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arXiv:cond-mat/9608138 · Physical review. B, Condensed matter

Ground states of the doped four-leg t-J ladder

Steven R. White, Douglas J. Scalapino

Published Jun 1, 1997• 47 citations
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Abstract

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.

DOI
10.1103/physrevb.55.r14701

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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 dx2\ensuremath−y2{\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}dx2\ensuremath−y2​ 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 dx2\ensuremath−y2{\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}dx2\ensuremath−y2​ pair field correlations.
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.
Holes weakly doped into a four-leg t-JJJ ladder bind in pairs. At dopings exceeding a critical doping of ${\ensuremath{\delta}}_{c}\ensuremath{\simeq}\frac{1}{8}$ four-hole clusters are observed to form in density-matrix renormalization-group calculations. The symmetry of the ground-state wave function does not change and we are able to reproduce this behavior qualitatively with an effective bosonic model in which the four-leg ladder is represented as two coupled two-leg ladders and hole pairs are mapped on hard-core bosons moving along and between these ladders. At lower dopings, $\ensuremath{\delta}<{\ensuremath{\delta}}_{c},$ a one-dimensional bosonic representation for hole pairs works and allows us to calculate accurately the Luttinger liquid parameter ${K}_{\ensuremath{\rho}}$ that takes the universal value ${K}_{\ensuremath{\rho}}=1$ as half-filling is approached.
The ground state of the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ ladder with four legs favors a striped charge distribution for the parameters corresponding to hole-doped cuprate superconductors. We investigate the dynamical spin and charge structure factors of the model by using the dynamical density matrix renormalization group (DMRG) and clarify the influence of the stripe on the structure factors. The dynamical charge structure factor along the momentum direction from q=(0,0)\mathbf{q}=(0,0)q=(0,0) to $(\ensuremath{\pi},0)$ clearly shows low-energy excitations corresponding to the stripe order in hole doping. On the other hand, the stripe order weakens in electron doping, resulting in fewer low-energy excitations in the charge channel. In the spin channel, we find incommensurate spin excitations near $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$ forming an hourglass behavior in hole doping, while in electron doping we find clearly spin-wave-like dispersions starting from $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$. Along the $(0,0)\text{\ensuremath{-}}(\ensuremath{\pi},0)$ direction, the spin excitations are strongly influenced by the stripes in hole doping, resulting in two branches that form a discontinuous behavior in the dispersion. In contrast, the electron-doped systems show a downward shift in energy toward $(\ensuremath{\pi},0)$. These behaviors along the $(0,0)\text{\ensuremath{-}}(\ensuremath{\pi},0)$ direction are qualitatively similar to momentum-dependent spin excitations recently observed by resonant inelastic x-ray scattering experiments in hole- and electron-doped cuprate superconductors.
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.
Inspired by the observation of a robust ddd-wave superconducting phase driven by tuning the next-nearest-neighbor (NNN) electron hopping in recent density matrix renormalization group (DMRG) studies of six- and eight-leg $t\text{\ensuremath{-}}J$ model, we systematically study the phase diagram of the two-leg $t\text{\ensuremath{-}}J$ ladder with the NNN couplings (including NNN hopping and spin interaction) in a large region of doping level, by means of the DMRG calculations. Upon doping from half filling, we identify the Luther-Emery liquid (LEL) phase, which can be distinguished as the pairing-dominant and charge density-dominant regime by comparing the Luttinger parameter ${K}_{\ensuremath{\rho}}$. With the growing NNN couplings, pairing correlations are enhanced and correspondingly ${K}_{\ensuremath{\rho}}$ increases, driving the system from the density-dominant to the pairing-dominant regime. In the Tomonaga-Luttinger liquid (TLL) phase in the larger doping region, we identify two TLL regimes with different features of charge density correlation. At the quarter filling (1/21/21/2 doping level), we find that the strong dimer orders of bond energy in the open system actually decay algebraically and thus do not indicate a spontaneous translational symmetry breaking. Our results show that in the LEL phase of the two-leg ladder, the NNN couplings seem to play the similar role as that on the wider $t\text{\ensuremath{-}}J$ cylinder, and studies on this more accessible system can be helpful toward understanding the emergence of the remarkable ddd-wave superconducting phase on the wider system.