Exact solution for the filling-induced thermalization transition in a one-dimensional 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 nc. Specifically, when evolution proceeds by operators that act on ℓ contiguous sites, the critical density is given by nc=1/(ℓ-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 zc≳6.