Fracton critical point at a higher-order topological phase transition

The theory of quantum phase transitions separating different phases with distinct symmetry patterns at zero temperature is one the foundations modern many-body physics. Here we demonstrate that existence a two-dimensional topological transition between higher-order insulator (HOTI) and trivial Mott same eludes this paradigm. We present critical point (QCP) driven by fluctuations percolation domain walls HOTI region. Due to spinon modes decorate rough corners walls, boundaries trigger spinon-dipole hopping term fracton dynamics. Hence find QCP characterized dipole liquid subsystem U(1) breakdown area law entanglement entropy which exhibits logarithmic enhancement: $Lln(L)$. Using density matrix renormalization group method, analyze stiffness together structure factor QCP, provides strong evidence Bose surface, UV-IR mixing, dispersion relation $\ensuremath{\omega}={k}_{x}{k}_{y}.$