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We identify a quantum critical point with fractal symmetry whose effectivetheory eludes the renormalization group framework. We consider the Newman-Mooremodel with three-body interaction subjected to an external transverse field,which exhibits a Kramers-Wannier type self-duality and a fractal $Z_2$ symmetrywith Ising charge conserved on a fractal subset of sites, i.e., on Sierpinskigaskets. Using large-scale quantum Monte Carlo simulations, we identify acontinuous quantum phase transition between a phase with spontaneous fractalsymmetry breaking and a paramagnetic phase. This phase transition ischaracterized 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 correlationfunction is related to the fractal dimension of the Sierpinski triangle. Wedevelop a field theory to elucidate such quantum criticality and denote thefractal scaling as a subsequence of UV-IR mixing, where the low energy modes atthe critical point are manipulated by short-wavelength physics due to thefractal symmetry.

Fractal Quantum Phase Transitions: Critical Phenomena Beyond Renormalization