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Physical Review B

Inhomogeneous state of the extended t - J model on a square lattice: A variational Monte Carlo and Gutzwiller approximation study

Chung-Pin Chou, Ting-Kuo Lee

Published Mar 14, 2012• 11 citations
Publisher

Abstract

We carry out the variational Monte Carlo calculation to examine spatially inhomogeneous states in hole- and electron-doped cuprates. By using Gutzwiller approximation, we consider the excitations, arising from charge density, spin density, and pair field, of the mean-field ground state of the $t$-$J$ model. It leads to the stripe patterns we have found numerically in a generalized $t$-$J$-type model including mass renormalization from the electron-phonon coupling. In the hole-doped side, a robust $d$-wave superconducting order results in the formation of the half-doped antiferromagnetic resonating-valence-bond (AF-RVB) stripes shown by the well-known Yamada plot. On the other hand, due to a long-range AF order in electron-doped materials, a stripe structure with the ``in-phase'' magnetic domain (IPMD) is obtained in the underdoped regime instead of the AF-RVB stripe. The IPMD stripe with the largest period permitted by lattice size is stabilized near the underdoped region and it excludes the Yamada plot from electron-doped cases. Based on finite lattice size to which we can reach, the existence of IPMD stripes may imply an electronic phase separation into an electron-rich and an insulating half-filled AF long-range ordered domains in electron-doped compounds.

DOI
10.1103/physrevb.85.104511

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We carry out the variational Monte Carlo calculation to examine spatially inhomogeneous states in hole- and electron-doped cuprates. By using Gutzwiller approximation, we consider the excitations, arising from charge density, spin density and pair field, of the mean-field ground state of the t-J model. It leads to the stripe patterns we have found numerically in a generalized t-J-type model including mass renormalization from the electron-phonon coupling. In the hole-doped side, a robust d-wave superconducting order results in the formation of the half-doped antiferromagnetic resonating-valence-bond (AF-RVB) stripes shown by the well-known Yamada plot. On the other hand, due to a long-range AF order in electron-doped materials, a stripe structure with the "in-phase" magnetic domain (IPMD) is obtained in the underdoped regime instead of the AF-RVB stripe. The IPMD stripe with the largest period permitted by lattice size is stabilized near the underdoped region and it excludes the Yamada plot from electron-doped cases. Based on finite lattice size to which we can reach, the existence of IPMD stripes may imply an electronic phase separation into an electron-rich and an insulating half-filled AF long-range ordered domains in electron-doped compounds.

