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arXiv:1610.04300 · Physical review. B./Physical review. B

Intertwined order in a frustrated four-leg t − J cylinder

John F. Dodaro, Hong‐Chen Jiang, Steven Allan Kivelson

Published Apr 12, 2017• 70 citations
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Abstract

We report a density-matrix renormalization group study of the $t\text{\ensuremath{-}}J$ model with nearest $({t}_{1}$ and ${J}_{1})$ and next-nearest $({t}_{2}$ and ${J}_{2})$ interactions on a four-leg cylinder with concentration $\ensuremath{\delta}=1/8$ of doped holes. We observe an astonishingly complex interplay between uniform $d$-wave superconductivity (SC) and strong spin and charge-density wave ordering tendencies (SDW and CDW). Depending on parameters, the CDWs can be commensurate with period 4 or 8. By comparing the charge ordering vectors with $2{k}_{F}$, we rule out Fermi surface nesting-induced density wave order in our model. Magnetic frustration (i.e., ${J}_{2}/{J}_{1}\ensuremath{\sim}1/2)$ significantly quenches SDW correlations with little effect on the CDW. Typically, the SC order is strongly modulated at the CDW ordering vector and exhibits $d$-wave symmetry around the cylinder. There is no evidence of a near-degenerate tendency to pair-density wave (PDW) ordering, charge $4e$ SC, or orbital current order.

DOI
10.1103/physrevb.95.155116

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Recently, a flurry of works have found strong competition between charge density wave (CDW) and superconductivity (SC) in the doped Hubbard and $t\text{\ensuremath{-}}J$ models on the square lattice. Interestingly, some recent results suggest that the electron-phonon coupling may suppress CDW order and enhance SC. In this work, we consider the square-lattice Hubbard model with the Holstein or Su-Schrieffer-Heeger electron-phonon coupling at the large-$U$ and antiadiabatic (infinite phonon frequency) limits, which gives an effective $t\text{\ensuremath{-}}J$ model with either a density attractive interaction $V$ or a ${J}_{P}$ term that contributes a larger spin exchange and a density repulsive interaction. To explore how these effective couplings may tune the competition between CDW and SC, we implement the density matrix renormalization group simulations on the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model with $V$ or ${J}_{P}$ coupling. We focus on the six-leg cylinder system with the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$, which hosts partially filled stripe and $d$-wave SC in phase diagram. By tuning ${t}^{\ensuremath{'}}/t>0$ and $V$ or ${J}_{P}$, we establish two quantum phase diagrams at $\frac{1}{12}$ doping ratio. In the SC phases, the increased $V$ or ${J}_{P}$ coupling can enhance the quasi-long-range SC order, consistent with some previous findings. Nonetheless, no SC emerges when the partially filled stripe phase disappears with increased $V$ or ${J}_{P}$. Instead, the system has a transition to either a phase-separation-like regime or a filled stripe phase. On the other hand, with increased ${t}^{\ensuremath{'}}/t$, not only the partially filled stripe but the phase separation and filled stripe can also be tuned to SC phase. Our results suggest that although $V$ and ${J}_{P}$ couplings may strengthen hole binding, the hole dynamics controlled by ${t}^{\ensuremath{'}}/t$ appears to play a more crucial role for obtaining a SC in the $t\text{\ensuremath{-}}J$ model.

