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arXiv:cond-mat/0603161 · Physical Review B

Charge susceptibility in the t−J model

D. N. Aristov, Giniyat Khaliullin

Published Jul 25, 2006• 10 citations
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Abstract

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 $J$-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.

DOI
10.1103/physrevb.74.045124

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Momentum and doping dependence of the static charge susceptibility \ensuremathχ(q)\ensuremath{\chi}(\mathbf{q})\ensuremathχ(q) in the t\ensuremath-t\ensuremath′\ensuremath-Jt\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}Jt\ensuremath-t\ensuremath′\ensuremath-J model is investigated. Correlations lead to a strongly momentum-dependent renormalization of \ensuremathχ(q)\ensuremath{\chi}(\mathbf{q})\ensuremathχ(q). The charge susceptibility near the (\ensuremathπ,\ensuremathπ)(\ensuremath{\pi},\ensuremath{\pi})(\ensuremathπ,\ensuremathπ) region of the Brillouin zone is strongly suppressed as the hole density \ensuremathδ\ensuremath{\delta}\ensuremathδ is decreased. However, contrary to naive expectations, \ensuremathχ(q)\ensuremath{\chi}(\mathbf{q})\ensuremathχ(q) around q=(\ensuremathπ,0)\mathbf{q}=(\ensuremath{\pi},0)q=(\ensuremathπ,0) and (0,\ensuremathπ)(0,\ensuremath{\pi})(0,\ensuremathπ) remains large and practically unchanged at \ensuremathδ\ensuremath∼0.1−−0.5\ensuremath{\delta}\ensuremath{\sim}0.1--0.5\ensuremathδ\ensuremath∼0.1−−0.5. This effect is consistent with a tendency towards low-energy charge fluctuations with the wave vectors along the \ensuremathΓ\ensuremath-X\ensuremath{\Gamma}\text{\ensuremath{-}}X\ensuremathΓ\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′{t}^{\ensuremath{'}}t\ensuremath′ leads to weakening of the above momentum-selective renormalizations of \ensuremathχ(q)\ensuremath{\chi}(\mathbf{q})\ensuremathχ(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.
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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.
Using a variational Monte Carlo approach with a recently proposed stripe wave function, we showed that the strong correlation included in a $t\text{\ensuremath{-}}J$-type model has essentially all the necessary ingredients to form these stripes with modulations of charge density, spin magnetization, and pair field. If a perturbative effect of electron-phonon coupling to renormalize the effective mass or the hopping rate of holes is considered with the model, we find the half-doped stripes, which has on the average one half of a hole in one period of charge modulation, to be most stable energetic wise in the underdoped region, $1/12\ensuremath{\le}\ensuremath{\delta}\ensuremath{\le}1/8$. This is in good agreement with the observation in the neutron-scattering experiments. We also find long-range Coulomb interaction to be less effective in the formation of half-doped stripes.
To clarify the effects of next-nearest-neighbor electron hopping ${t}^{\ensuremath{'}}$ on the spin, charge, and pairing correlations, we studied the two-dimensional $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}U$ Hubbard model using the constrained-path Monte Carlo technique. We found that a negative ${t}^{\ensuremath{'}}$ suppresses the spin fluctuations near the Brillouin zone point $M=(\ensuremath{\pi},\ensuremath{\pi})$ and the charge fluctuations near the point $X=(\ensuremath{\pi},0),$ while a positive ${t}^{\ensuremath{'}}$ has the opposite effect. For a negative ${t}^{\ensuremath{'}},$ we also found that the long-range vertex contributions to the pairing correlations are heavily suppressed, but the local contributions are enhanced. Over the entire hole doping range studied, the d-wave long-range vertex contributions to pairing dominated those of the extended s-wave vertex contributions. In general, we did not find any tendency for these correlations to grow with increasing system size. We also did a limited number of simulations with an attractive nearest neighbor density-density interaction added to the $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}U$ model. It enhanced the long-range vertex contributions to the pairing correlations only for weak values of U.
We study charge excitation spectra in the two-dimensional t - J model on a square lattice to explore a charge-order tendency recently found in electron-doped cuprates around the carrier density 0.15. The static susceptibility of d -wave charge density, which corresponds to the nematic susceptibility at the momentum transfer q = ( 0 , 0 ) , shows two characteristic peaks at momenta of the form q 1 = ( q ′ , q ′ ) and q 2 = ( q , 0 ) . These two peaks originate from the so-called 2 k F scattering processes enhanced by the d -wave character of the bond-charge density. The peak at q 1 is much broader, but develops to be very sharp in the vicinity of its instability, whereas the peak at q 2 becomes sharper with decreasing temperature, but does not diverge. The equal-time correlation function, which is measured by resonant x-ray scattering, exhibits a momentum dependence similar to the static susceptibility. We also present energy-resolved charge excitation spectra. The spectra show a V-shaped structure around q = ( 0 , 0 ) and bend back toward close-to-zero energy due to the charge-order tendency at q 1 and q 2 . The resulting spectra form gap-like features with a maximal gap at q ≈ q 1 / 2 and q 2 / 2 . We discuss implications for the recent experiments in electron-doped cuprates.
We study the density-density correlation function in a large-N scheme of the t-t'-J-V model. When the nearest-neighbor Coulomb interaction V is zero, our model exhibits phase separation in a wide doping region and we obtain large spectral weight near momentum q=(0,0) at low energy, which originates from the proximity to phase separation. These features are much stronger for electron doping than for hole doping. However, once phase separation is suppressed by including a finite V, the low-energy spectral weight around q=(0,0) is substantially suppressed. Instead a sharp zero-sound mode is stabilized above the particle-hole continuum. We discuss that the presence of a moderate value of V, which is frequently neglected in the t-J model, is important to understand low-energy charge excitations especially close to q=(0,0) for electron doping. This insight should be taken into account in a future study of x-ray scattering measurements.