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arXiv:cond-mat/0306545 · Physical Review Letters

Stripes on a 6-Leg Hubbard Ladder

Steven R. White, Douglas J. Scalapino

Published Sep 25, 2003• 114 citations
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Abstract

While density matrix renormalization group calculations find stripes on doped n-leg t-J ladders, little is known about the possible formation of stripes on n-leg Hubbard ladders. Here we report results for a 7x6 Hubbard model with four holes. We find that a stripe forms for values of U/t ranging from 6 to 20. For U/t approximately 3-4, the system exhibits the domain wall feature of a stripe, but the hole density is very broadened.

DOI
10.1103/physrevlett.91.136403

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While density matrix renormalization group calculations find stripes on doped n-leg t-J ladders, little is known about the possible formation of stripes on n-leg Hubbard ladders. Here we report results for a 7x6 Hubbard model with four holes. We find that a stripe forms for values of U/t ranging from 6 to 20. For U/t approximately 3-4, the system exhibits the domain wall feature of a stripe, but the hole density is very broadened.
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Using density-matrix renormalization-group techniques, we have studied the ground states of the four-leg t-J ladder doped near half-filling. Depending upon J/t and the hole doping x, three characteristic types of ground states are found: (1) a state containing ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$ pairs, (2) a striped charge-density wave (CDW) domain-wall state, and (3) a phase-separated regime. The CDW domain walls consist of fluctuating hole pairs and this state has significant ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$ pair field correlations.
As the number of legs nnn of an nnn-leg, $t\ensuremath{-}J$ ladder increases, density-matrix renormalization group calculations have shown that the doped state tends to be characterized by a static array of domain walls and that pairing correlations are suppressed. Here we present results for a ${t\ensuremath{-}t}^{\ensuremath{'}}\ensuremath{-}J$ model in which a diagonal, single-particle, next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$ is introduced. We find that this can suppress the formation of stripes and, for ${t}^{\ensuremath{'}}$ positive, enhance the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-like pairing correlations. The effect of ${t}^{\ensuremath{'}}>0$ is to cause the stripes to evaporate into pairs and for ${t}^{\ensuremath{'}}<0$ to evaporate into quasiparticles. Results for n=4n=4n=4 and 6 nnn-leg ladders are discussed.
Density matrix renormalization group (DMRG) calculations on four-leg $t\text{\ensuremath{-}}J$ and Hubbard ladders have found a phase exhibiting ``stripes'' at intermediate doping. Such behavior can be viewed as generalized Friedel oscillations, with wavelength equal to the inverse hole density, induced by the open boundary conditions. So far, this phase has not been understood using the conventional weak-coupling bosonization approach. Based on studies of a general bosonization proof, finite size spectrum, an improved analysis of weak-coupling renormalization group equations, and the decoupled two-leg ladder limit, we here find new types of phases of four-leg ladders, which exhibit stripes. They also inevitably exhibit ``bipairing,'' meaning that there is a gap to add one or two electrons (but not four) and that both single electron and electron pair correlation functions decay exponentially, while correlation functions of charge-4 operators exhibit a power-law decay. Whether or not bipairing occurs in the stripe phase found in DMRG is an important open question.
Results for a doped three-leg ttt-JJJ ladder obtained using the density-matrix renormalization group are reported. At low hole doping, the holes form a dilute gas with a uniform density. The momentum occupation of the odd band shows a sharp decrease at a large value of kF{k}_{F}kF​ similar to the behavior of a lightly doped ttt-JJJ chain, while the even modes appear gapped. The spin-spin correlations decay as a power law consistent with the absence of a spin gap, but the pair-field correlations are negligible. At larger doping we find evidence for a spin gap and as xxx increases further we find three-hole diagonal domain walls. In this regime there are significant pair-field correlations and the internal pair orbital has ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-like symmetry.
Using both the density-matrix renormalization group method and the constrained-path quantum Monte Carlo method, we have studied the ground-state energies and the spin and hole densities of a 12×312 \times 312×3 Hubbard model with open boundary conditions and 6 holes doped away from half-filling. Results obtained with these two methods agree well in the small and intermediate UUU regimes. For U/t≥6U/t \geq 6U/t≥6 we find a ground-state with stripes.