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arXiv:1805.11163 · Physical review. B./Physical review. B

Superconductivity in the doped t − J model: Results for four-leg cylinders

Hong‐Chen Jiang, Zheng-Yu Weng, Steven Allan Kivelson

Published Oct 19, 2018• 75 citations
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Abstract

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.

DOI
10.1103/physrevb.98.140505

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We report a density-matrix renormalization group study of the lightly doped t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model on a four-leg cylinder with doped hole concentrations per site \ensuremathδ=5\ensuremath{\delta}=5%\text{--}12.5%\ensuremathδ=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\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J or Hubbard system in which power-law superconducting correlations have been established.
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.
We revisit the ground state of the Hubbard model on two-legged ladders in this paper. We perform density matrix renormalization group (DMRG) calculations on large system sizes with large numbers of kept states and perform extrapolation of DMRG results with truncation errors in the converged region. We find that the superconducting correlation exponent Ksc{K}_{sc}Ksc​ extracted from the pair-pair correlation is very sensitive to the position of the reference bond, reflecting a huge boundary effect on it. By systematically removing the effects from boundary conditions, finite sizes, and truncation errors in DMRG, we obtain more accurate values of Ksc{K}_{sc}Ksc​ and ${K}_{\ensuremath{\rho}}$ with DMRG. With these exponents, we confirm that the two-legged Hubbard model is in the Luther-Emery liquid phase with ${K}_{sc}\ifmmode\cdot\else\textperiodcentered\fi{}{K}_{\ensuremath{\rho}}=1$ from tiny doping near half filling to 1/81/81/8 hole doping. When the doping is increased to $\ensuremath{\delta}⪆1/6$, the behaviors of charge, pairing, and spin correlations do not change qualitatively, but the relationship ${K}_{sc}\ifmmode\cdot\else\textperiodcentered\fi{}{K}_{\ensuremath{\rho}}=1$ is likely to be violated. With the further increase of the doping to $\ensuremath{\delta}=1/3$, the quasi-long-ranged charge correlation turns into a true long-ranged charge order, and the spin gap is closed, while the pair-pair correlation still decays algebraically. Our work provides a standard way to analyze the correlation functions when studying systems with open boundary conditions.
While the recent discovery of superconductivity in infinite-layer nickelates has drawn considerable attention, a common ingredient of the fundamental building blocks to describe their ground states has been lacking. A series of experimental and theoretical studies have suggested that an effective two-band Hubbard model with Ni 3dx2−y2d_{x^2-y^2}dx2−y2​ and rare-earth (RRR) 5ddd character may describe the low-energy physics. Here, we study the ground state properties of this two-band model on four-leg cylinders using the density-matrix renormalization group (DMRG). At half-filling, the ground state of the system is consistent with a Luttinger liquid, possessing quasi-long-range charge and spin correlations in the RRR layer, but short-range correlations in the Ni layer. At a hole doping concentration of δ=12.5%\delta=12.5\%δ=12.5%, the RRR layer is nearly empty and the ground state of the system is consistent with a Luther-Emery liquid, where we find power-law superconducting and charge density correlations in the Ni layer, but exponentially decaying spin correlations. The consistency of our results with experimental observations may help to reveal the microscopic mechanism for pairing in these nickelates and other unconventional superconductors.
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 ddd-wave superconducting ground state in the 2D limit.
Abstract We study the ground state properties of the Hubbard model on three-leg triangular cylinders using large-scale density-matrix renormalization group simulations. At half-filling, we identify an intermediate gapless spin liquid phase, which has one gapless spin mode and algebraic spin–spin correlations but exponential decay scalar chiral–chiral correlations, between a metallic phase at weak coupling and Mott insulating dimer phase at strong interaction. Upon light doping the gapless spin liquid, the system exhibits power-law charge-density-wave (CDW) correlations but short-range single-particle, spin–spin, and chiral–chiral correlations. Similar to CDW correlations, the superconducting correlations also decay in power-law but oscillate in sign as a function of distance, which is consistent with the striped pair-density wave. When further doping the gapless spin liquid phase or doping the dimer order phase, another phase takes over, which has similar CDW correlations but all other correlations decay exponentially.
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.