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Спектралне функције и покретљивост Холштајновог поларона

Petar Mitrić

Published Dec 21, 2023• 0 citations
OpenAlex

Abstract

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.

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Dynamical mean field theory of polarons and bipolarons in the half-filled Holstein model

Massimo Capone, P. Carta, Sergio Ciuchi

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.

arXiv:cond-mat/0509542•Physical Review B•Jul 10, 2006•38 citations

Cumulant expansion in the Holstein model: Spectral functions and mobility

Petar Mitrić, Veljko Janković, Nenad Vukmirović +1 more

We examine the range of validity of the second-order cumulant expansion (CE) for the calculation of spectral functions, quasiparticle properties, and mobility of the Holstein polaron. We devise an efficient numerical implementation that allows us to make comparisons in a broad interval of temperature, electron-phonon coupling, and phonon frequency. For a benchmark, we use the dynamical mean-field theory which gives, as we have recently shown, rather accurate spectral functions in the whole parameter space, even in low dimensions. We find that in one dimension, the CE resolves well both the quasiparticle and the first satellite peak in a regime of intermediate coupling. At high temperatures, the charge mobility assumes a power law $\ensuremath{\mu}\ensuremath{\propto}{T}^{\ensuremath{-}2}$ in the limit of weak coupling and $\ensuremath{\mu}\ensuremath{\propto}{T}^{\ensuremath{-}3/2}$ for stronger coupling. We find that, for stronger coupling, the CE gives slightly better results than the self-consistent Migdal approximation (SCMA), while the one-shot Migdal approximation is appropriate only for a very weak electron-phonon interaction. We also analyze the atomic limit and the spectral sum rules. We derive an analytical expression for the moments in CE and find that they are exact up to the fourth order, as opposed to the SCMA where they are exact to the third order. Finally, we analyze the results in higher dimensions.

arXiv:2212.13846•Physical review. B./Physical review. B•Mar 30, 2023•16 citations

Phonon spectral function of the Holstein polaron

J Loos, M Hohenadler, A Alvermann +1 more

The phonon spectral function of the one-dimensional Holstein model is obtained within weak and strong-coupling approximations based on analytical self-energy calculations. The characteristic excitations found in the limit of small charge-carrier density are related to the known (electronic) spectral properties of Holstein polarons such as the polaron band dispersion. Particular emphasis is laid on the different physics occurring in the adiabatic and anti-adiabatic regimes, respectively. Comparison is made with a cluster approach exploiting exact numerical results on small systems to yield an approximation for the thermodynamic limit. This method, similar to cluster perturbation theory, confirms the analytical findings, and yields accurate results also in the intermediate-coupling regime.

arXiv:cond-mat/0604040•Journal of Physics Condensed Matter•Jul 21, 2006•20 citations

Phonon spectral function of the Holstein polaron

Jan Loos, Martin Hohenadler, Andreas Alvermann +1 more

The phonon spectral function of the one-dimensional Holstein model is obtained within weak and strong-coupling approximations based on analytical self-energy calculations.The characteristic excitations found in the limit of small charge-carrier density are related to the known (electronic) spectral properties of Holstein polarons such as the polaron band dispersion.Particular emphasis is laid on the different physics occurring in the adiabatic and antiadiabatic regimes, respectively.Comparison is made with a cluster approach exploiting exact numerical results on small systems to yield an approximation for the thermodynamic limit.This method, similar to cluster perturbation theory, confirms the analytical findings, and yields accurate results also in the intermediate-coupling regime.

