  Spin models are widely studied in the natural sciences, from investigating
magnetic materials in condensed matter physics to studying neural networks.
Previous work has demonstrated that there exist simple classical spin models
that are universal: they can replicate -- in a precise and rigorous sense --
the complete physics of any other classical spin model, to any desired
accuracy. However, all previously known universal models break translational
invariance. In this paper we show that there exist translationally invariant
universal models. Our main result is an explicit construction of a
translationally invariant, 2D, nearest-neighbour, universal classical
Hamiltonian with a single free parameter. The proof draws on techniques from
theoretical computer science, in particular recent complexity theoretic results
on tiling problems. Our results imply that there exists a single Hamiltonian
which encompasses all classical spin physics, just by tuning a single parameter
and varying the size of the lattice. We also prove that our construction is
optimal in terms of the number of parameters in the Hamiltonian; there cannot
exist a translationally invariant universal Hamiltonian with only the lattice
size as a parameter.
