Full configuration interaction quantum Monte Carlo for coupled electron--boson systems and infinite spaces

We extend the scope of full configuration interaction quantum Monte Carlo (FCIQMC) to be applied to coupled fermion-boson hamiltonians, alleviating the a priori truncation in boson occupation which is necessary for many other wave function based approaches to be tractable. Detailing the required algorithmic changes for efficient excitation generation, we apply FCIQMC in two contrasting settings. The first is a sign-problem-free Hubbard--Holstein model of local electron-phonon interactions, where we show that with care to control for population bias via importance sampling and/or reweighting, the method can achieve unbiased energies extrapolated to the thermodynamic limit, without suffering additional computational overheads from relaxing boson occupation constraints. Secondly, we apply the method as a `solver' within a quantum embedding scheme which maps electronic systems to local electron-boson auxiliary models, with the bosons representing coupling to long-range plasmonic-like fluctuations. We are able to sample these general electron-boson hamiltonians with ease despite a formal sign problem, including a faithful reconstruction of converged reduced density matrices of the system.