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arXiv:2212.08794 · Physical review. B./Physical review. B

Ring-like shaped charge modulations in the t−J model with long-range Coulomb interaction

Matías Bejas, R. Zeyher, A. Greco

Published Dec 15, 2022• 7 citations
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Abstract

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 $\mathbf{q}=(0,0)$, 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 $\mathbf{q}=(0,0)$. If on the other hand a two-dimensional long-range Coulomb interaction is included the instability sets in at $\mathbf{q}$ momenta forming a ring around $\mathbf{q}=(0,0)$. 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.

DOI
10.1103/physrevb.106.224512

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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-NNN formulation of the two-dimensional t\ensuremath-Jt\text{\ensuremath{-}}Jt\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)\mathbf{q}=(0,0)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)\mathbf{q}=(0,0)q=(0,0). If on the other hand a two-dimensional long-range Coulomb interaction is included the instability sets in at q\mathbf{q}q momenta forming a ring around q=(0,0)\mathbf{q}=(0,0)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 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.
Motivated by the recent resonant x-ray scattering (RXS) and resonant inelastic x-ray scattering (RIXS) experiments for electron-doped cuprates, we study the charge excitation spectrum in a layered $t\text{\ensuremath{-}}J$ model with the long-range Coulomb interaction. We show that the spectrum is not dominated by a specific type of charge excitations, but by different kinds of charge fluctuations, and is characterized by a dual structure in the energy space. Low-energy charge excitations correspond to various types of bond-charge fluctuations driven by the exchange term (JJJ term), whereas high-energy charge excitations are due to usual on-site charge fluctuations and correspond to plasmon excitations above the particle-hole continuum. The interlayer coupling, which is frequently neglected in many theoretical studies, is particularly important to the high-energy charge excitations.
We study the density-density correlation function in a large-N scheme of the t-tA-J-V model.When the nearestneighbor Coulomb interaction V is zero, our model exhibits phase separation in a wide doping region and we obtain large spectral weight near momentum q ¼ ð0; 0Þ 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 ¼ ð0; 0Þ 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 ¼ ð0; 0Þ for electron doping.This insight should be taken into account in a future study of x-ray scattering measurements.
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.
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.
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.