Charge Density Wave Order in the Half-Filled Three Dimensional Holstein Model

The Holstein Model (HM) is a tight-binding Hamiltonian describing electrons coupled to local phonon degrees of freedom. The electron phonon interaction gives rise to charge density wave (CDW) order at half-filling and superconducting (SC) order away from half-filling. The HM has been a recent focus for Determinant Quantum Monte Carlo (DQMC) simulation studies. However, the cubic scaling with system size, and long autocorrelation times, have restricted simulations to one and two dimensional systems. Here we demonstrate that the recent development of Self-Learning Monte Carlo (SLMC)[1,2] has made it possible to significantly reduce autocorrelation times, thereby allowing for simulation studies of the Holstein model on a fully three-dimensional cubic lattice at half-filling. SLMC is able to control autocorrelation times by learning an effective model that is used to propose more effective monte carlo moves, while still satisfying detailed-balance. SLMC allows us to not only access larger system sizes, but also slower phonon modes where the phonon frequency ω is less than the inter-site hopping parameter t.