Bootstrap Embedding for Interacting Electron-Phonon Coupled Systems

Electron-phonon interactions are central for the formation of correlated phases like Mott insulators, polarons, and charge-density-wave states. However, accurately simulating these interactions is difficult because the Hilbert space for both electrons and bosons grows together. This work aims to create a scalable framework for handling coupled electron-phonon systems with accuracy, building on recent advancements in bootstrap embedding techniques for fermionic systems. We propose a fermi-bose bootstrap embedding (fb-BE) framework for studying the ground state of electron-phonon coupled systems. The approach integrates bootstrap embedding for correlated electrons with a self-consistent coherent state mean-field approximation for phonons. This approach represents the interacting electron-phonon problem as a system of correlated electrons moving within a self-consistently defined potential landscape, enabling efficient analysis of lattice systems. Benchmarking against density matrix renormalization group (DMRG) for the one-dimensional Hubbard-Holstein model at half-and quarter-filling indicates that the method exhibits optimal performance in regimes dominated by localization, such as the Mott insulating phase and the strong-coupling small polaron regime, where the local embedding ansatz remains applicable. However, we observe limitations in the weakly coupled delocalized region and at the Peierls transition, where quantum phonon fluctuations and long-range kinetic correlations become significant. This indicates that fb-BE is a scalable and precise method with significant potential for modeling strongly correlated materials where static lattice distortions are a crucial factor.