Translationally invariant universal classical Hamiltonians

TL;DR

This paper constructs a translationally invariant 2D classical spin Hamiltonian that is universal, capable of simulating any other classical spin model by tuning a single parameter and adjusting lattice size.

Contribution

It provides the first explicit construction of a translationally invariant universal classical Hamiltonian, advancing the understanding of universal models in physics.

Findings

Existence of a translationally invariant universal Hamiltonian

Construction of a 2D nearest-neighbour model with a single free parameter

Optimality of the number of parameters in the Hamiltonian

Abstract

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.