Fractal Quantum Phase Transitions: Critical Phenomena Beyond Renormalization

Understand

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.