The formation of density modulations observed in cuprates by means of scanning tunneling microscopy is analyzed in the framework of an effective model for hole-doped short-range two-dimensional antiferromagnets. That model has been derived from the $t\text{\ensuremath{-}}J$ model on the square lattice. It is defined in terms of hard-core bosons. The latter objects represent tightly bound hole pairs. Long-range Coulomb repulsion between them is assumed. The nature of the ground state for the effective model is determined by analyzing the density correlation function and the density structure factor. The exact diagonalization for four hard-core bosons in the 128-site cluster, which corresponds to eight holes in the $8\ifmmode\times\else\texttimes\fi{}8$ cluster of sites in the initial square lattice, is performed to derive these quantities. Signatures of the density modulations with the periodicity 4 spacings between nearest-neighbor sites in the initial square lattice in horizontal and vertical directions and with the full symmetry of that lattice may be seen both in the density correlation function and in the density structure factor. On the other hand, a strong tendency to form smectic order may be noticed in the results of the calculation. It turns out that the ground state is a symmetric combination of states representing stripes running either in the vertical or in the horizontal directions. In a real system, the same effect may have the formation of regions with short-range fluctuating stripe order oriented either vertically or horizontally.
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.
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.
Inspired by recent experimental findings, we investigate various scenarios of the doped Hubbard model with impurity potentials. We calculate the lattice Green's function in a finite-size cluster and then map it to the continuum real space, which allows for a direct comparison with scanning tunneling microscopy measurements on the local density of states. Our simulations successfully reproduce experimental data, including the characteristic stripe- and ladder-shaped structures observed in cuprate systems. Moreover, our results establish a connection between previous numerical findings on stripe-ordered ground states and experimental observations, thus providing new insights into microscopic mechanisms of the Mott insulator to superconductor transition in cuprates.
Recent scanning tunneling microscopy (STM) experiments on underdoped cuprates have displayed modulations in the local electronic density of states, which are centered on a Cu-O-Cu bond [Kohsaka et al., Science 315, 1380 (2007)]. As a paradigm of the pinning of such bond-centered ordering in strongly correlated systems, we present the theory of valence bond solid (VBS) correlations near a single impurity in a square lattice antiferromagnet. The antiferromagnet is assumed to be in the vicinity of a quantum transition from a magnetically ordered N\'eel state to a spin-gap state with long-range VBS order. We identify two distinct classes of impurities: (i) local modulation in the exchange constants and (ii) a missing or additional spin, for which the impurity perturbation is represented by an uncompensated Berry phase. The ``boundary'' critical theory for these classes is developed. In the second class, we find a VBS pinwheel around the impurity, accompanied by a suppression in the VBS susceptibility. Implications for numerical studies of quantum antiferromagnets and for STM experiments on the cuprates are noted.
Recent quantum-gas microscopy of ultracold atoms and scanning tunneling microscopy of the cuprates reveal new detailed information about doped Mott antiferromagnets, which can be compared with calculations. Using cellular dynamical mean-field theory, we map out the antiferromagnetic (AF) phase of the two-dimensional Hubbard model as a function of interaction strength
U, hole doping $\ensuremath{\delta}$, and temperature
T. The N\'eel phase boundary is nonmonotonic as a function of
U and $\ensuremath{\delta}$. Frustration induced by second-neighbor hopping reduces N\'eel order more effectively at small
U. The doped AF is stabilized at large
U by kinetic energy and at small
U by potential energy. The transition between the AF insulator and the doped metallic AF is continuous. At large
U, we find in-gap states similar to those observed in scanning tunneling microscopy. We predict that, contrary to the Hubbard bands, these states are only slightly spin polarized.