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We report a density-matrix renormalization group study of the t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model with nearest (t1({t}_{1}(t1​ and J1){J}_{1})J1​) and next-nearest (t2({t}_{2}(t2​ and J2){J}_{2})J2​) interactions on a four-leg cylinder with concentration \ensuremathδ=1/8\ensuremath{\delta}=1/8\ensuremathδ=1/8 of doped holes. We observe an astonishingly complex interplay between uniform ddd-wave superconductivity (SC) and strong spin and charge-density wave ordering tendencies (SDW and CDW). Depending on parameters, the CDWs can be commensurate with period 4 or 8. By comparing the charge ordering vectors with 2kF2{k}_{F}2kF​, we rule out Fermi surface nesting-induced density wave order in our model. Magnetic frustration (i.e., J2/J1\ensuremath∼1/2){J}_{2}/{J}_{1}\ensuremath{\sim}1/2)J2​/J1​\ensuremath∼1/2) significantly quenches SDW correlations with little effect on the CDW. Typically, the SC order is strongly modulated at the CDW ordering vector and exhibits ddd-wave symmetry around the cylinder. There is no evidence of a near-degenerate tendency to pair-density wave (PDW) ordering, charge 4e4e4e SC, or orbital current order.
Abstract The issue of whether d -wave superconductivity (SC) occurs in the square-lattice Hubbard model with U of order of the bandwidth has been one of the most debated issues to emerge from the study of high temperature SC. Here, we report variational results on eight-leg cylinders with next-nearest-neighbor hopping in the range −0.5t≤t′≤0.25t-0.5t\le t^{\prime} \le 0.25t−0.5t≤t′≤0.25t − 0.5 t ≤ t ′ ≤ 0.25 t with U = 8 t and 12 t and doped hole concentrations δ = 1/12 and 1/8. For t′≤0t^{\prime} \le 0t′≤0 t ′ ≤ 0 , the ground-state appears to be a charge-density wave (CDW) of one sort or another with SC correlations that are extremely short-ranged. In contrast, in some cases, the local magnetic order has a correlation length greater than half the cylinder width - suggestive that magnetic order might also arise in the 2D limit. For t′ > 0t^{\prime}\,>\,0t′>0 t ′ > 0 , our results depend more strongly on boundary conditions (periodic vs antiperiodic), making it still harder to correctly guess whether SC or CDW correlations dominate in the 2D limit. These results were obtained employing matrix-product states with bond dimensions large enough that energy differences as small as 10 −3 t per site can be resolved.
The square-lattice Hubbard and closely related t-J models are considered as basic paradigms for understanding strong correlation effects and unconventional superconductivity (SC). Recent large-scale density matrix renormalization group simulations on the extended t-J model have identified d-wave SC on the electron-doped side (with the next-nearest-neighbor hopping t_{2}>0) but a dominant charge density wave (CDW) order on the hole-doped side (t_{2}<0), which is inconsistent with the SC of hole-doped cuprate compounds. We re-examine the ground-state phase diagram of the extended t-J model by employing the state-of-the-art density matrix renormalization group calculations with much enhanced bond dimensions, allowing more accurate determination of the ground state. On six-leg cylinders, while different CDW phases are identified on the hole-doped side for the doping range δ=1/16-1/8, a SC phase emerges at a lower doping regime, with algebraically decaying pairing correlations and d-wave symmetry. On the wider eight-leg systems, the d-wave SC also emerges on the hole-doped side at the optimal 1/8 doping, demonstrating the winning of SC over CDW by increasing the system width. Our results not only suggest a new path to SC in general t-J model through weakening the competing charge orders, but also provide a unified understanding on the SC of both hole- and electron-doped cuprate superconductors.
A pair density wave (PDW) is a superconducting (SC) state with a spatially modulated order parameter. Although much is known about the properties of the PDW state, its realization in microscopic models with divergent susceptibility has been challenging. Here we report a density-matrix renormalization group study of a three-band Hubbard model (also known as the Emery model) for cuprates on long two-leg square cylinders. Upon light doping, we find that the ground state of the system is consistent with that of a PDW state with mutually commensurate and power-law SC, charge (CDW), and spin (SDW) density wave correlations. The SC correlations are dominant between neighboring Cu sites with ddd-wave pairing symmetry. Interestingly, we find that the near-neighbor interactions, especially the near-neighbor attractive Vpd{V}_{pd}Vpd​ interaction between neighboring Cu and oxygen sites, can notably enhance the SC correlations while simultaneously suppressing the CDW correlations. For a modestly strong attractive Vpd{V}_{pd}Vpd​, the SC correlations become quasi-long-ranged with a divergent PDW susceptibility.
