Adding machine learning within Hamiltonians: Renormalization group transformations, symmetry breaking and restoration

TL;DR

This paper integrates neural network functions into Hamiltonians to control phase transitions in statistical systems, demonstrating how machine learning can influence physical properties and critical phenomena.

Contribution

It introduces a novel approach to embed machine learning functions within Hamiltonians, enabling control over phase transitions and providing a new perspective on the intersection of ML and physics.

Findings

Neural network functions can induce symmetry-breaking phase transitions.

The method accurately estimates critical points and exponents.

The approach bridges machine learning and statistical physics.

Abstract

We present a physical interpretation of machine learning functions, opening up the possibility to control properties of statistical systems via the inclusion of these functions in Hamiltonians. In particular, we include the predictive function of a neural network, designed for phase classification, as a conjugate variable coupled to an external field within the Hamiltonian of a system. Results in the two-dimensional Ising model evidence that the field can induce an order-disorder phase transition by breaking or restoring the symmetry, in contrast with the field of the conventional order parameter which causes explicit symmetry breaking. The critical behavior is then studied by proposing a Hamiltonian-agnostic reweighting approach and forming a renormalization group mapping on quantities derived from the neural network. Accurate estimates of the critical point and of the critical exponents related to the operators that govern the divergence of the correlation length are provided. We conclude by discussing how the method provides an essential step toward bridging machine learning and physics.