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arXiv:cond-mat/9907375 · Physical review. B, Condensed matter

Phase separation and stripe formation in the two-dimensional t−J model: A comparison of numerical results

Steven R. White, Douglas J. Scalapino

Published Mar 1, 2000• 139 citations
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Abstract

We make a critical analysis of numerical results for and against phase separation and stripe formation in the $t\ensuremath{-}J$ model. We argue that the frustrated phase separation mechanism for stripe formation requires phase separation at too high a doping for it to be consistent with existing numerical studies of the $t\ensuremath{-}J$ model. We compare variational energies for various methods, and conclude that the most accurate calculations for large systems appear to be from the density-matrix renormalization group. These calculations imply that the ground state of the doped $t\ensuremath{-}J$ model is striped, not phase separated.

DOI
10.1103/physrevb.61.6320

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We report results of large-scale ground-state density matrix renormalization group (DMRG) calculations on t-[Formula: see text]-J cylinders with circumferences 6 and 8. We determine a rough phase diagram that appears to approximate the two-dimensional (2D) system. While for many properties, positive and negative [Formula: see text] values ([Formula: see text]) appear to correspond to electron- and hole-doped cuprate systems, respectively, the behavior of superconductivity itself shows an inconsistency between the model and the materials. The [Formula: see text] (hole-doped) region shows antiferromagnetism limited to very low doping, stripes more generally, and the familiar Fermi surface of the hole-doped cuprates. However, we find [Formula: see text] strongly suppresses superconductivity. The [Formula: see text] (electron-doped) region shows the expected circular Fermi pocket of holes around the [Formula: see text] point and a broad low-doped region of coexisting antiferromagnetism and d-wave pairing with a triplet p component at wavevector [Formula: see text] induced by the antiferromagnetism and d-wave pairing. The pairing for the electron low-doped system with [Formula: see text] is strong and unambiguous in the DMRG simulations. At larger doping another broad region with stripes in addition to weaker d-wave pairing and striped p-wave pairing appears. In a small doping region near [Formula: see text] for [Formula: see text], we find an unconventional type of stripe involving unpaired holes located predominantly on chains spaced three lattice spacings apart. The undoped two-leg ladder regions in between mimic the short-ranged spin correlations seen in two-leg Heisenberg ladders.

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We make a critical analysis of numerical results for and against phase separation and stripe formation in the t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model. We argue that the frustrated phase separation mechanism for stripe formation requires phase separation at too high a doping for it to be consistent with existing numerical studies of the t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model. We compare variational energies for various methods, and conclude that the most accurate calculations for large systems appear to be from the density-matrix renormalization group. These calculations imply that the ground state of the doped t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model is striped, not phase separated.
Using Gutzwiller-projected wave functions, I estimate the ground-state energy of the $t\text{\ensuremath{-}}J$ model for several variational states relevant for high-temperature cuprate superconductors. The results indicate that both the superconducting state and the staggered-flux normal state (proposed for vortex cores) are unstable at low doping towards antiferromagnetism and towards phase separation. While phase separation in the underdoped superconducting state may be relevant for the stripe formation mechanism, the results for the normal state suggest that similar charge inhomogeneities may also appear in vortex cores up to relatively high doping values.
We examine the two-dimensional $t\ensuremath{-}J$ model by using variational approach combined with well established quantum Monte Carlo techniques [S. Sorella et al., Phys. Rev. Lett. 88, 117002 (2002).] that are used to improve systematically the accuracy of the variational ansatz. Contrary to recent density-matrix renormalization group and projected entangled-pair state calculations [P. Corboz et al., Phys. Rev. B 84, 041108(R) (2011).], a uniform phase is found for J/t=0.4J/t=0.4J/t=0.4, even when the calculation is biased with an ansatz that explicitly contains stripe order. Moreover, in the small hole doping regime, that is, $\ensuremath{\delta}\ensuremath{\lesssim}0.1$, our results support the coexistence of antiferromagnetism and superconductivity.
Using the density matrix renormalization group, we study the 2D $t\ensuremath{-}J$ model at a hole doping of x=18x\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{8}x=81​ on clusters as large as $19\ifmmode\times\else\texttimes\fi{}8$. We find a striped phase consistent with recent neutron scattering experiments. We find that bond-centered and site-centered stripes have nearly the same energy, suggesting that in the absence of pinning effects the domain walls can fluctuate.
A theoretical investigation of boson versions of the $t\ensuremath{-}J$ and $t\ensuremath{-}{J}_{z}$ models on the square lattice is carried out, by means of Green's function Monte Carlo simulations. Accurate ground-state energy estimates as a function of hole doping are obtained, allowing one to investigate the stability of the uniform phase against separation of the system into hole-rich and hole-free phases. In the boson $t\ensuremath{-}{J}_{z}$ model, such a separation is found to occur for arbitrarily small values of Jz,{J}_{z},Jz​, at sufficiently low hole doping. Phase separation is suppressed in the boson $t\ensuremath{-}J$ model, which features a uniform ground state at any doping, for $J/t\ensuremath{\lesssim}1.5.$ Relevance of this study to the corresponding fermion models is discussed. Fermi statistics enhances the tendency toward phase separation; in particular, phase separation at low doping is predicted in the fermion $t\ensuremath{-}{J}_{z}$ model at any Jz>0.{J}_{z}>0.Jz​>0. The possible formation of stripes of holes is investigated for systems featuring both periodic and cylindrical boundary conditions. No evidence of a striped ground state is found in either the $t\ensuremath{-}J$ or $t\ensuremath{-}{J}_{z}$ boson models.
We report results of large-scale ground-state density matrix renormalization group (DMRG) calculations on t-[Formula: see text]-J cylinders with circumferences 6 and 8. We determine a rough phase diagram that appears to approximate the two-dimensional (2D) system. While for many properties, positive and negative [Formula: see text] values ([Formula: see text]) appear to correspond to electron- and hole-doped cuprate systems, respectively, the behavior of superconductivity itself shows an inconsistency between the model and the materials. The [Formula: see text] (hole-doped) region shows antiferromagnetism limited to very low doping, stripes more generally, and the familiar Fermi surface of the hole-doped cuprates. However, we find [Formula: see text] strongly suppresses superconductivity. The [Formula: see text] (electron-doped) region shows the expected circular Fermi pocket of holes around the [Formula: see text] point and a broad low-doped region of coexisting antiferromagnetism and d-wave pairing with a triplet p component at wavevector [Formula: see text] induced by the antiferromagnetism and d-wave pairing. The pairing for the electron low-doped system with [Formula: see text] is strong and unambiguous in the DMRG simulations. At larger doping another broad region with stripes in addition to weaker d-wave pairing and striped p-wave pairing appears. In a small doping region near [Formula: see text] for [Formula: see text], we find an unconventional type of stripe involving unpaired holes located predominantly on chains spaced three lattice spacings apart. The undoped two-leg ladder regions in between mimic the short-ranged spin correlations seen in two-leg Heisenberg ladders.