  We report a new attractive critical point occurring in the Anderson
localization scaling flow of symplectic models on fractals. The scaling theory
of Anderson localization predicts that in disordered symplectic two-dimensional
systems weak antilocalization effects lead to a metal-insulator transition.
This transition is characterized by a repulsive critical point above which the
system becomes metallic. Fractals possess a non-integer scaling of conductance
in the classical limit which can be continuously tuned by changing the fractal
structure. We demonstrate that in disordered symplectic Hamiltonians defined on
fractals with classical conductance scaling $g \sim L^{-\varepsilon}$, for $0 <
\varepsilon < \beta_\mathrm{max} \approx 0.15$, the metallic phase is replaced
by a critical phase with a scale invariant conductance dependent on the fractal
dimensionality. Our results show that disordered fractals allow an explicit
construction and verification of the $\varepsilon$ expansion.
