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Physical Review Letters

Density Matrix Renormalization Group Study of the Striped Phase in the 2D t − J Model

Steven R. White, Douglas J. Scalapino

Published Feb 9, 1998• 504 citations
Publisher

Abstract

Using the density matrix renormalization group, we study the 2D $t\ensuremath{-}J$ model at a hole doping of $x\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{8}$ 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.

DOI
10.1103/physrevlett.80.1272

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Using the density matrix renormalization group, we study the 2D t\ensuremath−Jt\ensuremath{-}Jt\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×\else\texttimes\fi819\ifmmode\times\else\texttimes\fi{}819\ifmmode×\else\texttimes\fi8. 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.
Using the density matrix renormalization group, we calculate the energy of a domain wall in the 2D $t\ensuremath{-}J$ model as a function of the linear hole density ${\ensuremath{\rho}}_{\ensuremath{\ell}}$, as well as the interaction energy between walls, for J/t=0.35J/t\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.35J/t=0.35. Based on these results, we conclude that the ground state always has domain walls for dopings $0<x\ensuremath{\lesssim}0.3$. For $x\ensuremath{\lesssim}0.125$, the system has (1, 0) domain walls with ${\ensuremath{\rho}}_{\ensuremath{\ell}}\ensuremath{\sim}0.5$, while for $0.125\ensuremath{\lesssim}x\ensuremath{\lesssim}0.17$ the system has a possibly phase-separated mixture of walls with ${\ensuremath{\rho}}_{\ensuremath{\ell}}\ensuremath{\sim}0.5$ and ${\ensuremath{\rho}}_{\ensuremath{\ell}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$. For $x\ensuremath{\gtrsim}0.17$, there are only walls with ${\ensuremath{\rho}}_{\ensuremath{\ell}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$.
Using the density matrix renormalization group, we study domain wall structures in the t-J model at a hole doping of x=1/8. We find that the domain walls are composed of d_{x^2-y^2} pairs and that the regions between the domain walls have antiferromagnetic correlations that are pi phase shifted across a domain wall. At x=1/8, the hole filling corresponds to one hole per two domain wall unit cells. When the pairs in a domain wall are pinned by an external field, the d_{x^2-y^2} pairing response is suppressed, but when the pinning is weakened, d_{x^2-y^2} pair-field correlations can develop.
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.
Hole-density correlation functions in the t-J model on lattices with square geometry are studied by the method of exact diagonalization for systems of various sizes with up to 26 sites. For the largest sizes we employ only a restricted basis, whereby for the extrapolation we introduce and test a scaling procedure. For two holes in an antiferromagnet our results confirm their binding for J>Jc{\mathit{J}}_{\mathit{c}}Jc​\ensuremath{\sim}0.2t. For four holes the two-point and four-point density correlations, calculated for N=18,20 sites, show clear evidence for hole-clustering instabilities for J>Js{\mathit{J}}_{\mathit{s}}Js​\ensuremath{\sim}1.5t; a first instability is likely to be towards a striped phase, with holes forming domain walls along the (0,1) or (1,0) direction. Signs for the phase separation into a hole-rich and a hole-free phase appear only at J>Js∗{\mathit{J}}_{\mathit{s}}^{\mathrm{*}}Js∗​\ensuremath{\sim}2.5t. Results in the intermediate regime 0.4J/t1.2 are consistent with the pairing picture, although there are also indications for a large weight of configurations representing domain walls along the (1,1) direction.