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arXiv:cond-mat/0004388

Phonon-Induced Renormalization and Interaction: An Improvement on Frohlich Transformation

Han Zheng

Published Apr 24, 2000• 0 citations
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Abstract

Starting from the Flohlich electron-phonon model, which has a history of 50 years, a new unitary transformation is proposed to implement the perturbation treatment. Our main results are: (1)The phonon-induced interaction shows a crossover from the BCS-like potential when the phonon frequency $ω_p$ is much smaller than the Fermi energy $E_F$ to that of the small polarons when $ω_p/E_F\ge 1$. (2)The jump of momentum distribution of electron number $n_{\bf k}$ at the Fermi surface goes to zero when the dimensionless coupling constant $λ$ increases to the critical value $λ_c\ge 1$, which means a possible broken down of the Fermi-liquid description.

DOI
10.48550/arxiv.cond-mat/0004388

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Starting from the Flohlich electron-phonon model, which has a history of 50 years, a new unitary transformation is proposed to implement the perturbation treatment. Our main results are: (1)The phonon-induced interaction shows a crossover from the BCS-like potential when the phonon frequency ωpω_pωp​ is much smaller than the Fermi energy EFE_FEF​ to that of the small polarons when ωp/EF≥1ω_p/E_F\ge 1ωp​/EF​≥1. (2)The jump of momentum distribution of electron number nkn_{\bf k}nk​ at the Fermi surface goes to zero when the dimensionless coupling constant λλλ increases to the critical value λc≥1λ_c\ge 1λc​≥1, which means a possible broken down of the Fermi-liquid description.
We have developed a new approach, starting from the Frohlich Hamiltonian and by means of the unitary transformation with the perturbation treatment, to deal with the momentum-dependent spectral functions (MDSF) of electron-phonon coupling system. The approach takes care of the fact that only those electrons near the Fermi surface within a layer of width ${\ensuremath{\omega}}_{ph}$ (phonon frequency) are scattered by phonons strongly. Our results show that the MDSF is mainly composed of two parts: the zero-phonon part and the one-phonon one. When the bare electron energy ϵk{ϵ}_{\mathbf{k}}ϵk​ is close to the Fermi energy EF{E}_{F}EF​, $\ensuremath{\mid}{ϵ}_{\mathbf{k}}\ensuremath{-}{E}_{F}\ensuremath{\mid}\ensuremath{\simeq}{\ensuremath{\omega}}_{ph}$, the one-phonon part is quite significant and the MDSF shows a two-headed peak. However, for $\ensuremath{\mid}{ϵ}_{\mathbf{k}}\ensuremath{-}{E}_{F}\ensuremath{\mid}⪢{\ensuremath{\omega}}_{ph}$, the one-phonon part disappears gradually and the MDSF shows a single broad peak. Our calculation for MDSF is qualitatively in agreement with recent measurement of angle-resolved photoemission spectroscopy on the Be(0001) surface and Bi2Sr2CaCu2O8{\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}\mathrm{Ca}{\mathrm{Cu}}_{2}{\mathrm{O}}_{8}Bi2​Sr2​CaCu2​O8​. Besides, the two-headed structure of MDSF indicates the ``hybridization'' between electrons and phonons.
Higher-order terms of the transformed electron-phonon Hamiltonian ${H}_{e\text{\ensuremath{-}}\mathit{ph}}$, obtained by performing the Fr\"ohlich transformation, are investigated. The influence of terms discarded by Fr\"ohlich (in particular those proportional to the third power of electron-phonon coupling) on the effective Hamiltonian is examined. To this end a second Fr\"ohlich-type transformation is performed, which yields, among others, an effective four-electron interaction. This interaction is reduced to a form admitting solution of thermodynamics. The form of the coupling of the four-electron interaction is found. By applying standard approximations, it is shown that this interaction is attractive with the interaction coupling given by $\ensuremath{-}{D}_{{\mathbf{k}}_{F}}^{6}∕{\ensuremath{\omega}}_{{\mathbf{k}}_{F}}^{5}$, where Dk{D}_{\mathbf{k}}Dk​ is electron-phonon coupling, ${\ensuremath{\omega}}_{\mathbf{k}}$ is the phonon energy, and kF{\mathbf{k}}_{F}kF​ is the Fermi momentum. The form of higher-order terms of the original Fr\"ohlich-transformed ${H}_{e\text{\ensuremath{-}}\mathit{ph}}$ is also found, up to terms proportional to the sixth power of the coupling---that is, up to those which yield the effective four-electron interactions.
An analytical approach has been developed for the electron–phonon coupling in the Holstein–Hubbard model. The Hubbard U is treated approximately by the slave-boson approach. We show that our approach is flexible and physically clear because it is based on perturbation theory. When the dimensionless ratio ω p /2 Dt is small, ω p /2 Dt ≪ 1, our result is similar to that of the Migdal–Eliashberg theory. When ω p /2 Dt > 1, our result approaches that of the Lang–Firsov transformation and small polaron theory. We have calculated the effective mass of electrons at the Fermi surface to show that the new perturbation approach works well in the intermediate region where ω p /2 Dt ∼ 1.
When a mobile impurity interacts with a many-body system, such as a phonon bath, a polaron is formed. Despite the importance of the polaron problem for a wide range of physical systems, a unified theoretical description valid for arbitrary coupling strengths is still lacking. Here we develop a renormalization group approach for analyzing a paradigmatic model of polarons, the so-called Fröhlich model, and apply it to a problem of impurity atoms immersed in a Bose-Einstein condensate of ultra cold atoms. Polaron energies obtained by our method are in excellent agreement with recent diagrammatic Monte Carlo calculations for a wide range of interaction strengths. They are found to be logarithmically divergent with the ultra-violet cut-off, but physically meaningful regularized polaron energies are also presented. Moreover, we calculate the effective mass of polarons and find a smooth crossover from weak to strong coupling regimes. Possible experimental tests of our results in current experiments with ultra cold atoms are discussed.
The electron-phonon interaction is one of the fundamental interactions in almost all condensed matter materials. In conventional superconductors, the electron-phonon interaction is the glue that attracts two electrons to one another to form a pair. A strong electron-phonon interaction leads to the concept of a polaron, which is an electron with lattice distortions around it. The small polaron is a polaron with spatial extent comparable to an interatomic dimension of the solid. Evidence for polarons has been identified in many experiments in superconductors and semiconductors. In this thesis we present exact calculations of the polaron. Specifically we have refined Trugman's method to solve the ground state of an electron-phonon coupled system in the whole parameter regime, and we also generalized this method to treat spin-orbit coupled systems. The most difficult regimes, which is the strong-coupling regime and the small phonon frequency limit, have been solved by these refinements. There are three representative kinds of electron-phonon interaction, the Holstein model, the Fr"ohlich model and the BLF-SSH model. In this thesis we have addressed the first and the third one. The second one, the Fr"ohlich model, is very similar to the Holstein model but the interaction is nonlocal. For the Holstein model we have observed the expected smooth crossover from free electrons to small polarons, while for the BLF-SSH model, we have studied the weak coupling regime with perturbation theory and derived a new analytical result for the one-dimensional problem.