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arXiv:cond-mat/0211395 · Physical review. B, Condensed matter

Numerical renormalization-group study of the symmetric Anderson-Holstein model: Phonon and electron spectral functions

Gun Sang Jeon, Tae-Ho Park, Han-Yong Choi

Published Jul 3, 2003• 45 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.68.045106

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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≳(\ensuremath≲)Us(\ensuremathΔ)U\ensuremath{\gtrsim}(\ensuremath{\lesssim}{)U}_{s}(\ensuremath{\Delta})U\ensuremath≳(\ensuremath≲)Us​(\ensuremathΔ) depending on the parameters of the AH model, U, and the hybridization \ensuremathΔ.\ensuremath{\Delta}.\ensuremathΔ. And, for g\ensuremath≳gco,g\ensuremath{\gtrsim}{g}_{\mathrm{co}},g\ensuremath≳gco​, 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 study the symmetric Anderson–Holstein (AH) model at zero temperature with Wilson's numerical renormalization group (NRG) technique to study the interplay between the electron-electron and electron-phonon interactions. An improved method for calculating the phonon propagator using the NRG technique is presented and applied to the AH model as the onsite Coulomb repulsion U and electron-phonon coupling constant g are varied. As g is increased, the phonon propagator is successively renormalized, and for g≳gco crosses over to the regime where the mode splits into two components, one of which approaches back to the bare frequency and the other develops into a soft mode. The initial renormalization of the phonon mode, as g is increased from 0, can be either positive or negative depending on U and the hybridization Δ.
We present results on the dynamical correlation functions of the particle-hole symmetric Holstein-Hubbard model at zero temperature, calculated using the dynamical mean-field theory which is solved by the numerical renormalization group method. We clarify the competing influences of the electron-electron and electron-phonon interactions particularity at the different metal to insulator transitions. The Coulomb repulsion is found to dominate the behavior in large parts of the metallic regime. By suppressing charge fluctuations, it effectively decouples electrons from phonons. The phonon propagator shows a characteristic softening near the metal to bipolaronic transition but there is very little softening on the approach to the Mott transition.
We use the continuous-time interaction expansion (CT-INT) quantum Monte Carlo method to calculate the phonon spectral function of the one-dimensional Holstein-Hubbard model at half-filling. Our results are consistent with a soft-mode Peierls transition in the adiabatic regime, and the existence of a central peak related to long-range order in the Peierls phase. We explain a previously observed feature at small momenta in terms of a hybridization of charge and phonon excitations. Tuning the system from a Peierls to a metallic phase with a nonzero Hubbard interaction suppresses the central peak, but a significant renormalization of the phonon dispersion remains. In contrast, the dispersion is only weakly modified in the Mott phase. We discuss finite-size effects, the relation to the dynamic charge structure factor, as well as additional sum rules and their implications. Finally, we reveal the existence of a discrete symmetry in a continuum field theory of the Holstein model, which is spontaneously broken in the Peierls phase.
The temperature dependence of the electric dipolar susceptibility χP is discussed on the basis of the Anderson-Holstein model with the use of a numerical renormalization group (NRG) technique. Note that χ P is related to the phonon Green’s function D. In order to obtain a correct temperature dependence for χ P at low temperatures, we propose a method to evaluate χ P through the Dyson equation from the charge susceptibility χ c calculated by using the NRG instead of the direct NRG calculation of D. We find that the irreducible charge susceptibility estimated from χ c agree with the perturbation calculation, suggesting that our method works well.
Using the functional renormalization group (FRG) and the numerical renormalization group (NRG), we calculate the spectral function of the Anderson impurity model at zero and finite temperatures. In our FRG scheme spin fluctuations are treated nonperturbatively via a suitable Hubbard-Stratonovich field, but vertex corrections are neglected. A comparison with our highly accurate NRG results shows that this FRG scheme provides a good description of the spectral line shape at zero and finite temperatures both in the weak- and strong-coupling regimes, although at zero temperature the FRG is not able to reproduce the known exponential narrowing of the Kondo resonance at strong coupling.
We investigate the interaction of strongly correlated electrons with phonons in the frame of the Hubbard-Holstein model. The electron-phonon interaction is considered to be strong and is an important parameter of the model besides the Coulomb repulsion of electrons and band filling. This interaction with the nondispersive optical phonons has been transformed to the problem of mobile polarons by using the canonical transformation of Lang and Firsov. A new diagram technique is used in order to handle the strong Coulomb repulsion of the electrons, which leads to a closed set of integral equations for the irreducible on-site many-particle Green's functions. The coupling to the phonons leads to an essential renormalization of the one- and two-particle irreducible Green's functions. It also influences the parameters which control the metal-insulator or superconducting phase transition. In addition we find a collective mode of phonon clouds which surround the polarons. The physics of the emission and absorption of this mode by the polarons has been investigated.