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We carry out the variational Monte Carlo calculation to examine spatially inhomogeneous states in hole- and electron-doped cuprates. By using Gutzwiller approximation, we consider the excitations, arising from charge density, spin density, and pair field, of the mean-field ground state of the ttt-JJJ model. It leads to the stripe patterns we have found numerically in a generalized ttt-JJJ-type model including mass renormalization from the electron-phonon coupling. In the hole-doped side, a robust ddd-wave superconducting order results in the formation of the half-doped antiferromagnetic resonating-valence-bond (AF-RVB) stripes shown by the well-known Yamada plot. On the other hand, due to a long-range AF order in electron-doped materials, a stripe structure with the ``in-phase'' magnetic domain (IPMD) is obtained in the underdoped regime instead of the AF-RVB stripe. The IPMD stripe with the largest period permitted by lattice size is stabilized near the underdoped region and it excludes the Yamada plot from electron-doped cases. Based on finite lattice size to which we can reach, the existence of IPMD stripes may imply an electronic phase separation into an electron-rich and an insulating half-filled AF long-range ordered domains in electron-doped compounds.
We carry out the variational Monte Carlo calculation to examine spatially inhomogeneous states in hole- and electron-doped cuprates. By using Gutzwiller approximation, we consider the excitations, arising from charge density, spin density and pair field, of the mean-field ground state of the t-J model. It leads to the stripe patterns we have found numerically in a generalized t-J-type model including mass renormalization from the electron-phonon coupling. In the hole-doped side, a robust d-wave superconducting order results in the formation of the half-doped antiferromagnetic resonating-valence-bond (AF-RVB) stripes shown by the well-known Yamada plot. On the other hand, due to a long-range AF order in electron-doped materials, a stripe structure with the "in-phase" magnetic domain (IPMD) is obtained in the underdoped regime instead of the AF-RVB stripe. The IPMD stripe with the largest period permitted by lattice size is stabilized near the underdoped region and it excludes the Yamada plot from electron-doped cases. Based on finite lattice size to which we can reach, the existence of IPMD stripes may imply an electronic phase separation into an electron-rich and an insulating half-filled AF long-range ordered domains in electron-doped compounds.
Motivated by the observation of inhomogeneous patterns in some high-Tc{T}_{c}Tc​ cuprate compounds, several variational Gutzwiller-projected wave functions with built-in charge and bond-order parameters are proposed for the extended $t\text{\ensuremath{-}}J\text{\ensuremath{-}}V$ model on the square lattice at low doping. First, following a recent Gutzwiller-projected mean-field approach by one of us [D. Poilblanc, Phys. Rev. B 72, 060508(R) (2005)], we investigate, as a function of doping and Coulomb repulsion the relative stability of a wide variety of modulated structures with square unit cells of size $2\ifmmode\times\else\texttimes\fi{}2$, $\sqrt{8}\ifmmode\times\else\texttimes\fi{}\sqrt{8}$, $4\ifmmode\times\else\texttimes\fi{}4$, and $\sqrt{32}\ifmmode\times\else\texttimes\fi{}\sqrt{32}$. It is found that the $4\ifmmode\times\else\texttimes\fi{}4$ bond-order wave function with staggered-flux pattern (and small charge and spin current density wave) is a remarkable competitive candidate for hole doping around 1∕81∕81∕8 in agreement with scanning tunneling microscopy observations in the underdoped regime of some cuprates. This wave function is then optimized accurately and its properties studied extensively using a variational Monte Carlo scheme. Moreover, we find that under increasing the Coulomb repulsion, the ddd-wave superconducting RVB wave function is rapidly destabilized with respect to the $4\ifmmode\times\else\texttimes\fi{}4$ bond-order wave function. The stability of the bond-modulated wave function is connected to a gain of Coulomb and exchange energies. We suggest that such ordering patterns could be dynamical or could spontaneously appear in the vicinity of an impurity or a vortex in the mixed phase of the cuprates. Finally, we consider also a commensurate flux phase, but this wave function turns out not to be competitive because of its rather poor kinetic energy. However, we find it has very competitive exchange and Coulomb energies.
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.
The long-studied Hubbard model is one of the simplest models of copper-oxide superconductors. However, the connection between the model and the experimental phase diagram is still under debate, in particular regarding the existence and extent of the ddd-wave superconducting phase. Recent rapid progress in improving the accuracy of numerical solvers has opened a way to answer this question reliably. Here, we study the hole-doping concentration ($\ensuremath{\delta}$) dependence of the Hubbard model in the ground states on a square lattice at strong coupling U/t=10U/t=10U/t=10, for the on-site interaction UUU and the transfer ttt, using a variational Monte Carlo method. The method, which combines tensor network and Lanczos methods on top of Pfaffian wave functions, reveals a rich phase diagram, in which many orders compete severely and degenerate within the energy range of 0.01t0.01t0.01t. We have identified distinct phases including a uniform ddd-wave superconducting phase for $0.17\ensuremath{\lesssim}\ensuremath{\delta}\ensuremath{\lesssim}0.22$ and a stripe charge/spin ordered phase for $\ensuremath{\delta}\ensuremath{\lesssim}0.17$ with the stripe period depending on $\ensuremath{\delta}$, together with presumable spatially coexisting antiferromagnetic and stripe order for $\ensuremath{\delta}\ensuremath{\lesssim}0.07$ and coexisting stripe and ddd-wave superconductivity for $0.07\ensuremath{\lesssim}\ensuremath{\delta}\ensuremath{\lesssim}0.17$. The present, improved method revealed a wider region of a charge uniform superconducting phase than the previous studies and shows a qualitative similarity to the phase diagram of the cuprate superconductors. The superconducting order parameter is largest at doping of around $\ensuremath{\delta}=0.17$ in the ground state, which undergoes phase transitions from an inhomogeneous to a uniform state.
Striped phases, in which spin, charge, and pairing correlations vary inhomogeneously in the CuO2{}_{2}2​ planes, are a known experimental feature of cuprate superconductors, and are also found in a variety of numerical treatments of the two-dimensional Hubbard Hamiltonian. In this paper, we use determinant quantum Monte Carlo to show that if a stripe density pattern is imposed on the model, the ddd-wave pairing vertex is significantly enhanced. We attribute this enhancement to an increase in antiferromagnetic order which is caused by the appearance of more nearly half-filled regions when the doped holes are confined to the stripes. We also observe a $\ensuremath{\pi}$-phase shift in the magnetic order.
The possibility of coexistence of superconductivity (SC) and antiferromagnetic long-range order (AFLRO) of the two-dimensional extended $t\text{\ensuremath{-}}J$ model in the very underdoped region is studied by the variational Monte Carlo (VMC) method. In addition to using previously studied wave functions, a recently proposed wave function generated from the half-filled Mott insulator is used. For hole-doped systems, the phase boundary between AFLRO and ddd-wave SC for the physical parameters, J∕t=0.3J∕t=0.3J∕t=0.3, ${t}^{\ensuremath{'}}∕t=\ensuremath{-}0.3$ and ${t}^{\ensuremath{''}}∕t=0.2$, is located near hole density ${\ensuremath{\delta}}_{c}\ensuremath{\cong}0.06$, and there is no coexistence. The phase transition is first order between these two homogeneous phases at ${\ensuremath{\delta}}_{c}$.