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arXiv:cond-mat/0509072 · Europhysics Letters (EPL)

Renormalization group study of the electron-phonon interaction in high- T c cuprates

Henry Fu, Carsten Honerkamp A, D. H. Lee

Published Jun 9, 2006• 25 citations
arXivHTMLPDF

Abstract

We generalize the numerical renormalization group (RG) scheme of Honerkamp et al. ( Phys. Rev. B , 63 (2001) 035109), Zanchi et al. ( Europhys. Lett. , 44 (1997) 235; Phys. Rev. B , 61 (2000) 13609) and Halboth et al. ( Phys. Rev. Lett. , 85 (2000) 5162) to study the phonon-mediated retarded interactions in the high- T c cuprates. We find that three sets of phonon-mediated retarded quasiparticle scatterings grow under RG flow. These scatterings share the following common features: 1) the initial and final quasiparticle momenta are in the antinodal regions, and 2) the scattering amplitudes have a x 2 - y 2 symmetry. All three sets of retarded interaction are driven to strong coupling by the magnetic fluctuations around (π/ a ,π/ a ). After growing strong, these retarded interaction will trigger density wave orders with d -wave symmetry. However, due to the d -wave form factor they will leave the nodal quasiparticle unaffected. We conclude that the main effect of electron-phonon coupling in the cuprates is to promote these density wave orders.

