Gravitational Anderson Localization

We present a higher dimensional model where gravity is bound to a brane due to Anderson localization. The extra dimensions are taken to be a disordered crystal of branes, with randomly distributed tensions of order the fundamental scale. Such geometries bind the graviton and thus allow for arbitrarily large extra dimensions even when the curvature is small. Thus this model is quite distinct from that of Randall and Sundrum where localization is a consequence of curvature effects in the bulk. The hierarchy problem can be solved by having the standard model brane live a distance away from the brane on which the graviton is localized. The statistical properties of the system are worked out and it is shown that the scenario leads to a continuum of four dimensional theories with differing strengths of gravitational interactions. We live on one particular brane whose gravitational constant is $G_N$.

Figure 1

The volcano potential generated in the RS scenario which has a zero mode bound state and repels the KK modes.

Figure 2

The propagation from $A$ to $B$ receives contributions from multiple paths each of which contributes a random phase to the probability amplitude. Closed paths on the other hand all have time reversed paths which have identical phases, leading to constructive interference and localization.

Figure 3

The band structure with vanishing disorder. On the scale of $M_{\star }$ the bands are nearly continuous. The separation between levels if of order $M_{\star }/N$.

Figure 4

The density of states in the strong disorder limit. The width of the flat part of the band is of order $f$. At the edges the density of states $n(m)$ dies off exponentially fast and vanishes at the origin.