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arXiv:1208.3210 · Physical Review B

Lindhard and RPA susceptibility computations in extended momentum space in electron-doped cuprates

Yung Jui Wang, B. Barbiellini, Hsin Lin, Tanmoy Das, Susmita Basak, Peter E. Mijnarends, Stanislaw Kaprzyk, R.S. Markiewicz, Arun Bansil

Published Jun 25, 2012• 12 citations
arXivHTMLPDF

Abstract

We present an approximation for efficient calculation of the Lindhard susceptibility ${\ensuremath{\chi}}^{L}(\mathbf{q},\ensuremath{\omega})$ in a periodic system through the use of simple products of real space functions and the fast Fourier transform (FFT). The method is illustrated by providing ${\ensuremath{\chi}}^{L}(\mathbf{q},\ensuremath{\omega})$ results for the electron doped cuprate Nd${}_{2\ensuremath{-}x}$Ce${}_{x}$CuO${}_{4}$ extended over several Brillouin zones. These results are relevant for interpreting inelastic x-ray scattering spectra from cuprates.

DOI
10.1103/physrevb.85.224529

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We present an approximation for efficient calculation of the Lindhard susceptibility \ensuremathχL(q,\ensuremathω){\ensuremath{\chi}}^{L}(\mathbf{q},\ensuremath{\omega})\ensuremathχL(q,\ensuremathω) in a periodic system through the use of simple products of real space functions and the fast Fourier transform (FFT). The method is illustrated by providing \ensuremathχL(q,\ensuremathω){\ensuremath{\chi}}^{L}(\mathbf{q},\ensuremath{\omega})\ensuremathχL(q,\ensuremathω) results for the electron doped cuprate Nd2\ensuremath−x{}_{2\ensuremath{-}x}2\ensuremath−x​Cex{}_{x}x​CuO4{}_{4}4​ extended over several Brillouin zones. These results are relevant for interpreting inelastic x-ray scattering spectra from cuprates.
© Kazan Federal University (KFU). We derive the dynamical spin susceptibility in the t-J-G model combining the random phase approximation (RPA) and projection operator method, which allows describing the mutual interplay between the local and the itinerant components of susceptibility. Near the antifer- romagnetic wave vector the calculated dispersion of the spin excitations reproduces well the so-called hour-glass dispersion, characteristic for several layered cuprates. It is formed as a result of competition between the original spin-gap in magnon-like excitations spectrum and the superconducting gap, which affects the itinerant component of the susceptibility. Further- more, the calculated collective spin excitations along (0, 0)-(0, π) are in agreement with the positions of the absorption peaks in the inelastic X-ray scattering spectra. They refer to the paramagnon-like modes, characteristic to the itinerant spin system, rather than magnon-like excitations that originate from short range order effect in the system of local spins at Cu sites.
The physics of high-Tc superconducting cuprates is obscured by the effect of strong electronic correlations. One way to overcome this problem is to seek an exact solution at least within a small cluster and expand it to the whole crystal. Such an approach is at the heart of cluster perturbation theory (CPT). Here, we developed CPT for the dynamic spin and charge susceptibilities (spin-CPT and charge-CPT), with the correlation effects explicitly taken into account by the exact diagonalization. We applied spin-CPT and charge-CPT to the effective two-band Hubbard model for the cuprates obtained from the three-band Emery model and calculated one- and two-particle correlation functions, namely, a spectral function and spin and charge susceptibilities. The doping dependence of the spin susceptibility was studied within spin-CPT and CPT-RPA, that is, the CPT generalization of the random phase approximation (RPA). In the underdoped region, both our methods resulted in the signatures of the upper branch of the spin excitation dispersion with the lowest excitation energy at the (π,π) wave vector and no presence of low-energy incommensurate excitations. In the high doping region, both methods produced a low energy response at four incommensurate wave vectors in qualitative agreement with the results of the inelastic neutron scattering experiments on overdoped cuprates.
We present a nonperturbative, divergence-free series expansion of Green's functions using effective operators. The method is especially suited for computing correlators of complex operators as a series of correlation functions of simpler forms. We apply the method to study low-energy excitations in resonant inelastic x-ray scattering (RIXS) in doped one- and two-dimensional single-band Hubbard models. The RIXS operator is expanded into polynomials of spin, density, and current operators weighted by fundamental x-ray spectral functions. These operators couple to different polarization channels resulting in simple selection rules. The incident photon energy dependent coefficients help to pinpoint main RIXS contributions from different degrees of freedom. We show in particular that, with parameters pertaining to cuprate superconductors, local spin excitation dominates the RIXS spectral weight over a wide doping range in the cross-polarization channel.
Using a generalized response theory we derive the electronic Raman response function for metals with anisotropic relaxation rates. The calculations account for the long-range Coulomb interaction and treat the collision operator within a charge conserving relaxation time approximation. We extend earlier treatments to finite wave numbers $(\ensuremath{\mid}\mathbf{q}\ensuremath{\mid}⪡{k}_{\mathrm{F}})$ and incorporate inelastic electron-electron scattering besides elastic impurity scattering. Moreover we generalize the Lindhard density response function to the Raman case. Numerical results for the quasiparticle scattering rate and the Raman response function for cuprate superconductors are presented.
We study the dynamical, momentum-dependent two- and four-spin response functions in doped and undoped one-dimensional cuprates, as probed by resonant inelastic x-ray scattering, using an exact numerical diagonalization procedure. In the undoped ttt--JJJ system, the four-spin response vanishes at $\ensuremath{\pi}$, whereas the two-spin correlator is peaked around $\ensuremath{\pi}/2$, with generally larger spectral weight. Upon doping, spectra tend to soften and broaden, with a transfer of spectral weight toward higher energy. However, the total spectral weight and average peak position of either response are only weakly affected by doping up to a concentration of 1/8. Only the two-spin response at $\ensuremath{\pi}$ changes strongly, with a large reduction of spectral weight and enhancement of excitation energy. At other momenta, the higher energy, generic features of the magnetic response are robust against doping. It signals the presence of strong short-range antiferromagnetic correlations, even after doping mobile holes into the system. We expect this to hold also in higher dimensions.
Momentum and doping dependence of the static charge susceptibility $\ensuremath{\chi}(\mathbf{q})$ in the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model is investigated. Correlations lead to a strongly momentum-dependent renormalization of $\ensuremath{\chi}(\mathbf{q})$. The charge susceptibility near the $(\ensuremath{\pi},\ensuremath{\pi})$ region of the Brillouin zone is strongly suppressed as the hole density $\ensuremath{\delta}$ is decreased. However, contrary to naive expectations, $\ensuremath{\chi}(\mathbf{q})$ around $\mathbf{q}=(\ensuremath{\pi},0)$ and $(0,\ensuremath{\pi})$ remains large and practically unchanged at $\ensuremath{\delta}\ensuremath{\sim}0.1--0.5$. This effect is consistent with a tendency towards low-energy charge fluctuations with the wave vectors along the $\ensuremath{\Gamma}\text{\ensuremath{-}}X$ direction, reported in earlier studies. Our main finding is that the above trends are amplified by JJJ-driven pairing effects, indicating that the pseudogap formation may promote the charge inhomogeneity. The next-nearest hopping ${t}^{\ensuremath{'}}$ leads to weakening of the above momentum-selective renormalizations of $\ensuremath{\chi}(\mathbf{q})$. We analyze the effects of long-range Coulomb interaction, taking into account a layered structure of cuprates. As an application, the results are discussed in the context of bond-stretching phonon softening in hole-doped cuprates. In particular, a peculiar doping and momentum dependence of the electron-phonon coupling constant is found.