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

arXiv: https://arxiv.org/abs/2010.00054 · HTML: https://mlsanity.com/paper/2010.00054/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Sep 2020.

## Built on by (most cited first, 7 of 7)

- [Quantum field-theoretic machine learning](https://mlsanity.com/paper/2102.09449/) (2021)
- [Applications of Machine Learning to Lattice Quantum Field Theory](https://mlsanity.com/paper/2202.05838/) (2022)
- [A simple framework for contrastive learning phases of matter](https://mlsanity.com/paper/2205.05607/) (2022)
- [Machine learning with quantum field theories](https://mlsanity.com/paper/2109.07730/) (2021)
- [Quantum field theories, Markov random fields and machine learning](https://mlsanity.com/paper/2110.10928/) (2021)
- [Quantitative analysis of phase transitions in two-dimensional XY models using persistent homology](https://mlsanity.com/paper/2109.10960/) (2021)
- [Feature extraction of machine learning and phase transition point of Ising model](https://mlsanity.com/paper/2111.11166/) (2021)

## Abstract (arXiv)

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.