  We identify a quantum critical point with fractal symmetry whose effective
theory eludes the renormalization group framework. We consider the Newman-Moore
model with three-body interaction subjected to an external transverse field,
which exhibits a Kramers-Wannier type self-duality and a fractal $Z_2$ symmetry
with Ising charge conserved on a fractal subset of sites, i.e., on Sierpinski
gaskets. Using large-scale quantum Monte Carlo simulations, we identify a
continuous quantum phase transition between a phase with spontaneous fractal
symmetry breaking and a paramagnetic phase. This phase transition is
characterized by the emergence of a fractal scaling dimension $d=\ln(3)/\ln(2)$
at the quantum critical point, where the power-law exponent of the correlation
function is related to the fractal dimension of the Sierpinski triangle. We
develop a field theory to elucidate such quantum criticality and denote the
fractal scaling as a subsequence of UV-IR mixing, where the low energy modes at
the critical point are manipulated by short-wavelength physics due to the
fractal symmetry.
