In this thesis, I investigate the transport and dynamical properties of electron-phonon coupled systems using numerical methods. I focus on the one-dimensional Holstein model, which incorporates both polaron and charge density wave (CDW) physics, and use matrix-product-state-based algorithms. In the first part, I present work where we combine purification with local basis optimization to study the finite-temperature properties of the model. This gives access to polaron spectral functions, polaron and bipolaron optical conductivity, and the energy-transport coefficient at finite filling. The second part focuses on the Holstein model coupled to two tight-binding leads. There, we investigate how current is transported through the structure and how CDW states break down. In the last part, we analyze the spread of an electron in a lattice and the decay of CDWs. We aim to understand if a full quantum mechanical treatment of the phonons is necessary or if trajectory-based algorithms are applicable.
A comprehensive picture of polaron and bipolaron physics is essential to understand the optical absorption spectrum in many materials with electron-phonon interactions. In particular, the finite-temperature properties are of interest since they play an important role in many experiments. Here, we combine the parallel two-site time-dependent variational principle algorithm (p2TDVP) with local basis optimization (LBO) and purification to calculate time-dependent current-current correlation functions. From this information, we extract the optical conductivity for the Holstein polaron and bipolaron with dispersive phonons at finite temperatures. For the polaron in the weak and intermediate electron-phonon coupling regimes, we analyze the influence of phonon dispersion relations on the spectra. For strong electron-phonon coupling, the known result of an asymmetric Gaussian is reproduced for a flat phonon band. For a finite phonon bandwidth, the center of the Gaussian is either shifted to larger or smaller frequencies, depending on the sign of the phonon hopping. We illustrate that this can be well understood by considering the Born-Oppenheimer surfaces. Similar behavior is seen for the bipolaron for strong coupling. For the bipolaron with weak and intermediate coupling strengths and a flat phonon band, we obtain two very different spectra. The latter also has a temperature-dependent resonance at a frequency below the phonon frequency.
We develop a process tensor approach for calculating the finite-temperature spectral functions of lattice polarons. We show that the finite-temperature phonon influence functional can be expressed as the elementwise exponential of a low-rank matrix product state, enabling an efficient summation of electron worldlines. An embedding construction embeds worldlines of different propagation times into a single matrix product state. This construction yields the propagator over the full simulated time interval in a single calculation. We benchmark this approach against available finite-temperature density-matrix renormalization group results for the one-dimensional Holstein polaron. We further obtain numerically converged spectra for the two-dimensional Holstein polaron. The spectra reveal a progressive transfer of spectral weight from coherent spectral features into an incoherent background with increasing temperature.
We study the symmetric Anderson-Holstein (AH) model at zero temperature with Wilson's numerical renormalization-group (NRG) technique to investigate the interplay between the electron-electron and electron-phonon interactions. An improved method for calculating the phonon propagator using the NRG technique is presented. It is more accurate and reliable than the previous methods in that it is written in the form of the Dyson equation which calculates the phonon renormalization explicitly, and satisfies the boson sum rule better. The method is applied to calculate the renormalized phonon propagators along with the electron propagators as the on-site Coulomb repulsion U and electron-phonon coupling constant g are varied. As g is increased from 0, the phonon mode gets softened (hardened) for $U\ensuremath{\gtrsim}(\ensuremath{\lesssim}{)U}_{s}(\ensuremath{\Delta})$ depending on the parameters of the AH model, U, and the hybridization $\ensuremath{\Delta}.$ And, for $g\ensuremath{\gtrsim}{g}_{\mathrm{co}},$ one crosses over to the regime where the phonon mode splits into two components, one of which developing into a soft mode and the other approaching back to the bare frequency as g is further increased. Correlated with the emergence of the soft mode, the central peak of the electron spectral function is severely suppressed. These NRG calculations are compared with the standard Green's-function results for the weak-coupling regime to understand the phonon renormalization and soft mode.
We use the density matrix renormalization group (DMRG) to compute the phonon spectral function of a one-dimensional spinless Holstein model doped with a low but finite carrier concentration,
x≤0.15, as a function of the electron-phonon coupling
λ. To the best of our knowledge, these are the first such results in this regime, complementing extensive prior work at the single-polaron level (
x→0). We find that significant phonon spectral weight is transferred both below, all the way down to
ω=0, and above the bare phonon energy
Ω, in stark contrast with the Kohn-anomaly phenomenology expected in the Migdal limit, where weight remains centered near
Ω with a kink at
q=2kF. No signature of this
2kF kink appears in our results. This behavior is captured qualitatively by the Random Phase Approximation (RPA), and semi-quantitatively, at negligible extra computational cost, by a ``dressed RPA'' scheme in which the electron addition propagator is renormalized using the Momentum Average (MA) approximation for the low-density electron-polaron. By contrast, adding the lowest-order vertex correction to this dressed scheme produces unphysical negative spectral weight, signaling that vertex and propagator dressings must be treated consistently once the propagators are dressed nonperturbatively. Our results provide an efficient approximation for the phonon spectral functions of low-density polaron metals, a regime relevant to weakly doped insulators.
We propose a density-matrix renormalization-group (DMRG) approach to study lattices including bosons. The key to the approach is an exact mapping of a boson site containing
2N states to
N pseudosites, each with 2 states. The pseudosites can be viewed as the binary digits of a boson level. We apply the pseudosite DMRG method to the polaron problem in the one- and two-dimensional Holstein models. Ground-state results are presented for a wide range of electron-phonon coupling strengths and phonon frequencies on lattices large enough (up to 80 sites in one dimension and up to $20\ifmmode\times\else\texttimes\fi{}20$ sites in two dimensions) to eliminate finite-size effects, with up to 128 phonon states per phonon mode. We find a smooth but quite abrupt crossover from a quasi-free-electron ground state with a slightly renormalized mass at weak electron-phonon coupling to a polaronic ground state with a large effective mass at strong coupling, in agreement with previous studies.