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arXiv:2004.03001 · Physical review. B./Physical review. B

Plaquette versus ordinary d -wave pairing in the t′ -Hubbard model on a width-4 cylinder

Chia-Min Chung, Mingpu Qin, Shiwei Zhang, Ulrich Schollwöck, Steven R. White

Published Jul 6, 2020• 86 citations
arXivHTMLPDF

Abstract

The Hubbard model and its extensions are important microscopic models for understanding high-${T}_{c}$ superconductivity in cuprates. In the model with next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$ (the ${t}^{\ensuremath{'}}$-Hubbard model), pairing is strongly influenced by ${t}^{\ensuremath{'}}$. In particular, a recent study on a width-4 cylinder observed quasi-long-range superconducting order, associated with a negative ${t}^{\ensuremath{'}}$, which was taken to imply superconductivity in the two-dimensional (2D) limit. In this work we study more carefully pairing in the width-$4\phantom{\rule{4pt}{0ex}}{t}^{\ensuremath{'}}$-Hubbard model. We show that in this specific system, the pairing symmetry with ${t}^{\ensuremath{'}}<0$ is not the ordinary $d$-wave one would expect in the 2D limit. Instead we observe a so-called plaquette $d$-wave pairing. We show that the plaquette $d$-wave or its extension is difficult to generalize in other geometries, for example a 4-leg ladder with open boundaries or a width-6 cylinder. We find that a negative ${t}^{\ensuremath{'}}$ suppresses the conventional $d$-wave, leading to plaquette pairing. In contrast, a different ${t}^{\ensuremath{'}\ensuremath{'}}$ coupling acting diagonally on the plaquettes suppresses plaquette pairing, leading to conventional $d$-wave pairing.

