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.
Highlights
At the heart of our understanding of phase transitions lies a mathematical apparatus called the renormalization group [1,2,3,4,5,6]
By giving a physical interpretation to the function of a neural network as a Hamiltonian term, we explore the effect of the coupled field on the considered system, demonstrate that it can break or restore the reflection symmetry by inducing a phase transition, and extract with high accuracy the location of the critical inverse temperature and the operators of the renormalization group transformation that govern the divergence of the correlation length
The inclusion of the predictive function in the Hamiltonian enables the calculation of a relevant operator, namely, the magnetic field exponent θ, that was previously inaccessible through supervised machine learning methods which are agnostic to the symmetries of the system
Summary
At the heart of our understanding of phase transitions lies a mathematical apparatus called the renormalization group [1,2,3,4,5,6]. We consider the predictive function of a neural network, designed for phase classification, as a conjugate variable coupled to an external field, and introduce it as a term in the Hamiltonian of a system Given this formulation, we propose reweighting that is agnostic to the original Hamiltonian to explore if the external field generates a richer structure than the one associated with the conventional order parameter, and if it can induce a phase transition by breaking or restoring the system’s symmetry. By giving a physical interpretation to the function of a neural network as a Hamiltonian term, we explore the effect of the coupled field on the considered system, demonstrate that it can break or restore the reflection symmetry by inducing a phase transition, and extract with high accuracy the location of the critical inverse temperature and the operators of the renormalization group transformation that govern the divergence of the correlation length
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