A Machine Learning Approach to Duality in Statistical Physics

The notion of duality -- that a given physical system can have two different mathematical descriptions -- is a key idea in modern theoretical physics. Establishing a duality in lattice statistical mechanics models requires the construction of a dual Hamiltonian and a map from the original to the dual observables. By using neural networks to parameterize these maps and introducing a loss function that penalises the difference between correlation functions in original and dual models, we formulate the process of duality discovery as an optimization problem. We numerically solve this problem and show that our framework can rediscover the celebrated Kramers-Wannier duality for the 2d Ising model, numerically reconstructing the known mapping of temperatures. We further investigate the 2d Ising model deformed by a plaquette coupling and find families of ``approximate duals''. We discuss future directions and prospects for discovering new dualities within this framework.

Must there be a unique, efficient mathematical description of a physical system? In the last century, we've learned that in general, this is not the case. A physical system may have several inequivalent descriptions that appear very different and are useful for different purposes; this phenomenon is referred to by physicists as a "duality". In this work, we explore the potential of generating entirely new types of "dual" descriptions using machine learning. The main new insight is that we use neural networks to parametrise potential descriptions of a given system, and train them so that many perfect ones can be found. We study this in some simple examples from statistical physics (closely related to how phenomena such as the magnetism of macroscopic objects are described at an atomic level) and show that we can automatically rediscover existing results, as well as make some progress towards discovering new dualities.