Adding machine learning within Hamiltonians: Renormalization group transformations, symmetry breaking and restoration
This work bridges machine learning and physics by offering a novel approach to manipulate statistical systems, potentially impacting theoretical physics and computational modeling.
The authors introduced a method to incorporate neural network functions into Hamiltonians, enabling control over phase transitions in the two-dimensional Ising model, where an external field induced order-disorder transitions and provided accurate critical point and exponent estimates.
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.