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arXiv:1706.05023 · Physical review. B./Physical review. B

Optimal inhomogeneity for pairing in Hubbard systems with next-nearest-neighbor hopping

Gideon Wachtel, Shirit Baruch, Dror Orgad

Published Aug 30, 2017• 11 citations
arXivHTMLPDF

Abstract

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, $t$, and next-nearest-neighbor (diagonal) hopping, ${t}_{d}$. 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 ${t}_{d}/t$, it is suppressed and appears at weaker repulsion strengths and lower hole concentrations when ${t}_{d}/t$ is negative. We point out a possible connection between the pairing maximum and the magnetic properties of the system.

DOI
10.1103/physrevb.96.064527

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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×\else\texttimes\fi44\ifmmode\times\else\texttimes\fi{}44\ifmmode×\else\texttimes\fi4 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.
We report the results of exact-diagonalization studies of Hubbard models on a $4\ifmmode\times\else\texttimes\fi{}4$ square lattice with periodic boundary conditions and various degrees and patterns of inhomogeneity, which are represented by inequivalent hopping integrals ttt and ${t}^{\ensuremath{'}}$. We focus primarily on two patterns, the checkerboard and the striped cases, for a large range of values of the on-site repulsion UUU and doped hole concentration xxx. We present evidence that superconductivity is strongest for UUU of the order of the bandwidth and intermediate inhomogeneity $0<{t}^{\ensuremath{'}}<t$. The maximum value of the ``pair-binding energy'' we have found with purely repulsive interactions is ${\ensuremath{\Delta}}_{pb}=0.32t$ for the checkerboard Hubbard model with U=8tU=8tU=8t and ${t}^{\ensuremath{'}}=0.5t$. Moreover, for near-optimal values, our results are insensitive to changes in boundary conditions, suggesting that the correlation length is sufficiently short that finite-size effects are already unimportant.
Constrained-path quantum Monte Carlo method is applied to study the pairing correlation in the checkerboard Hubbard model with inhomogeneous nearest-neighbor hopping at a low doping of holes. The inhomogeneous hopping can enhance the pairing correlation among different plaquette clusters. An obvious maximum for the pairing correlation is observed at a certain inhomogeneous hopping. The cluster pairing correlation shows the strongest long-range behavior at the optimal inhomogeneity. The enhancement of cluster pairing correlation might be associated with the transition of the Fermi surface structure. This work indicates that the inhomogeneous hopping could tailor the pairing correlation effectively.
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 ${t}^{\ensuremath{'}}$, 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.
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.
The question of whether spatially inhomogeneous hopping in the two dimensional Hubbard model can lead to enhancement of superconductivity has been tackled by a number of authors in the context of the checkerboard Hubbard model (CHM). We address the effects of disorder on superconducting properties of the CHM by using exact diagonalization calculations for both potential and hopping disorder. We characterize the superconducting tendencies of the model by focusing on the pair-binding energy, the spin gap, and ddd-wave pairing order parameter. We find that superconducting tendencies, particularly the pair-binding energy, are more robust to disorder when there is inhomogeneous hopping than for the uniform Hubbard model. We also study all possible staggered potentials for an eight-site CHM cluster and relate the behavior of these configurations to the disordered system.
We implement the contractor-renormalization method to study the checkerboard Hubbard model on various finite-size clusters as function of the inter-plaquette hopping ${t}^{\ensuremath{'}}$ and the on-site repulsion UUU at low hole doping. We find that the pair-binding energy and the spin gap exhibit a pronounced maximum at intermediate values of ${t}^{\ensuremath{'}}$ and UUU, thus indicating that moderate inhomogeneity of the type considered here substantially enhances the formation of hole pairs. The rise of the pair-binding energy for ${t}^{\ensuremath{'}}<{t}_{\text{max}}^{\ensuremath{'}}$ is kinetic-energy driven and reflects the strong resonating valence-bond correlations in the ground state that facilitate the motion of bound pairs as compared to single holes. Conversely, as ${t}^{\ensuremath{'}}$ is increased beyond ${t}_{\text{max}}^{\ensuremath{'}}$ antiferromagnetic magnons proliferate and reduce the potential energy of unpaired holes and with it the pairing strength. For the periodic clusters that we study the estimated phase-ordering temperature at ${t}^{\ensuremath{'}}={t}_{\text{max}}^{\ensuremath{'}}$ is a factor of 2--6 smaller than the pairing temperature.