Extrapolation of polaron properties to low phonon frequencies by Bayesian machine learning

Feasibility of accurate quantum calculations is often restricted by the dimensionality of the truncated Hilbert space required for the numerical computations. The present work demonstrates Bayesian machine learning (ML) models that use quantum properties in an effectively lower-dimensional Hilbert space to make predictions for the Hamiltonian parameters that require a larger basis set as applied to a classical problem in quantum statistical mechanics, the polaron problem. We consider two polaron models: the Su-Schrieffer-Heeger (SSH) model and the mixed SSH-Holstein model. We demonstrate ML models that can extrapolate polaron properties in the phonon frequency. We consider the sharp transition in the ground-state momentum of the SSH polaron and examine the evolution of this transition from the anti-adiabatic regime to the adiabatic regime. We also demonstrate Bayesian models that use the posterior distributions of highly approximate quantum calculations as the prior distribution for models of more accurate quantum results. This drastically reduces the number of fully converged quantum calculations required to map out the polaron dispersion relations for the full range of Hamiltonian parameters of interest.