Fractal Quantum Phase Transitions: Critical Phenomena Beyond Renormalization

TL;DR

This paper explores a novel quantum phase transition with fractal symmetry that cannot be described by traditional renormalization group methods, revealing unique critical phenomena associated with fractal structures.

Contribution

It introduces a new class of quantum critical points with fractal symmetry, supported by large-scale simulations and a developed field theory, expanding understanding beyond conventional frameworks.

Findings

Identified a continuous quantum phase transition with fractal symmetry breaking.

Discovered fractal scaling dimension at the critical point related to Sierpinski gaskets.

Developed a field theory explaining UV-IR mixing due to fractal symmetry.

Abstract

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.