  We study the spreading of initially-local operators under unitary time
evolution in a 1d random quantum circuit model which is constrained to conserve
a $U(1)$ charge and its dipole moment, motivated by the quantum dynamics of
fracton phases. We discover that charge remains localized at its initial
position, providing a crisp example of a non-ergodic dynamical phase of random
circuit dynamics. This localization can be understood as a consequence of the
return properties of low dimensional random walks, through a mechanism
reminiscent of weak localization, but insensitive to dephasing. The charge
dynamics is well-described by a system of coupled hydrodynamic equations, which
makes several nontrivial predictions in good agreement with numerics.
Importantly, these equations also predict localization in 2d fractonic
circuits. Immobile fractonic charge emits non-conserved operators, whose
spreading is governed by exponents distinct to non-fractonic circuits.
Fractonic operators exhibit a short time linear growth of observable
entanglement with saturation to an area law, as well as a subthermal volume law
for operator entanglement. The entanglement spectrum follows semi-Poisson
statistics, similar to eigenstates of MBL systems. The non-ergodic
phenomenology persists to initial conditions containing non-zero density of
dipolar or fractonic charge. Our work implies that low-dimensional fracton
systems preserve forever a memory of their initial conditions in local
observables under noisy quantum dynamics, thereby constituting ideal memories.
It also implies that 1d and 2d fracton systems should realize true MBL under
Hamiltonian dynamics, even in the absence of disorder, with the obstructions to
MBL in translation invariant systems and in d>1 being evaded by the nature of
the mechanism responsible for localization. We also suggest a possible route to
new non-ergodic phases in high dimensions.