DOI
10.1103/physrevb.102.041106

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The Hubbard model and its extensions are important microscopic models for understanding high-Tc{T}_{c}Tc​ superconductivity in cuprates. In the model with next-nearest-neighbor hopping t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′ (the t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′-Hubbard model), pairing is strongly influenced by t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′. In particular, a recent study on a width-4 cylinder observed quasi-long-range superconducting order, associated with a negative t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′, which was taken to imply superconductivity in the two-dimensional (2D) limit. In this work we study more carefully pairing in the width-4t\ensuremath′4\phantom{\rule{4pt}{0ex}}{t}^{\ensuremath{'}}4t\ensuremath′-Hubbard model. We show that in this specific system, the pairing symmetry with t\ensuremath′<0{t}^{\ensuremath{'}}<0t\ensuremath′<0 is not the ordinary ddd-wave one would expect in the 2D limit. Instead we observe a so-called plaquette ddd-wave pairing. We show that the plaquette ddd-wave or its extension is difficult to generalize in other geometries, for example a 4-leg ladder with open boundaries or a width-6 cylinder. We find that a negative t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′ suppresses the conventional ddd-wave, leading to plaquette pairing. In contrast, a different t\ensuremath′\ensuremath′{t}^{\ensuremath{'}\ensuremath{'}}t\ensuremath′\ensuremath′ coupling acting diagonally on the plaquettes suppresses plaquette pairing, leading to conventional ddd-wave pairing.
The Hubbard and closely related t-J models are exciting platforms for unconventional superconductivity (SC). Through state-of-the-art density matrix renormalization group calculations using the grand canonical ensemble, we address open issues regarding the ground-state phase diagram of the extended t-J model on a square lattice in the parameter regime relevant to cuprate superconductors. On large 8-leg cylinders, we demonstrate that the pure t-J model with only nearest-neighbor hoppings and superexchange interactions, for a wide range of doping ( δ = 0.1 to 0.2 ), hosts robust d-wave superconductivity possibly coexisting with weak unidirectional pair density wave. Furthermore, a small next nearest neighbor hopping t 2 suppresses pair and charge density waves, resulting in a uniform d-wave SC phase in both electron- and hole-doped cuprate model systems. Our work validates the t-J model as a proper minimum model for the emergence of superconductivity in cuprate superconductors.
Extensive numerical studies have demonstrated that the two-dimensional single-band Hubbard model contains much of the key physics in cuprate high-temperature superconductors. However, there is no definitive proof that the Hubbard model truly possesses a superconducting ground state or, if it does, of how it depends on model parameters. To answer these longstanding questions, we study an extension of the Hubbard model including an infinite-range ddd-wave pair field term, which precipitates a superconducting state in the ddd-wave channel. Using exact diagonalization on 16-site square clusters, we study the evolution of the ground state as a function of the strength of the pairing term. This is achieved by monitoring the fidelity metric of the ground state, as well as determining the ratio between the two largest eigenvalues of the ddd-wave pair/spin/charge-density matrices. The calculations show a ddd-wave superconducting ground state in doped clusters bracketed by a strong antiferromagnetic state at half filling controlled by the Coulomb repulsion UUU and a weak short-range checkerboard charge ordered state at larger hole doping controlled by the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$. We also demonstrate that negative ${t}^{\ensuremath{'}}$ plays an important role in facilitating ddd-wave superconductivity.
Motivated by the recent discovery of anomalously strong nearest-neighbor attraction in the one-dimensional hole-doped cuprate ${\text{Ba}}_{2+x}{\text{Sr}}_{x}{\text{CuO}}_{3+\ensuremath{\delta}}$, we investigate the ddd-wave superconducting property of an extended checkerboard Hubbard model, which includes an intraplaquette nearest-neighbor attraction VVV. Our quantum Monte Carlo simulations reveal that, for small interplaquette hopping integral ${t}^{\ensuremath{'}}$, VVV induces a drastic enhancement of long-range ddd-wave pairing correlations in the parameter range of $0.0\ensuremath{\le}U\ensuremath{\le}6.0t$ (ttt is the intraplaquette hopping integral) when the electron density lies between half-filling and quarter-filling. At $V=\ensuremath{-}0.4t$, one order of magnitude larger ddd-wave pairing correlations are observed compared to the ones at V=0.0tV=0.0tV=0.0t and, more importantly, they exceed the values of the free electron system, signifying the existence of ddd-wave superconductivity. The exact diagonalization calculations on the $2\ifmmode\times\else\texttimes\fi{}2$ plaquette indicate that the enhancement of ddd-wave pairing correlations can be attributed to the VVV-induced enhancement of pair-binding energy. Our findings provide a new perspective on the realization of high-temperature ddd-wave superconductivity via the combination of nearest-neighbor attraction and electronic inhomogeneity.
We study the competition between stripe states with different periods and a uniform ddd-wave superconducting state in the extended two-dimensional Hubbard model at 1/8 hole doping using infinite projected entangled-pair states (iPEPS). With increasing strength of negative next-nearest neighbor hopping ${t}^{\ensuremath{'}}$, the preferred period of the stripe decreases. For the values of ${t}^{\ensuremath{'}}$ predicted for cuprate high-Tc{T}_{c}Tc​ superconductors, we find stripes with a period 4 in the charge order, in agreement with experiments. Superconductivity in the period 4 stripe is suppressed at 1/81/81/8 doping. Only at larger doping, $0.18\ensuremath{\lesssim}\ensuremath{\delta}<0.25$, the period 4 stripe exhibits coexisting ddd-wave superconducting order. The uniform ddd-wave state is only favored for sufficiently large positive ${t}^{\ensuremath{'}}$.
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.
Recently, robust d-wave superconductive (SC) order has been unveiled in the ground state of the 2D t-t^{'}-J model-with both nearest-neighbor (t) and next-nearest-neighbor (t^{'}) hoppings-by density matrix renormalization group studies. However, there is currently a debate on whether the d-wave SC holds up strong on both t^{'}/t>0 and t^{'}/t<0 cases for the t-t^{'}-J model, which correspond to the electron- and hole-doped sides of the cuprate phase diagram, respectively. Here, we exploit state-of-the-art thermal tensor network approach to accurately obtain the phase diagram of the t-t^{'}-J model on cylinders with widths up to W=6 and down to low temperature as T/J≃0.06, pushing the boundaries of contemporary finite-T calculations. For t^{'}/t>0, we find a domelike SC regime with a diverging d-wave pairing susceptibility, χ_{SC}∝1/T^{α} below a characteristic temperature T_{c}^{*}. Near optimal doping, T_{c}^{*} reaches its highest value of about 0.15J. Above T_{c}^{*} yet below a higher crossover temperature T^{*}, the magnetic susceptibility becomes suppressed, which can be related to the onset of pseudogap (PG) behaviors. On the other hand, for t^{'}/t<0, we find the pairing correlations are much weaker, although there exhibits a node-antinode structure in the PG regime as observed in the hole-doped cuprates. The thermal tensor network calculations of the t-t^{'}-J model underscore both the similarities and differences in the finite-temperature phase diagram between the fundamental model and cuprates, yielding unique insights into their intricate behaviors.