Momentum average approximation for models with boson-modulated hopping: Role of closed loops in the dynamical generation of a finite quasiparticle mass
We generalize the momentum average approximation to study the properties of single polarons in models with boson affected hopping, where the fermion-boson scattering depends explicitly on both the fermion's and the boson's momentum. As a specific example, we investigate the Edwards fermion-boson model in both one and two dimensions. In one dimension, this allows us to compare our results with exact diagonalization results, to validate the accuracy of our approximation. The generalization to two-dimensional lattices allows us to calculate the polaron's quasiparticle weight and dispersion throughout the Brillouin zone and to demonstrate the importance of Trugman loops in generating a finite effective mass even when the free fermion has an infinite mass.
We generalize the momentum average approximation to study the properties of single polarons in models with boson affected hopping, where the fermion-boson scattering depends explicitly on both the fermion's and the boson's momentum. As a specific example, we investigate the Edwards fermion-boson model in both one and two dimensions. In one dimension, this allows us to compare our results with exact diagonalization results, to validate the accuracy of our approximation. The generalization to two-dimensional lattices allows us to calculate the polaron's quasiparticle weight and dispersion throughout the Brillouin zone and to demonstrate the importance of Trugman loops in generating a finite effective mass even when the free fermion has an infinite mass.
Employing a self-consistent (optimized) variational diagonalization scheme, we investigate the formation of polaronic quasiparticles in a spinless fermion-boson transport model that couples the movement of charge carriers to fluctuations and correlations of a background medium. The background is parameterized by bosonic degrees of freedom. The variational fermion-boson Hilbert space is constructed to achieve high accuracy in one to three spatial dimensions with modest computational requirements. To characterize the ground-state properties of the Edwards model in the single-particle sector, we present exact numerical results for the polaron band dispersion, quasiparticle weight, Drude weight, mass enhancement, and the particle-boson correlations in a wide parameter regime. In the Edwards model, transport will be quasifree, diffusive or boson-assisted in the weakly fermion-boson coupled, fluctuation-dominated or strongly correlated regimes, respectively. Thereby correlated transport is not only assisted but also limited by the bosonic excitations. As a result, the Drude weight remains finite even in the limit of very small boson frequencies. For a strongly correlated background, closed loops are important, in any dimension, to generate a finite effective particle mass even when the free fermion has an infinite mass.
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.
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.
We explore the quasiparticle properties of lattice polarons on the basis of a quite general electron-phonon Hamiltonian with a long-range displacement-type interaction. To treat the dynamical quantum phonons without significant loss of accuracy we adapt an exact Lanczos diagonalization method and compute various static and dynamical quantities, such as the electron-lattice correlation function, the polaron band dispersion, the effective polaron mass, the kinetic energy, the single-particle spectral function, and the optical conductivity, on finite one-dimensional lattices for a wide range of model parameters. We compare the results with those obtained for the standard Holstein model with short-range electron-phonon interaction only.
Allowing a single electron to hop to next-nearest neighbours (NNN) in addition to the closest atomic sites in the Holstein model, a modified Trugman method is applied to exactly calculate the effect on the polaronic effective mass in one, two, and three dimensions, building on the previous study of the one-dimensional NNN Holstein model. We also present perturbative calculations and a heuristic scaling factor for the coupling strength and ion frequency to nearly map the NNN Holstein model back onto the original Holstein model. When account is taken of the modified electronic bandwidth near the electron energy, we find that including NNN hopping effectively increases the polaron effective mass.
Many condensed matter problems involve a particle coupled to its environment. The polaron, originally introduced to describe electrons in a polarizable medium, describes a particle coupled to a bosonic field. The Holstein polaron model, although simple, including only optical Einstein phonons and an interaction that couples them to the electron density, captures almost all of the standard polaronic properties. We herein investigate polarons that differ significantly from this behaviour. We study a model with phonon-modulated hopping, and find a radically different behaviour at strong couplings. We report a sharp transition, not a crossover, with a diverging effective mass at the critical coupling. We also look at a model with acoustic phonons, away from the perturbative limit, and again discover unusual polaron properties. Our work relies on the Bold Diagrammatic Monte Carlo (BDMC) method, which samples Feynman diagrammatic expansions efficiently, even those with weak sign problems. Proposed by Prokof'ev and Svistunov, it is extended to lattice polarons for the first time here. We also use the Momentum Average (MA) approximation, an analytical method proposed by Berciu, and find an excellent agreement with the BDMC results. A novel MA approximation able to treat dispersive phonons is also presented, along with a new exact solution for finite systems, inspired by the same formalism.