Machine learning effective models from a Boltzmann perspective

We investigate the derivation of effective models for quantum impurity type problems using machine learning methods. Parameters of the effective model are optimized with respect to the parent Hamiltonian by using a classical probability distribution extracted from a diagrammatic expansion of the partition function. The classical probability distributions are naturally in the form of an energy-based model within this framework, making clear the connection to Boltzmann machines. In this case, the energy-based model is the effective model and has a physical meaning. The information geometry inspired derivation of the cost function predicts that the best fitting effective model has the same thermal expectation value of effective interactions as in the parent model. However, we show that this does not necessarily yield an effective model with the same low-energy physics as the parent model due to information monotonicity along RG flow [1].

[1] J. B. Rigo and A. K. Mitchell, arXiv:1910.11300 (2019)