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arXiv:1002.4367 · Physical Review B

Pair density waves in coupled doped two-leg ladders

Javier Almeida, Guillaume Roux, Didier Poilblanc

Published Jul 13, 2010• 9 citations
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Abstract

Motivated by resonant x-ray scattering experiments in cuprate ladder materials showing charge-order modulation of period $\ensuremath{\lambda}=3$ and 5 at specific hole densities, we investigate models involving the electronic $t\text{\ensuremath{-}}J$ ladders and bosonic chains coupled via screened Coulomb repulsion. Extensive density-matrix renormalization-group calculations applied to the ladders/chains supplemented by a self-consistent mean-field treatment of the interladder/chain coupling provide quantitative estimates of the charge order for $\ensuremath{\lambda}=3$, 4, and 5. As previously proposed, such patterns correspond to the emergence of pair density waves which stem from the strong electronic correlations. We comment on the existence of a $\ensuremath{\lambda}=4$ modulation not seen so far in experiment.

DOI
10.1103/physrevb.82.041102

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Motivated by resonant x-ray scattering experiments in cuprate ladder materials showing charge-order modulation of period \ensuremathλ=3\ensuremath{\lambda}=3\ensuremathλ=3 and 5 at specific hole densities, we investigate models involving the electronic t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J ladders and bosonic chains coupled via screened Coulomb repulsion. Extensive density-matrix renormalization-group calculations applied to the ladders/chains supplemented by a self-consistent mean-field treatment of the interladder/chain coupling provide quantitative estimates of the charge order for \ensuremathλ=3\ensuremath{\lambda}=3\ensuremathλ=3, 4, and 5. As previously proposed, such patterns correspond to the emergence of pair density waves which stem from the strong electronic correlations. We comment on the existence of a \ensuremathλ=4\ensuremath{\lambda}=4\ensuremathλ=4 modulation not seen so far in experiment.
We study two charge-ordered states in a $t\ensuremath{-}J\ensuremath{-}U$ model with the method of renormalized mean-field theory. Compared with the $t\ensuremath{-}J$ model, which excludes the existence of one site occupied by two electrons, finite Coulomb interaction UUU will allow the appearance of doublons and produce more charge fluctuations. The charge density wave states accompanied by Cooper pair density modulations have similar features with the states found in the $t\ensuremath{-}J$ model. The difference is that the modulation of doublons exists, and it has the same wavelength and phase as holes. Charge-ordered states disappear at the $U/t\ensuremath{\le}9$ region when the system does not correspond to the strongly correlated resonating valence-bond state. This feature indicates that local Coulomb interactions are the origin of charge-ordered states observed in cuprates. In addition, we calculate the strength of superconducting coherence and the gap depth of the nodal pair-density-wave state based on the local density of states. These two quantities also vary with lattice sites periodically. The oscillations have same wave vector and phase as the modulation of holes. These results are consistent with recent experiments.
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.
We propose a doped two-leg spin-1/21/21/2 diagonal ladder model (or a composite spin model) to simulate a doped spin-1 chain. By using the density matrix renormalization group method for an open chain, we find that the ground state hosts a dominant singlet dxy{d}_{xy}dxy​-wave superconducting correlation with features similar to those of the spin-1 Haldane phase. Meanwhile, we apply the renormalized mean-field theory to the model and obtain a dxy{d}_{xy}dxy​-wave superconducting state. The superconducting order exhibits a domelike shape as a function of hole concentration $\ensuremath{\delta}$. Our work provides a physical understanding of the origin of dxy{d}_{xy}dxy​ pairing symmetry, which originates from antiferromagnetic Heisenberg interaction in the diagonal direction, and the intrachain hopping term makes it stable.
Magnetic interactions are widely believed to play a crucial role in the microscopic mechanism leading to high critical temperature superconductivity. It is therefore important to study the signatures of pairing in the magnetic excitation spectrum of simple models known to show unconventional superconducting tendencies. Using the density matrix renormalization group technique, we calculate the dynamical spin structure factor $S(\mathbf{k},\ensuremath{\omega})$ of a generalized $t\ensuremath{-}U\ensuremath{-}J$ Hubbard model away from half filling in a two-leg ladder geometry. The addition of JJJ enhances pairing tendencies. We analyze quantitatively the signatures of pairing in the magnetic excitation spectra. We found that the superconducting pair-correlation strength, that can be estimated independently from ground state properties, is closely correlated with the integrated low-energy magnetic spectral weight in the vicinity of $(\ensuremath{\pi},\ensuremath{\pi})$. In this wave-vector region, robust spin incommensurate features develop with increasing doping. The branch of the spectrum with rung direction wave vector krung=0{k}_{\mathrm{rung}}=0krung​=0 does not change substantially with doping where pairing dominates and thus plays a minor role. We discuss the implications of our results for neutron scattering experiments, where the spin excitation dynamics of hole-doped quasi-one-dimensional magnetic materials can be measured and also address implications for recent resonant inelastic x-ray scattering experiments.
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.
We provide, for the first time, in a doped strongly correlated system (two-leg ladder), a controlled theoretical demonstration of the existence of a state in which long-range ordered orbital currents are arranged in a staggered pattern, coexisting with a charge density wave. The method used is the highly accurate density-matrix renormalization group technique. This brings us closer to recent proposals that this order is realized in the enigmatic pseudogap phase of the cuprate high temperature superconductors.