Fractonic critical point proximate to a higher-order topological insulator: How does UV blend with IR?

We propose an unconventional topological quantum phase transition connecting a higher-order topological insulator (HOTI) and a featureless Mott insulator sharing the same symmetry patterns. We construct an effective theory description of the quantum critical point (QCP) by combining a bosonization approach and the coupled-stripe construction of 1D critical spin ladders. The phase transition theory is characterized by a critical dipole liquid theory with subsystem $U(1)$ symmetry whose low energy modes contain a Bose surface along the $k_x,k_y$ axis. Such a quantum critical point manifests fracton dynamics and the breakdown of the area law entanglement entropy due to the existence of a Bose surface. We numerically confirm our findings by measuring the entanglement entropy, topological rank-2 Berry phase, and the static structure factor throughout the topological transition and compare it with our previous approach obtained from the percolation picture. A significant new element of our phase transition theory is that the infrared~(IR) effective theory is controlled by short wave-length fluctuations with peculiar UV-IR mixing.