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

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 can only cause explicit symmetry breaking. The critical behaviour is then studied by proposing reweighting that is agnostic to the original Hamiltonian and forming a renormalization group mapping on quantities derived from the neural network. Accurate estimates of the critical fixed point and the operators that govern the divergence of the correlation length are provided. We conclude by discussing how the method provides an essential step towards bridging machine learning and physics.