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Physical Review B

Checkerboard or stripes: Hard-core bosons on the checkerboard lattice as a model of charge ordering in planar cuprates

P. Wróbel

Published Jul 12, 2006• 6 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.74.014507

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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\ensuremath-Jt\text{\ensuremath{-}}Jt\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×\else\texttimes\fi88\ifmmode\times\else\texttimes\fi{}88\ifmmode×\else\texttimes\fi8 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.
Understanding the microscopic mechanism of coexisting long-range orders (such as lattice supersolidity) in strongly correlated systems is a subject of immense interest. We study the possible manifestations of long-range orders, including lattice-supersolid phases with differently broken symmetry, in a two-dimensional square lattice system of hard-core bosons (HCBs) coupled to archetypal cooperative/coherent normal-mode distortions such as those in perovskites. At strong HCB-phonon coupling, using a duality transformation to map the strong-coupling problem to a weak-coupling one, we obtain an effective Hamiltonian involving nearest-neighbor, next-nearest-neighbor, and next-to-next-nearest-neighbor hoppings and repulsions. Using stochastic series expansion quantum Monte Carlo, we construct the phase diagram of the system. As coupling strength is increased, we find that the system undergoes a first-order quantum phase transition from a superfluid to a checkerboard solid at half-filling and from a superfluid to a diagonal striped solid [with crystalline ordering wave vector $\stackrel{P\vec}{Q}=(2\ensuremath{\pi}/3,2\ensuremath{\pi}/3)$ or $(2\ensuremath{\pi}/3,4\ensuremath{\pi}/3)]$ at one-third filling without showing any evidence of supersolidity. On tuning the system away from these commensurate fillings, checkerboard supersolid is generated near half-filling whereas a rare diagonal striped supersolid is realized near one-third filling. Interestingly, there is an asymmetry in the extent of supersolidity about one-third filling. Within our framework, we also provide an explanation for the observed checkerboard and stripe formations in ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{NiO}}_{4}$ at x=1/2x=1/2x=1/2 and x=1/3x=1/3x=1/3.
We study the ground-state phase diagram of the hard-core extended boson Hubbard model on the square lattice with both nearest- and next-nearest-neighbor hoppings and repulsions, using Gutzwiller mean-field theory and quantum Monte Carlo simulations. We observe the formation of supersolid states with checkerboard, striped, and quarter-filled crystal structures when the system is doped away from commensurate fillings. In the striped supersolid phase, a strong anisotropy in the superfluid density is obtained from the simulations; however, the transverse component remains finite, indicating a true two-dimensional superflow. We find that upon doping, the striped supersolid transitions directly into the supersolid with quarter-filled crystal structure, via a first-order stripe melting transition.
We study the $t\text{\ensuremath{-}}J$ model with four holes on a 32-site square lattice using exact diagonalization. This system corresponds to doping level x=18x=\frac{1}{8}x=81​. At the ``realistic'' parameter J∕t=0.3J∕t=0.3J∕t=0.3, holes in the ground state of this system are unbound. They have short range repulsion due to the lowering of the kinetic energy. There is no antiferromagnetic spin order and the electron momentum distribution function resembles hole pockets. Furthermore, we show evidence that in case antiferromagnetic order exists, holes form ddd-wave bound pairs and there is mutual repulsion among hole pairs. This presumably will occur at low doping levels. This scenario is compatible with a checkerboard-type charge density state proposed to explain the ``18\frac{1}{8}81​ anomaly'' in the lanthanum cuprate family, except that it is the ground state only when the system possesses strong antiferromagnetic order.
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.
We determine the spin and charge orders in the ground state of the doped two-dimensional (2D) Hubbard model in its simplest form, namely with only nearest-neighbor hopping and on-site repulsion. At half-filling, the ground state is known to be an antiferromagnetic Mott insulator. Doping Mott insulators is believed to be relevant to the superconductivity observed in cuprates. A variety of candidates have been proposed for the ground state of the doped 2D Hubbard model. A recent work employing a combination of several state-of-the-art numerical many-body methods established the stripe order as the ground state near 1/81/81/8 doping at strong interactions. In this paper, we apply one of these methods, the cutting-edge constrained-path auxiliary field quantum Monte Carlo (AFQMC) method with self-consistently optimized gauge constraints, to systematically study the model as a function of doping and interaction strength. With careful finite size scaling based on large-scale computations, we map out the ground state phase diagram in terms of its spin and charge order. We find that modulated antiferromagnetic order persists from near half-filling to about 1/51/51/5 doping. At lower interaction strengths or larger doping, these ordered states are best described as spin-density waves, with essentially delocalized holes and modest oscillations in charge correlations. When the charge correlations are stronger (large interaction or small doping), they are best described as stripe states, with the holes more localized near the node in the antiferromagnetic spin order. In both cases, we find that the wavelength in the charge correlations is consistent with so-called filled stripes.