In the context of the Holstein polaron problem, it is shown that the dynamical mean-field theory (DMFT) corresponds to the summation of a special class of local diagrams in the skeleton expansion of the self-energy. In the real space representation, these local diagrams are characterized by the absence of vertex corrections involving phonons at different lattice sites. Such corrections vanish in the limit of infinite dimensions, for which the DMFT provides the exact solution of the Holstein polaron problem. However, for finite dimensional systems, the accuracy of the DMFT is limited. In particular, it cannot describe correctly the adiabatic spreading of the polaron over multiple lattice sites. Arguments are given that the DMFT limitations on vertex corrections found for the Holstein polaron problem persist for finite electron densities and arbitrary phonon dispersion.
In the context of the Holstein polaron problem, it is shown that the dynamical mean-field theory (DMFT) corresponds to the summation of a special class of local diagrams in the skeleton expansion of the self-energy. In the real space representation, these local diagrams are characterized by the absence of vertex corrections involving phonons at different lattice sites. Such corrections vanish in the limit of infinite dimensions, for which the DMFT provides the exact solution of the Holstein polaron problem. However, for finite dimensional systems, the accuracy of the DMFT is limited. In particular, it cannot describe correctly the adiabatic spreading of the polaron over multiple lattice sites. Arguments are given that the DMFT limitations on vertex corrections found for the Holstein polaron problem persist for finite electron densities and arbitrary phonon dispersion.
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 (DMFT) 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 scales) 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.
We present an inhomogeneous dynamical mean field theory (I-DMFT) that is suitable to investigate electron-lattice interactions in non-translationally invariant and/or inhomogeneous systems. The presented approach, whose only assumption is that of a local, site-dependent self-energy, recovers both the exact solution of an electron for a generic random tight-binding Hamiltonian in the non-interacting limit and the DMFT solution for the small polaron problem in translationally invariant systems. To illustrate its full capabilities, we use I-DMFT to study the effects of defects embedded on a two-dimensional surface. The computed maps of the local density of states reveal Friedel oscillations, whose periodicity is determined by the polaron mass. This can be of direct relevance for the interpretation of scanning-tunneling microscopy experiments on systems with sizable electron-lattice interactions. Overall, the easy numerical implementation of the method, yet full self-consistency, allows one to study problems in real-space that were previously difficult to access.
The structure of the translationally-invariant diagrammatic perturbation theory for one polaron is examined on the 1D discrete lattice described by the Holstein Hamiltonian. The latter is characterized by the electron hopping t, the phonon frequency ω_0 and the electron-phonon coupling g. It is shown that the polaron localization (and translation) properties are contained in the electron propagator of one electron, intermittently added to the lattice, and/or in the phonon correlation function with one electron permanently present in the lattice. The order by order analysis in g/ω_0 shows that the expansion of the irreducible electron self-energy corresponds to the expansion of the phonon correlation function, rather than of the irreducible phonon self-energy. The range of polaronic correlations is determined in this way. For small t/ω_0 and already to the second order g/ω_0 small, the electron-lattice correlation becomes very short ranged, i.e. the polaron is already localized to one site, although the overall translational symmetry remains unbroken. For large t/ω0, the second order result is meaningful up to large g/ω_0 ≈ (t/ω_0)^\frac14, where it becomes degenerate with the results for the large adiabatic Holstein polaron. This suggests that the translationally invariant perturbation theory crosses then over smoothly, without symmetry breaking, into the adiabatic, continuous quantum limit, as rigorously demonstrated in the companion paper. Thus the quantum theory of the large adiabatic Holstein polaron provides a simple, instructive example of the quantum crossover which replaces the behavior in the quantum critical point.
