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arXiv:1307.5347 · Physical Review B

Disorder effects on superconducting tendencies in the checkerboard Hubbard model

Peter M. Smith, Malcolm P. Kennett

Published Dec 31, 2013• 8 citations
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Abstract

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 $d$-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.

DOI
10.1103/physrevb.88.214518

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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 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.
Inhomogeneous sss-wave superconductivity is studied in the two-dimensional, square lattice attractive Hubbard Hamiltonian using the Bogoliubov-de Gennes (BdG) mean field approximation. We find that at weak coupling, and for densities mainly below half-filling, an inhomogeneous interaction in which the on-site interaction Ui{U}_{i}Ui​ takes on two values, Ui=0,2U{U}_{i}=0,\phantom{\rule{0.3em}{0ex}}2UUi​=0,2U results in a larger zero temperature pairing amplitude, and that the superconducting Tc{T}_{c}Tc​ can also be significantly increased, relative to a uniform system with Ui=U{U}_{i}=UUi​=U on all sites. These effects are observed for stripe, checkerboard, and even random patterns of the attractive centers, suggesting that the pattern of inhomogeneity is unimportant. Monte Carlo calculations which reintroduce some of the fluctuations neglected within the BdG approach see the same effect, both for the attractive Hubbard model and a Hamiltonian with ddd-wave pairing symmetry.
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.
Using a dynamical cluster quantum Monte Carlo approximation, we investigate the ddd-wave superconducting transition temperature Tc{T}_{c}Tc​ in the doped two-dimensional repulsive Hubbard model with a weak inhomogeneity. The inhomogeneity is introduced in the hoppings ${t}^{\ensuremath{'}}$ and ttt in the form of a checkerboard pattern where ttt is the hopping within a $2\ifmmode\times\else\texttimes\fi{}2$ plaquette, and ${t}^{\ensuremath{'}}$ is the hopping between the plaquettes. We find inhomogeneity suppresses Tc{T}_{c}Tc​. The characteristic spin excitation energy and the strength of ddd-wave pairing interaction decrease with decreasing Tc{T}_{c}Tc​, suggesting a strong correlation between these quantities.
We present results of three studies done using a dynamical cluster quantum Monte Carlo approximation.First, we investigate the d-wave superconducting transition temperature T c in the doped 2D repulsive Hubbard model with a weak inhomogeneity in hopping in the form of checkerboard pattern or a lattice of 2 × 2 plaquettes.Near neighbor hoppings within a plaquette is t and that of between the plaquettes is t ′ .We investigate T c in the weak inhomogeneous limit 0.8t < t ′ < 1.2t.We find inhomogeneity (t ′ = t) suppresses T c .The characteristic spin excitation energy (effective exchange energy) and the strength of d-wave pairing interaction decrease with decreasing T c .The latter observations suggest a strong correlation among effective exchange interaction, T c and the d-wave pairing interaction of the system.Second 1 , we further find that enhancement of effective exchange interaction causes a slight increase in T c of a weakly disordered system with low impurity concentration, compared to the homogeneous system.Here the disorder is introduced to homogeneous repulsive 2D Hubbard model as a weak local potential disorder.Third, we present an improved maximum entropy method to analytically continue quantum Monte Carlo data with a severe sign problem.1 A result from a collaborative study done with A. Kemper of
We study sss-wave superconductivity in the two-dimensional square-lattice attractive Hubbard Hamiltonian for various inhomogeneous patterns of interacting sites. Using the Bogoliubov--de Gennes mean-field approximation, we obtain the phase diagram for inhomogeneous patterns in which the on-site attractive interaction Ui{U}_{i}Ui​ between the electrons takes on two values Ui=0{U}_{i}=0Ui​=0 and $\ensuremath{-}U∕(1\ensuremath{-}f)$ (with fff the concentration of noninteracting sites) as a function of average electron occupation per site, nnn, and study the evolution of the phase diagram as fff varies. In certain regions of the phase diagram, inhomogeneity results in a larger zero-temperature average pairing amplitude (order parameter) and also a higher superconducting critical temperature Tc{T}_{c}Tc​, relative to a uniform system with the same mean interaction strength (${U}_{i}=\ensuremath{-}U$ on all sites). These effects are observed for stripes, checkerboard, and even random patterns of the attractive centers, suggesting that the pattern of inhomogeneity is unimportant. The phase diagrams also include regions where superconductivity is obliterated due to the formation of various charge-ordered phases. The enhancement of Tc{T}_{c}Tc​ due to inhomogeneity is robust as long as the electron doping per site, nnn, is less than twice the fraction of interacting sites $[2(1\ensuremath{-}f)]$ regardless of the pattern. We also show that for certain inhomogeneous patterns, when $n=2(1\ensuremath{-}f)$, increasing temperature can work against the stability of existing charge-ordered phases for large fff and, as a result, enhance Tc{T}_{c}Tc​.