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

## TL;DR

This paper introduces a translationally invariant spin model that mimics multiplication and factorization, exhibiting ultra-slow glassy dynamics with double exponential relaxation times despite lacking a finite-temperature phase transition.

## Contribution

The study constructs a novel 2D Ising spin model that reproduces reversible multiplication circuits and reveals glassy dynamics with unprecedented slow relaxation behavior.

## Key findings

- No finite-temperature phase transition with open boundaries.
- Glassy dynamics with double exponential relaxation times.
- Energy barriers scale linearly with exponentially diverging correlation length.

## Abstract

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. We argue that the energy barrier that needs to be overcome in order to heal such defects scales linearly with the correlation length, which diverges exponentially with inverse temperature, thus yielding the double exponential behavior of the relaxation time.