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arXiv:2302.11633 · Physical review. B./Physical review. B

Superconducting valence bond fluid in lightly doped eight-leg t − J cylinders

Hong‐Chen Jiang, Steven Allan Kivelson, D. H. Lee

Published Aug 4, 2023• 27 citations
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Abstract

Superconductivity in doped quantum paramagnets has been a subject of long theoretical inquiry. In this work, we report a density matrix renormalization group study of lightly doped $t\text{\ensuremath{-}}J$ models on the finite-width square lattice (doped hole densities $\ensuremath{\delta}=1/12$ and 1/8) with parameters for which previous studies have suggested that the undoped system in 2D is either a quantum spin liquid or a valence bond crystal. Our studies are performed on cylinders with width up to 8. Ground-state correlations are found to be nearly identical for the ``doped quantum spin liquid'' and ``doped valence bond crystal.'' Upon increasing the cylinder widths from 4 to 8, we observed a significant strengthening of the quasi-long-range superconducting correlations and a dramatic suppression of any ``competing'' charge density wave order. Extrapolating from the observed behavior of the width eight cylinders, we speculate that the system has a nodeless $d$-wave superconducting ground state in the 2D limit.

DOI
10.1103/physrevb.108.054505

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Superconductivity in doped quantum paramagnets has been a subject of long theoretical inquiry. In this work, we report a density matrix renormalization group study of lightly doped t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J models on the finite-width square lattice (doped hole densities \ensuremathδ=1/12\ensuremath{\delta}=1/12\ensuremathδ=1/12 and 1/8) with parameters for which previous studies have suggested that the undoped system in 2D is either a quantum spin liquid or a valence bond crystal. Our studies are performed on cylinders with width up to 8. Ground-state correlations are found to be nearly identical for the doped quantum spin liquid'' and doped valence bond crystal.'' Upon increasing the cylinder widths from 4 to 8, we observed a significant strengthening of the quasi-long-range superconducting correlations and a dramatic suppression of any ``competing'' charge density wave order. Extrapolating from the observed behavior of the width eight cylinders, we speculate that the system has a nodeless ddd-wave superconducting ground state in the 2D limit.
We have performed density-matrix renormalization group studies of a square lattice $t\text{\ensuremath{-}}J$ model with small hole doping, $\ensuremath{\delta}\ensuremath{\ll}1$, on long four and six-leg cylinders. We include frustration in the form of a second-neighbor exchange coupling, J2=J1/2{J}_{2}={J}_{1}/2J2​=J1​/2, such that the undoped ($\ensuremath{\delta}=0$) ``parent'' state is a quantum spin liquid. In contrast to the relatively short range superconducting (SC) correlations that have been observed in recent studies of the six-leg cylinder in the absence of frustration, we find power-law SC correlations with a Luttinger exponent, ${K}_{\mathrm{SC}}\ensuremath{\approx}1$, consistent with a strongly diverging SC susceptibility, $\ensuremath{\chi}\ensuremath{\sim}{T}^{\ensuremath{-}(2\ensuremath{-}{K}_{\mathrm{SC}})}$ as the temperature $T\ensuremath{\rightarrow}0$. The spin-spin correlations---as in the undoped state---fall exponentially suggesting that the SC ``pairing'' correlations evolve smoothly from the insulating parent state.
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 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.
The authors employ the density-matrix renormalization group to investigate the doped Hubbard model on six-leg square cylinders. It uncovers a rich quantum phase diagram, intricately sensitive to next-nearest-neighbor electron hopping ($t\ensuremath{'}$). The positive-ttt\ensuremath{'} region shows a robust ddd-wave superconducting phase with intertwined superconducting and charge-density-wave orders. In contrast, the negative-ttt\ensuremath{'} side remains insulating, where doped holes form either long-range charge stripe order at small ttt\ensuremath{'} or a holon Wigner crystal with one doped hole per emergent unit cell and short-range spin correlations at larger ttt'.
We have performed large-scale density-matrix renormalization group studies of the lightly doped Hubbard model on the honeycomb lattice on long three- and four-leg cylinders. We find that the ground state of the system upon lightly doping is consistent with that of a superconducting state with coexisting quasi-long-range superconducting and charge density wave orders. Both the superconducting and charge density wave correlations decay as a power law at long distances with corresponding exponents Ksc<2{K}_{sc}<2Ksc​<2 and Kc<2{K}_{c}<2Kc​<2. On the contrary, the spin-spin and single-particle correlations decay exponentially, although with relatively long correlation lengths.
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.