•Jan 1, 2006•12 citations

Diagrammatic content of the dynamical mean-field theory for the Holstein polaron problem in finite dimensions

O. S. Barišić

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.

arXiv:0706.2582•Physical Review B•Nov 26, 2007•9 citations

Polarons and slow quantum phonons

Andreas Alvermann, Holger Fehske, S. A. Trugman

We describe the formation and properties of Holstein polarons in the entire parameter regime. Our presentation focuses on the polaron mass and radius, which we obtain with an improved numerical technique. It is based on the combination of variational exact diagonalization with an improved construction of phonon states, providing results even for the strong coupling adiabatic regime. In particular we can describe the formation of large and heavy adiabatic polarons. A comparison of the polaron mass for the one- and three-dimensional situation explains how the different properties in the static oscillator limit determine the behavior in the adiabatic regime. The transport properties of large and small polarons are characterized by the $f$-sum rule and the optical conductivity. Our calculations are approximation free and have negligible numerical error. This allows us to give a conclusive and impartial description of polaron formation. We finally discuss the implications of our results for situations beyond the Holstein model.

arXiv:1001.2482•Physical Review B•Apr 21, 2010•49 citations
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.
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 examine the range of validity of the second-order cumulant expansion (CE) for the calculation of spectral functions, quasiparticle properties, and mobility of the Holstein polaron. We devise an efficient numerical implementation that allows us to make comparisons in a broad interval of temperature, electron-phonon coupling, and phonon frequency. For a benchmark, we use the dynamical mean-field theory which gives, as we have recently shown, rather accurate spectral functions in the whole parameter space, even in low dimensions. We find that in one dimension, the CE resolves well both the quasiparticle and the first satellite peak in a regime of intermediate coupling. At high temperatures, the charge mobility assumes a power law $\ensuremath{\mu}\ensuremath{\propto}{T}^{\ensuremath{-}2}$ in the limit of weak coupling and $\ensuremath{\mu}\ensuremath{\propto}{T}^{\ensuremath{-}3/2}$ for stronger coupling. We find that, for stronger coupling, the CE gives slightly better results than the self-consistent Migdal approximation (SCMA), while the one-shot Migdal approximation is appropriate only for a very weak electron-phonon interaction. We also analyze the atomic limit and the spectral sum rules. We derive an analytical expression for the moments in CE and find that they are exact up to the fourth order, as opposed to the SCMA where they are exact to the third order. Finally, we analyze the results in higher dimensions.
The phonon spectral function of the one-dimensional Holstein model is obtained within weak and strong-coupling approximations based on analytical self-energy calculations. The characteristic excitations found in the limit of small charge-carrier density are related to the known (electronic) spectral properties of Holstein polarons such as the polaron band dispersion. Particular emphasis is laid on the different physics occurring in the adiabatic and anti-adiabatic regimes, respectively. Comparison is made with a cluster approach exploiting exact numerical results on small systems to yield an approximation for the thermodynamic limit. This method, similar to cluster perturbation theory, confirms the analytical findings, and yields accurate results also in the intermediate-coupling regime.
The phonon spectral function of the one-dimensional Holstein model is obtained within weak and strong-coupling approximations based on analytical self-energy calculations.The characteristic excitations found in the limit of small charge-carrier density are related to the known (electronic) spectral properties of Holstein polarons such as the polaron band dispersion.Particular emphasis is laid on the different physics occurring in the adiabatic and antiadiabatic regimes, respectively.Comparison is made with a cluster approach exploiting exact numerical results on small systems to yield an approximation for the thermodynamic limit.This method, similar to cluster perturbation theory, confirms the analytical findings, and yields accurate results also in the intermediate-coupling regime.
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.
We describe the formation and properties of Holstein polarons in the entire parameter regime. Our presentation focuses on the polaron mass and radius, which we obtain with an improved numerical technique. It is based on the combination of variational exact diagonalization with an improved construction of phonon states, providing results even for the strong coupling adiabatic regime. In particular we can describe the formation of large and heavy adiabatic polarons. A comparison of the polaron mass for the one- and three-dimensional situation explains how the different properties in the static oscillator limit determine the behavior in the adiabatic regime. The transport properties of large and small polarons are characterized by the fff-sum rule and the optical conductivity. Our calculations are approximation free and have negligible numerical error. This allows us to give a conclusive and impartial description of polaron formation. We finally discuss the implications of our results for situations beyond the Holstein model.