DOI
10.1209/epl/i2006-10082-0

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We generalize the numerical renormalization group (RG) scheme of Honerkamp et al. ( Phys. Rev. B , 63 (2001) 035109), Zanchi et al. ( Europhys. Lett. , 44 (1997) 235; Phys. Rev. B , 61 (2000) 13609) and Halboth et al. ( Phys. Rev. Lett. , 85 (2000) 5162) to study the phonon-mediated retarded interactions in the high- T c cuprates. We find that three sets of phonon-mediated retarded quasiparticle scatterings grow under RG flow. These scatterings share the following common features: 1) the initial and final quasiparticle momenta are in the antinodal regions, and 2) the scattering amplitudes have a x 2 - y 2 symmetry. All three sets of retarded interaction are driven to strong coupling by the magnetic fluctuations around (π/ a ,π/ a ). After growing strong, these retarded interaction will trigger density wave orders with d -wave symmetry. However, due to the d -wave form factor they will leave the nodal quasiparticle unaffected. We conclude that the main effect of electron-phonon coupling in the cuprates is to promote these density wave orders.
We present the numerical solution of the renormalization group (RG) equations derived in Tsai, Castro Neto, Shankar, et al. [Phys. Rev. B to appear], for the problem of superconductivity in the presence of both electron–electron and electron–phonon coupling at zero temperature. We study the instability of a Fermi liquid to a superconductor and the RG flow of the couplings in the presence of retardation effects and the crossover from weak to strong coupling. We show that our numerical results provide an ansatz for the analytic solution to the problem in the asymptotic limits of weak and strong coupling.
The ground-state phase diagram of the half filled one-dimensional Holstein-Hubbard model contains a charge-density-wave (CDW) phase, driven by the electron-phonon $(e\text{\ensuremath{-}}ph)$ coupling, and a spin-density-wave (SDW) phase, driven by the on-site electron-electron $(e\text{\ensuremath{-}}e)$ repulsion. Recently, the existence of a third phase, which is metallic and lies in a finite region of parameter space between these two gapped phases, has been claimed. We study this claim using a renormalization-group method for interacting electrons that has been extended to include also $e\text{\ensuremath{-}}ph$ couplings. Our method [Tsai et al., Phys. Rev. B 72, 054531 (2005); Philos. Mag. B 86, 2631 (2006)] treats $e\text{\ensuremath{-}}e$ and $e\text{\ensuremath{-}}ph$ interactions on an equal footing and takes retardation effects fully into account. We find a direct transition between the SDW and CDW states. We study the effects of retardation, which are particularly important near the transition, and find that umklapp processes at finite frequencies drive the CDW instability close to the transition.
The renormalization of the electron-phonon interaction by strong electronic correlations is studied using a one-band Hubbard model with infinite repulsion and nearest (t model) and nearest and second-nearest (tt\ensuremath{'} model) neighbor hopping terms and an on-site electron-phonon coupling. Using Hubbard's X operators and an extension from 2 to N degrees of freedom for the electrons the leading contributions for the electron self-energy and the vertex function in 1/N and the electron-phonon coupling constant are given and numerically evaluated for a square lattice. We find that the momentum dependence of the vertex function depends strongly on doping: For large dopings it is rather weak, with decreasing doping it becomes more and more pronounced leading to a strong reduction of the vertex at larger momentum transfers until, for very small dopings, the vertex essentially consists of a forward scattering peak with a width proportional to the doping. This behavior occurs both in the t and the tt\ensuremath{'} models and also in one dimension where analytic expressions are derived. Correlation effects also change ${\mathrm{\ensuremath{\alpha}}}^{2}$F in general: The full-symmetric component ${\mathrm{\ensuremath{\alpha}}}^{2}$F1{\mathit{F}}_{1}F1​ is in the t model somewhat, in the tt\ensuremath{'} model rather strongly suppressed, especially near half-filling; the other symmetry components ${\mathrm{\ensuremath{\alpha}}}^{2}$Fi{\mathit{F}}_{\mathit{i}}Fi​ with i=2,...,5 increase strongly with decreasing doping and are of similar magnitude as ${\mathrm{\ensuremath{\alpha}}}^{2}$F1{\mathit{F}}_{1}F1​ near half-filling.Including also direct Coulomb repulsion nontrivial symmetries such as d wave become more stable than the s-wave order parameter below a critical value for the doping even for the considered phonon-mediated superconductivity. Most dramatic, however, is the quenching of the resistivity due to electron-phonon scattering both in the t and the tt\ensuremath{'} models at intermediate and small dopings. This result may explain the absence of phonon features in the experimental transport coefficients of high-Tc{\mathit{T}}_{\mathit{c}}Tc​ compounds. \textcopyright{} 1996 The American Physical Society.
Recent resonant x-ray scattering (RXS) experiments revealed a novel charge order in extremely overdoped ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4}$ (LSCO) [Li et al., Phys. Rev. Lett. 131, 116002 (2023)]. The observed charge order appears around the $(\ensuremath{\pi}/3,0)$ wave vector, distinct from the well-known stripe fluctuations near 1/8 doping, and persists from cryogenic temperatures to room temperature. To investigate the origin of this charge order in the overdoped regime, we use determinant quantum Monte Carlo (DQMC) simulations to examine correlated models with various interactions. We demonstrate that this distinctive CDW originates from remnant correlations in extremely overdoped cuprates, with its specific pattern shaped by interactions beyond the Hubbard model, particularly electron-phonon couplings. The persistence of the $(\ensuremath{\pi}/3,0)$ wave vector across different doping levels indicates the presence of nonlocal couplings. Our study reveals the significant role of phonons in cuprates, which assist correlated electrons in the formation of unconventional phases.
We obtain the phase diagram of the half-filled two-dimensional Hubbard model on a square lattice in the presence of Einstein phonons. We find that the interplay between the instantaneous electron-electron repulsion and electron-phonon interaction leads to new phases. In particular, a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superconducting phase emerges when both anisotropic phonons and repulsive Hubbard interaction are present. For large electron-phonon couplings, charge-density-wave and sss-wave superconducting regions also appear in the phase diagram, and the widths of these regions are strongly dependent on the phonon frequency, indicating that retardation effects play an important role. Since at half filling the Fermi surface is nested, a spin-density wave is recovered when the repulsive interaction dominates. We employ a functional multiscale renormalization-group method [Tsai et al., Phys. Rev. B 72, 054531 (2005)] that includes both electron-electron and electron-phonon interactions, and take retardation effects fully into account.
We extend the dynamical vertex approximation (DΓΓΓA), in its most commonly used ladder-version, to systems with retarded electronic interactions. Such an extended scheme, which we name "dynamical-UUU DΓΓΓA", allows us to study strongly interacting electrons coupled to bosons including non-local spatial correlations on all length scales on top of the purely local ones captured by dynamical mean-field theory. We demonstrate the applicability of the method in the specific case of the coupling to a single dispersionless phononic mode, by studying the Hubbard-Holstein model on the square and cubic lattices, focusing on regimes of sizable Hubbard interactions where strong electronic correlations compete with phonon-mediated interactions. As illustrative examples, we consider the antiferromagnetic phase in the three-dimensional model and the d-wave superconductor in two dimensions, analyzing the effect of the electron-phonon coupling, including its frequency dependence, on both instabilities.