales. In this Letter, we present an argument providing a mechanism for disrupting localization: by tuning the underlying curvature of the manifold on which diffusion takes place. We show that negative curvature manifolds contain a natural infrared cutoff for the probability of self-returning paths. We calculate the Cooperon-directly related to the weak-localization corrections to the conductivity-in hyperbolic space. It is shown that negative curvature leads to a rapid growth in the number of available trajectories a particle can coherently traverse in a given time, reducing the importance of interference effects and restoring diffusive behavior even in the absence of inelastic collisions. We further argue that on certain manifolds with variable curvature the hyperbolic regions dominate due to intermittency and arrest weak localization, which may be of relevance to two-dimensional materials due to surface roughness. This result may be amenable to experimental verification through the use of quantum simulators. Finally, we use results on mixed-curvature gravity by Zel'dovich to argue that our generic result on the absence of weak localization in two dimensions also applies to surfaces of mixed curvature, which may be of relevance to realistic experiments on weakly disordered metallic films on rough substrates.