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

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.

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• Phase Transitions in Particle Physics -- Results and Perspectives from Lattice Quantum Chromo-Dynamics
• Efficient Modelling of Trivializing Maps for Lattice ${\varphi }^4$ Theory Using Normalizing Flows: A First Look at Scalability
• Quantum field-theoretic machine learning
• Quantitative analysis of phase transitions in two-dimensional XY models using persistent homology
• Inverse Renormalization Group in Quantum Field Theory
• Signal detection in nearly continuous spectra and symmetry breaking
• A simple framework for contrastive learning phases of matter
• Quantum field theories, Markov random fields and machine learning
• Reducing finite-size effects with reweighted renormalization group transformations
• Overlap renormalization group transformations for disordered systems