We report a density-matrix renormalization group study of the lightly doped $t\text{\ensuremath{-}}J$ model on a four-leg cylinder with doped hole concentrations per site $\ensuremath{\delta}=5%\text{--}12.5%$. By keeping an unusually large number of states and long system sizes, we are able to accurately document the interplay between superconductivity, spin, and charge-density-wave orders. The long-distance behavior is consistent with that of a Luther-Emery liquid with a spin gap and power-law charge-density-wave and superconducting correlations. This is the widest $t\text{\ensuremath{-}}J$ or Hubbard system in which power-law superconducting correlations have been established.
The quest to understand the nature of superconductivity in the cuprates has spotlighted the pair density wave (PDW)–a superconducting state characterized by a spatially modulated order parameter. Despite significant advances in understanding PDW properties, conclusively demonstrating its presence in systems pertinent to cuprate superconductors remains elusive. In this study, we present a systematic density-matrix renormalization group study to investigate the Emery model (or the three-band Hubbard model) on two-leg square cylinders with negative electron hopping term tpp between adjacent oxygen sites. Kinetic frustration - introduced by changing the sign of oxygen-oxygen hopping - leads to a much reduced Cu-Cu antiferromagnetic exchange along with an enlarged charge transfer energy that changes the local properties of the model. At light doping levels, our findings reveal a ground state remarkably consistent with a PDW, exhibiting mutually commensurate superconducting (SC), charge, and spin density wave correlations. Intriguingly, the dominant SC pairing is observed between neighboring oxygen sites, diverging from the expected Cu sites in the positive tpp case. When the system incorporates moderate near-neighbor interactions, particularly an attractive Vpp between adjacent oxygen sites, the SC correlations become quasi-long-ranged, accompanied by a pronounced divergence in the PDW susceptibility. When the attractive Vpp increases further, the system gives way to an unconventional d-wave superconductivity.
Recently, a flurry of works have found strong competition between charge density wave (CDW) and superconductivity (SC) in the doped Hubbard and $t\text{\ensuremath{-}}J$ models on the square lattice. Interestingly, some recent results suggest that the electron-phonon coupling may suppress CDW order and enhance SC. In this work, we consider the square-lattice Hubbard model with the Holstein or Su-Schrieffer-Heeger electron-phonon coupling at the large-UUU and antiadiabatic (infinite phonon frequency) limits, which gives an effective $t\text{\ensuremath{-}}J$ model with either a density attractive interaction VVV or a JP{J}_{P}JP​ term that contributes a larger spin exchange and a density repulsive interaction. To explore how these effective couplings may tune the competition between CDW and SC, we implement the density matrix renormalization group simulations on the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model with VVV or JP{J}_{P}JP​ coupling. We focus on the six-leg cylinder system with the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$, which hosts partially filled stripe and ddd-wave SC in phase diagram. By tuning ${t}^{\ensuremath{'}}/t>0$ and VVV or JP{J}_{P}JP​, we establish two quantum phase diagrams at 112\frac{1}{12}121​ doping ratio. In the SC phases, the increased VVV or JP{J}_{P}JP​ coupling can enhance the quasi-long-range SC order, consistent with some previous findings. Nonetheless, no SC emerges when the partially filled stripe phase disappears with increased VVV or JP{J}_{P}JP​. Instead, the system has a transition to either a phase-separation-like regime or a filled stripe phase. On the other hand, with increased ${t}^{\ensuremath{'}}/t$, not only the partially filled stripe but the phase separation and filled stripe can also be tuned to SC phase. Our results suggest that although VVV and JP{J}_{P}JP​ couplings may strengthen hole binding, the hole dynamics controlled by ${t}^{\ensuremath{'}}/t$ appears to play a more crucial role for obtaining a SC in the $t\text{\ensuremath{-}}J$ model.