The electron-phonon interaction significantly affects the properties of semiconducting materials. Because of it, the phononic cloud can renormalize electrons, which leads to the emergence of polarons - a new quasiparticle that now, instead of the electron, plays the role of the current carrier in our system. The consequences of polaron formation are most easily studied using simplified models of electronphonon systems. Among these models, the simplest one is the Holstein model, which successfully reproduces the most important polaronic effects. In practice, the Holstein model is used for testing and developing various theoretical methods that can subsequently be applied to more complex models or even real materials. The goal of this dissertation is to investigate the single-particle and transport properties of the Holstein model using different methods. Until recently, it was widely accepted that the dynamical mean-field theory (DMFT) provides a good description of the single-particle properties of the Holstein model only in the cases of threedimensional or even higher-dimensional systems. However, our results show that DMFT actually provides an excellent description of single-particle properties even in the one-dimensional case, regardless of the regime, which is determined by temperature, phonon frequency, and electron-phonon coupling strength. We have reached these conclusions by comparing the results obtained using this method, with the most reliable results currently available in the literature. Although DMFT is approximate, it is also a nonperturbative method that is exact in two different limits: in the weak coupling limit and in the atomic limit. Having in mind that DMFT neglects non-local correlations, which are most pronounced in the one-dimensional case, our conclusions about the high reliability of this method are expected to continue to hold in an arbitrary number of dimensions as well. This has been explicitly verified on the example of the effective mass in one-, two-, and three-dimensional cases. In addition, we have also presented a numerical procedure for the application of DMFT that requires very little computational resources. Therefore, this method allows us to easily generate a large amount of reliable results in different regimes, which can now be used to assess the quality of any other method. One such method that we intend to investigate more thoroughly is the cumulant expansion (CE) Method. In contrast to DMFT, the CE is a perturbative method that does not rely on Dyson’s equation for the calculation of the single-particle properties. Although CE does not provide reliable results in all regimes, the advantage of this method in comparison to the DMFT is that it can be easily applied to significantly more complex models, and even to real materials. Therefore, it is very important to determine in which parameter regimes can CE be expected to give an adequate description of the observed physical system. In this dissertation, this was investigated on the example of the Holstein model, by comparing the CE with the DMFT results, which we have already established as reliable. It turns out that, although CE is exact only in the weak coupling and atomic limits, reliable approximate predictions of this method are possible even for moderate interactions, where the corresponding spectral function accurately reproduces both the quasiparticle and the first satellite peak. This is signifix icantly better than what would be obtained using the lowest-order perturbation theory. In addition, the high-temperature results of the CE look promising, although we proved, using the spectral sum rules, that this method cannot be exact in the limit T → ∞. For the study of transport properties, we focused on calculating mobility and a somewhat more general quantity, the optical conductivity.Within the framework of linear response theory, both of these quantities can be represented as the sum of the so-called bubble term, determined by the single-particle properties, and vertex corrections. The bubble term for mobility μ was calculated numerically, and detailed comparisons were made between the DMFT and CE predictions. We established that at high temperatures, the charge mobility assumes a power law μ ∝ T−2 in the case of very weak coupling, and μ ∝ T−3/2 for somewhat stronger coupling. We analytically proved that in the weak coupling and atomic limits of the Holstein model, the vertex corrections of mobility are vanishing. In all other regimes, the contribution of vertex corrections was examined numerically, by calculating the bubble term using the DMFT and by comparing it to the exact result from the literature.
We derive sum rules for the phonon self-energy and the electron-phonon contribution to the electron self-energy of the Holstein-Hubbard model in the limit of large Coulomb interaction U. Their relevance for finite U is investigated using exact diagonalization and dynamical mean-field theory. Based on these sum rules, we study the importance of vertex corrections to the electron-phonon interaction in a diagrammatic approach. We show that they are crucial for a sum rule for the electron self-energy in the undoped system while a sum rule related to the phonon self-energy of doped systems is satisfied even if vertex corrections are neglected. We provide explicit results for the vertex function of a two-site model.
We introduce the first bold diagrammatic Monte Carlo approach to deal with polaron problems at a finite electron density nonperturbatively, i.e., by including vertex corrections to high orders. Using the Holstein model on a square lattice as a prototypical example, we demonstrate that our method is capable of providing accurate results in the thermodynamic limit in all regimes from a renormalized Fermi liquid to a single polaron, across the nonadiabatic region where Fermi and Debye energies are of the same order of magnitude. By accounting for vertex corrections, the accuracy of the theoretical description is increased by orders of magnitude relative to the lowest-order self-consistent Born approximation employed in most studies. We also find that for the electron-phonon coupling typical for real materials, the quasiparticle effective mass increases and the quasiparticle residue decreases with increasing the electron density at constant electron-phonon coupling strength.