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arXiv:1008.3908 · Physical Review B

Enhanced pairing in the checkerboard Hubbard ladder

George Karakonstantakis, Erez Berg, Steven R. White, Steven Allan Kivelson

Published Feb 22, 2011• 30 citations
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Abstract

We study signatures of superconductivity in a two-leg ``checkerboard'' Hubbard ladder model, defined as a one-dimensional (period 2) array of square plaquettes with an intraplaquette hopping $t$ and interplaquette hopping ${t}^{\ensuremath{'}}$, using the density-matrix renormalization-group method. The highest pairing scale (characterized by the spin gap or the pair binding energy, extrapolated to the thermodynamic limit) is found for doping levels close to half-filling, $U\ensuremath{\approx}6t$ and ${t}^{\ensuremath{'}}/t\ensuremath{\approx}0.6$. Other forms of modulated hopping parameters, with periods of either one or three lattice constants, are also found to enhance pairing relative to the uniform two-leg ladder, although to a lesser degree. A calculation of the phase stiffness of the ladder reveals that in the regime with the strongest pairing, the energy scale associated with phase ordering is comparable to the pairing scale.

DOI
10.1103/physrevb.83.054508

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We study signatures of superconductivity in a two-leg ``checkerboard'' Hubbard ladder model, defined as a one-dimensional (period 2) array of square plaquettes with an intraplaquette hopping ttt and interplaquette hopping t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′, using the density-matrix renormalization-group method. The highest pairing scale (characterized by the spin gap or the pair binding energy, extrapolated to the thermodynamic limit) is found for doping levels close to half-filling, U\ensuremath≈6tU\ensuremath{\approx}6tU\ensuremath≈6t and t\ensuremath′/t\ensuremath≈0.6{t}^{\ensuremath{'}}/t\ensuremath{\approx}0.6t\ensuremath′/t\ensuremath≈0.6. Other forms of modulated hopping parameters, with periods of either one or three lattice constants, are also found to enhance pairing relative to the uniform two-leg ladder, although to a lesser degree. A calculation of the phase stiffness of the ladder reveals that in the regime with the strongest pairing, the energy scale associated with phase ordering is comparable to the pairing scale.
By employing the density-matrix renormalization group method, we study an extended checkerboard Hubbard model on the two-leg ladder, which includes an intraplaquette nearest-neighbour attraction V. The simulated results show that V plays a significant role in enhancing the d-wave superconductivity when the electron density is close to half-filling. In the homogeneous case t'=t (t and t' are the intraplaquette and interplaquette hopping integrals), large critical |Vc| is required to induce the superconducting ground state. With decreasing t', |Vc| is substantially diminished and the pair state has a nearly C4 symmetry. In the extremely inhomogeneous case t'<0.2t, the system transits to the d-wave superconducting phase at V\sim-0.3t and V\sim-0.4t for U=8t and U=12t, respectively, accompanying with a shift of spin and single-particle excitations from gapless to gapped type.
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.
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 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.
Asymmetric two-leg Hubbard ladders with different on-site interactions Uy{U}_{y}Uy​ and hoppings ty{t}_{y}ty​ on each leg are investigated using the density-matrix renormalization group method and exact diagonalizations. The pairing found in symmetric ladders is robust against the introduction of the leg asymmetry. When studying pairing, one-band Hubbard ladder models are better described as one-dimensional correlated two-band models than as sublattices of higher-dimensional systems. The asymmetric Hubbard ladder provides us with a simple model for studying pairing in the crossover regime between charge-transfer and Mott insulators.
Previous studies have shown that bipartite Hubbard systems with inhomogeneous hopping amplitudes can exhibit higher pair-binding energies than the uniform model. Here we examine whether this result holds for systems with a more generic band structure. To this end, we use exact diagonalization and the density matrix renormalization-group method to study the $4\ifmmode\times\else\texttimes\fi{}4$ Hubbard cluster and the two-leg Hubbard ladder with checkerboard-modulated nearest-neighbor hopping, ttt, and next-nearest-neighbor (diagonal) hopping, td{t}_{d}td​. We find that the strongest pairing continues to occur at an intermediate level of inhomogeneity. While the maximal pair-binding energy is enhanced by a positive td/t{t}_{d}/ttd​/t, it is suppressed and appears at weaker repulsion strengths and lower hole concentrations when td/t{t}_{d}/ttd​/t is negative. We point out a possible connection between the pairing maximum and the magnetic properties of the system.