Numerical solution of the Holstein polaron problem

Physical Review B

We investigate the thermodynamics and finite-temperature spectral functions of the Holstein polaron using a density-matrix renormalization group method. Our method combines purification and local basis optimization (LBO) as an efficient treatment of phonon modes. LBO is a scheme which relies on finding the optimal local basis by diagonalizing the local reduced density matrix. By transforming the state into this basis, one can truncate the local Hilbert space with a negligible loss of accuracy for a wide range of parameters. In this work, we focus on the crossover regime between large and small polarons of the Holstein model. Here, no analytical solution exists and we show that the thermal expectation values at low temperatures are independent of the phonon Hilbert space truncation provided the basis is chosen large enough. We then demonstrate that we can extract the electron spectral function and establish consistency with results from a finite-temperature Lanczos method. We additionally calculate the electron emission spectrum and the phonon spectral function and show that all the computations are significantly simplified by the local basis optimization. We observe that the electron emission spectrum shifts spectral weight to both lower frequencies and larger momenta as the temperature is increased. The phonon spectral function experiences a large broadening and the polaron peak at large momenta gets significantly flattened and merges almost completely into the free-phonon peak.

2013

Noninteracting itinerant electrons in a solid occupy Bloch one-electron states. Phonons are collective vibrational excitations of the crystal lattice. The basic electron-phonon (EP) interaction process is the absorption or emission of a phonon by the electron with a simultaneous change of the electron state. From

EPL (Europhysics Letters), 2013

Polarons and electron-phonon interactions PACS 72.10.Di -Scattering by phonons, magnons, and other nonlocalized excitations PACS 63.20.kd -Phonon-electron interactions

Europhysics Letters (EPL), 2007

We generalize the Momentum Average approximations MA (0) and MA to study the effects of coupling to multiple optical phonons on the properties of a Holstein polaron. As for a single phonon mode, these approximations are numerically very efficient. They become exact for very weak or very strong couplings, and are highly accurate in the intermediate regimes, e.g. the spectral weights obey exactly the first six, respectively eight, sum rules. Our results show that the effect on ground-state properties is cumulative in nature. In particular, if the effective coupling to one mode is much larger than to the others, this mode effectively determines the GS properties. However, even very weak coupling to a second phonon mode has important non-perturbational effects on the higher energy spectrum, in particular on the dispersion and the phonon statistics of the polaron band.

European Physical Journal B, 2005

Polaron formation is investigated in a one-dimensional chain by taking into account both the local Holstein and the non-local SSH electron-phonon interactions. The study of the adiabatic regime points out that the combined effects of the two interactions are important mainly in the weak coupling regime. Thus, using the weak-coupling perturbation theory, spectral weights, effective masses, polaronic phase-diagram, and band structures are discussed. Contrarily to what happens in the Fröhlich and Holstein models, we find that the ratio between the coherent spectral weight and the mass renormalization ratio is greater than 1. Moreover, we show that the non-local electron-phonon interaction is responsible for the largest deviations of the band structure from the cosine shape of the free energy band.

2005

In system with strong electron-phonon (e-ph) interaction, the carriers lose mobility, ultimately acquiring polaronic character. A polaron is a state in which the phonon and electron degrees of freedom are strongly entangled, and the presence of an electron is associated to a finite lattice distortion, which in turn binds the electron leading to the so-called self-trapping effect. Polarons also tend to create bound pairs, called bipolarons of course the presence of Coulomb repulsion destabilize bi-polarons in favor of a pure polaronic state at finite densities [1,. Typical signatures of polarons are seen in multi-peaked photoemission spectra[3] and transport measurements, where an activated behavior with a characteristic energy given by the polaronic binding energy is observed. The polaronic peak found in the mid infrared measurements of optical conductivity [4] may also not only detect the polaronic binding energy [5] but also other subtle polaronic transitions at very low energy . Another less classical indication of polaronic formation comes from the analysis of lattice displacements associated to the excess charge as obtained by the distribution of distances between atoms[3].

Physical Review B, 2005

The polaron features due to electron-phonon interactions with different coupling ranges are investigated by adopting a variational approach. The ground-state energy, the spectral weight, the average kinetic energy, the mean number of phonons, and the electron-lattice correlation function are discussed for the system with coupling to local and nearest neighbor lattice displacements comparing the results with the long range case. For large values of the coupling with nearest neighbor sites, most physical quantities show a strong resemblance with those obtained for the long range electron-phonon interaction. Moreover, for intermediate values of interaction strength, the correlation function between electron and nearest neighbor lattice displacements is characterized by an upturn as function of the electron-phonon coupling constant.

Physical Review B

We use the variational exact diagonalization to investigate the single polaron properties for four different dual models, combining a short-range off-diagonal (Peierls) plus a longer-range diagonal (Holstein or breathingmode) coupling. This allows us to investigate the sensitivity of various polaron properties both to the range of the diagonal coupling and to the specific diagonal coupling chosen. We find strong sensitivity to the range for all dual models as the adiabatic limit is approached; however, considerable sensitivity is observed for some quantities even in the antiadiabatic limit. Also, strong dependence of the results on the specific form of the diagonal coupling is observed everywhere in the parameter space. Taken together, these results suggest that a careful consideration must be given to the specific coupling and its proper range, when quantitative comparisons with experiments are sought.

Physical Review B, 2006

The evolution of the properties of a finite density electronic system as a function of the electron-phonon coupling is investigated in the Holstein model using the dynamical mean-field theory ͑DMFT͒ that becomes exact in infinite dimensions. We compare the spinless fermion case, in which only isolated polarons can be formed, with the spinful model in which the polarons can bind and form bipolarons. In the latter case, the bipolaronic binding occurs within DMFT as a metal-insulator transition. In the adiabatic regime in which the phonon energy is small with respect to the electron hopping we compare numerically exact DMFT results with an analytical scheme inspired by the Born-Oppenheimer procedure. Within the latter approach, a truncation of the phononic Hilbert space leads to a mapping of the original model onto an Anderson spin-fermion model. In the anti-adiabatic regime ͑where the phonon energy exceeds the electronic scales͒ the standard treatment based on Lang-Firsov canonical transformation allows one to map the original model on to an attractive Hubbard model in the spinful case. The separate analysis of the two regimes supports the numerical evidence that the presence of well-defined polaronic lattice displacements is not necessarily associated to a metal-insulator transition, which is instead due to pairing between the carriers. The finite-dimensionality effects neglected in DMFT may lead to a finite conductivity in the bipolaronic state which is, however, not always associated with polaronic distortions. At the polaron crossover the Born-Oppenheimer approximation is shown to break down due to the entanglement of the electron-phonon state.

2021

The cross over from low to high carrier densities in a many-polaron system is studied in the framework of the one-dimensional spinless Holstein model, using unbiased numerical methods. Combining a novel quantum Monte Carlo approach and exact diagonalization, accurate results for the singleparticle spectrum and the electronic kinetic energy on fairly large systems are obtained. A detailed investigation of the quality of the Monte Carlo data is presented. In the physically most important adiabatic intermediate electron-phonon coupling regime, for which no analytical results are available, we observe a dissociation of polarons with increasing band filling, leading to normal metallic behavior, while for parameters favoring small polarons, no such density-driven changes occur. The present work points towards the inadequacy of single-polaron theories for a number of polaronic materials such as the manganites.