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.
Magnetic and bond-charge interactions can arise from the same microscopic interaction. Motivated by this observation, we compute magnetic and bond-charge excitation spectra on an equal footing by introducing a simple effective model on a square lattice, which describes antiferromagnetic and
d-wave superconducting phases around half-filling on the electron-doped side. The magnetic excitation spectrum $\mathrm{Im}\ensuremath{\chi}(\mathbf{q},\ensuremath{\omega})$ has strong weight around $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$ in low energy, and its intensity map exhibits a pencil-tip-like shape in $\mathbf{q}\text{\ensuremath{-}}\ensuremath{\omega}$ space. Around
q=(0,0) magnetic excitations show a steep dispersion toward the $(\ensuremath{\pi},\ensuremath{\pi})$ and $(\ensuremath{\pi},0)$ directions, which is very similar to a spin-wave dispersion although the system is nonmagnetic. Bond-charge excitations are characterized by four different symmetries and studied for all possible couplings. Bond-charge fluctuations with three different symmetries have large spectral weight around $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$ in a relatively low-energy region and extend widely more than the magnetic excitation spectrum. The
d-wave symmetry of bond-charge excitations also has sizable spectral weight along the direction $(\ensuremath{\pi}/2,\ensuremath{\pi}/2)\text{\ensuremath{-}}(0,0)\text{\ensuremath{-}}(\ensuremath{\pi}/2,0)$ in a low-energy region and exhibits softening around $\mathbf{q}\ensuremath{\approx}(0.5\ensuremath{\pi},0)$, whereas no such softening is present in the other symmetries. These results capture the essential features observed in electron-doped cuprates and may motivate an experimental test of bond-charge excitations around $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$ on top of the strong magnetic excitations there as well as additional softening in the
d-wave channel in the $(\ensuremath{\pi},\ensuremath{\pi})\text{\ensuremath{-}}(\ensuremath{\pi}/2,\ensuremath{\pi}/2)$ region at low temperatures near the magnetic phase. We extend the present analysis to the hole-doped side and highlight a contrast to the electron-doped side, which includes incommensurate correlations, electronic nematic correlations, and spin and bond-charge resonance modes in the superconducting state.
The charge susceptibility of the two-dimensional repulsive Hubbard model is investigated by using the diagram technique developed for the case of strong correlations. In this technique, a power series in the hopping constant is used. It is shown that once the Fermi level crosses one of the Hubbard subbands, a sharp peak appears in the momentum dependence of the static susceptibility. With further departure from half-filling, the peak transforms to a ridge around the $\ensuremath{\Gamma}$ point. Within the considered range $0\ensuremath{\le}|1\ensuremath{-}\overline{n}|\ensuremath{\lesssim}0.2$ of the electron filling
n, the static susceptibility is finite, which points to the absence of long-range charge ordering. However, for $|1\ensuremath{-}\overline{n}|\ensuremath{\approx}0.12$, the susceptibility maxima are located halfway between the center and the boundaries of the Brillouin zone. In this case, an interaction of the carriers with the tetragonal distortions can stabilize the charge-density wave with the wavelength of four lattice spacings, as experimentally observed in the low-temperature tetragonal phase of lanthanum cuprates. Within the range of parameters inherent in cuprate perovskites, the character of the susceptibility evolution with
n depends only weakly on the ratio of the nearest-neighbor hopping constant to the Hubbard repulsion and on details of the initial band structure. The location of the susceptibility maxima in the Brillouin zone is mainly determined by the value of
n.
Motivated by the growing evidence of the importance of charge fluctuations in the pseudogap phase in high-temperature cuprate superconductors, we apply a large-
N expansion formulated in a path integral representation of the two-dimensional
t-
J model on a square lattice. We study all possible charge instabilities of the paramagnetic state in leading order of the
1/N expansion. While the
d-wave charge density wave (flux phase) becomes the leading instability for various choices of model parameters, we find that a
d-wave Pomeranchuk (electronic nematic phase) instability occurs as a next leading one. In particular, the nematic state has a strong tendency to become inhomogeneous. In the presence of a large second nearest-neighbor hopping integral, the flux phase is suppressed and the electronic nematic instability becomes leading in a high doping region. Besides these two major instabilities, bond-order phases occur as weaker instabilities close to half-filling. Phase separation is also detected in a finite temperature region near half-filling.
The study of the charge excitations in cuprates is presently an interesting topic because of the development of new and precise x-ray experiments. Based on a large-
N formulation of the two-dimensional $t\text{\ensuremath{-}}J$ model, which allows us to consider all possible charge excitations on an equal footing, we investigate the charge spectrum for both electron- and hole-doped cases. In both cases, the instability toward phase separation, which has momentum modulation
q=(0,0), is found to be robust in a large region of the doping-temperature phase diagram. If a short-range Coulomb repulsion is included the phase separation region shrinks, but the instability remains at
q=(0,0). If on the other hand a two-dimensional long-range Coulomb interaction is included the instability sets in at
q momenta forming a ring around
q=(0,0). The computed charge spectrum in the translation-invariant phase shows well-formed rings. We discuss our results in the light of recent x-ray experiments in electron- and hole-doped cuprates, where ring-like shaped charge modulations have been reported.
We study possible charge instabilities in doped Mott insulators by employing the two-dimensional t - J model with a positive value of the next nearest-neighbor hopping integral on a square lattice, which is applicable to electron-doped cuprates. Although the d -wave charge density wave (flux phase) and d -wave Pomeranchuk instability (nematic order) are dominant instabilities for a negative that corresponds to hole-doped cuprates, we find that those instabilities are strongly suppressed and become relevant only rather close to half filling. Instead, various types of bond orders with modulation vectors close to are dominant in a moderate doping region. Phase separation is also enhanced, but it can be suppressed substantially by the nearest-neighbor Coulomb repulsion without affecting the aforementioned charge instabilities.
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.