Exact solution for the filling-induced thermalization transition in a 1D fracton system

We study a random circuit model of constrained fracton dynamics, in which particles on a one-dimensional lattice undergo random local motion subject to both charge and dipole moment conservation. The configuration space of this system exhibits a continuous phase transition between a weakly fragmented ("thermalizing") phase and a strongly fragmented ("nonthermalizing") phase as a function of the number density of particles. Here, by mapping to two different problems in combinatorics, we identify an exact solution for the critical density n_c. Specifically, when evolution proceeds by operators that act on s contiguous sites, the critical density is given by n_c = 1/(s−2). We identify the critical scaling near the transition, and we show that there is a universal value of the correlation length exponent ν = 2. We confirm our theoretical results with numeric simulations. In the thermalizing phase the dynamical exponent is subdiffusive: z = 4, while at the critical point it increases to z_c > 6.