O.-Y. Tong
The Momentum Average (MA) approximation is a variational approximation that has been applied to analytically calculate the spectral function of single polarons and bipolarons, and was found to quantitatively match numerical results apart from the strongly adiabatic limit. The MA approximation was recently extended to a low carrier concentration of spinless fermions in 1D by making the additional assumption that the N-fermion ground-state is approximated well by the mean-field ground-state. This allowed the spectral function to be calculated analytically outside the Migdal limit. Good agreement was found with Density Matrix Renormalization Group simulations for fermion concentrations up to 0.15. In this thesis, the same approximation is extended to calculate the spectral function of a Holstein polaron for low carrier concentrations of spinless fermions on square and cubic lattices. Polaron effective mass and quasiparticle weight were found to depend strongly on the concentration x, especially in 3D. The 2D mass renormalizations were found to give a close match (< 5% error) to Bold Diagrammatic Monte Carlo (BDMC) results from the literature at all x < 0.2 when coupling is weak. However, there were significant quantitative differences between MA and BDMC results at large coupling. A possible explanation for this discrepancy is the presence of a soft excitation close to the polaron ground state. In addition, an efficient numerical scheme to calculate the real-space free electron propagator for a noninteracting Fermi sea is presented, which is used to calculate the self-energy for the 3D Green’s function at finite concentrations.