Fractal lattices let physicists dial up dimensions between 2 and 3

Anderson localization — the quantum freezing of electron transport by disorder — is known to behave qualitatively differently in 2D versus 3D, but what happens at fractional dimensions in between has been hard to test. This work constructs a family of 3D fractal lattices whose spectral dimension can be tuned continuously from 2 to 3, then uses large-scale finite-size scaling to map out the full transition. The critical disorder strength rises smoothly from zero to 16.6 as the dimension climbs from 2 to 3, and the critical exponent follows an approximate inverse dependence on spectral dimension. Spectral dimension controls the universality class, while microscopic geometry shifts the precise critical point.