Ultra-slow dynamics in a translationally invariant spin model for multiplication and factorization

Understand

We construct a model of short-range interacting Ising spins on a translationally invariant two-dimensional lattice that mimics a reversible circuit that multiplies or factorizes integers, depending on the choice of boundary conditions.

• We prove that, for open boundary conditions, the model exhibits no finite-temperature phase transition.
• Yet we find that it displays glassy dynamics with astronomically slow relaxation times, numerically consistent with a double exponential dependence on the inverse temperature.
• The slowness of the dynamics arises due to errors that occur during thermal annealing that cost little energy but flip an extensive number of spins.