Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0803.0933 · Physical Review B

Optimal inhomogeneity for superconductivity: Finite-size studies

Wei-Feng Tsai, Hong Yao, Andreas Martin Läuchli, Steven Allan Kivelson

Published Jun 4, 2008• 58 citations
arXivHTMLPDF

Abstract

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 $t$ 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 $U$ and doped hole concentration $x$. We present evidence that superconductivity is strongest for $U$ 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 $U=8t$ 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.

DOI
10.1103/physrevb.77.214502

Related papers

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

Gideon Wachtel, Shirit Baruch, Dror Orgad

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.

arXiv:1706.05023•Physical review. B./Physical review. B•Aug 30, 2017•11 citations

s -wave superconductivity phase diagram in the inhomogeneous two-dimensional attractive Hubbard model

Karan Aryanpour, Thereza Paiva, Warren E. Pickett +1 more

We study $s$-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 ${U}_{i}$ between the electrons takes on two values ${U}_{i}=0$ and $\ensuremath{-}U∕(1\ensuremath{-}f)$ (with $f$ the concentration of noninteracting sites) as a function of average electron occupation per site, $n$, and study the evolution of the phase diagram as $f$ 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 ${T}_{c}$, 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 ${T}_{c}$ due to inhomogeneity is robust as long as the electron doping per site, $n$, 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 $f$ and, as a result, enhance ${T}_{c}$.

arXiv:0708.0451•Physical Review B•Nov 30, 2007•23 citations

Disorder effects on superconducting tendencies in the checkerboard Hubbard model

Peter M. Smith, Malcolm P. Kennett

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.

arXiv:1307.5347•Physical Review B•Dec 31, 2013•8 citations

Contractor-renormalization study of Hubbard plaquette clusters

Shirit Baruch, Dror Orgad

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 $U$ 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 $U$, 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.

arXiv:1005.0978•Physical Review B•Oct 29, 2010•21 citations

Effect of inhomogeneity on s -wave superconductivity in the attractive Hubbard model

Karan Aryanpour, Elbio R. Dagotto, Matthias Mayr +3 more

Inhomogeneous $s$-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 ${U}_{i}$ takes on two values, ${U}_{i}=0,\phantom{\rule{0.3em}{0ex}}2U$ results in a larger zero temperature pairing amplitude, and that the superconducting ${T}_{c}$ can also be significantly increased, relative to a uniform system with ${U}_{i}=U$ 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 $d$-wave pairing symmetry.

arXiv:cond-mat/0507588•Physical Review B•Mar 29, 2006•37 citations

d -wave superconductivity on the checkerboard Hubbard model at weak and strong coupling

Shiladitya Chakraborty, David Sénéchal, A.–M. S. Tremblay

It has been argued that inhomogeneity generally can enhance superconductivity (SC) in the cuprate high-${T}_{c}$ materials. To check the validity of this claim, we study $d$-wave SC on the checkerboard Hubbard model on a square lattice using the cellular dynamical mean-field theory method with an exact diagonalization solver at zero temperature. The $d$-wave order parameter is computed for various inhomogeneity levels over the entire doping range of interest in both strong- and weak-coupling regimes. At a given doping, the size of the $d$-wave order parameter manifests itself directly in the height of the coherence peaks and, hence, is an appropriate measure of the strength of SC. The weak-coupling results reveal a suppression of the order parameter in the presence of inhomogeneity for small-to-intermediate hole dopings, while it is enhanced for large dopings. In contrast, for strong coupling, there is a monotonic decrease in the maximum amplitude of the SC order parameter with inhomogeneity over the entire doping range of interest. Furthermore, at moderately high inhomogeneity, the system undergoes a first-order transition from the SC to the normal state in the underdoped regime. In the overdoped regime, the change in the value of the SC order parameter correlates with the height of the lowest-energy peak in the spectral weight of antiferromagnetic spin fluctuations, confirming the connection between antiferromagnetic fluctuations and $d$-wave SC found in earlier papers on the homogeneous case. Our results are benchmarked by comparisons with numerically exact results on the checkerboard Hubbard ladder.

arXiv:1104.5540•Physical Review B•Aug 26, 2011•14 citations
We report the results of exact-diagonalization studies of Hubbard models on a 4\ifmmode×\else\texttimes\fi44\ifmmode\times\else\texttimes\fi{}44\ifmmode×\else\texttimes\fi4 square lattice with periodic boundary conditions and various degrees and patterns of inhomogeneity, which are represented by inequivalent hopping integrals ttt and t\ensuremath′{t}^{\ensuremath{'}}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′<t0<{t}^{\ensuremath{'}}<t0<t\ensuremath′<t. The maximum value of the ``pair-binding energy'' we have found with purely repulsive interactions is \ensuremathΔpb=0.32t{\ensuremath{\Delta}}_{pb}=0.32t\ensuremathΔpb​=0.32t for the checkerboard Hubbard model with U=8tU=8tU=8t and t\ensuremath′=0.5t{t}^{\ensuremath{'}}=0.5tt\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.
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.
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​.
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.
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.
It has been argued that inhomogeneity generally can enhance superconductivity (SC) in the cuprate high-Tc{T}_{c}Tc​ materials. To check the validity of this claim, we study ddd-wave SC on the checkerboard Hubbard model on a square lattice using the cellular dynamical mean-field theory method with an exact diagonalization solver at zero temperature. The ddd-wave order parameter is computed for various inhomogeneity levels over the entire doping range of interest in both strong- and weak-coupling regimes. At a given doping, the size of the ddd-wave order parameter manifests itself directly in the height of the coherence peaks and, hence, is an appropriate measure of the strength of SC. The weak-coupling results reveal a suppression of the order parameter in the presence of inhomogeneity for small-to-intermediate hole dopings, while it is enhanced for large dopings. In contrast, for strong coupling, there is a monotonic decrease in the maximum amplitude of the SC order parameter with inhomogeneity over the entire doping range of interest. Furthermore, at moderately high inhomogeneity, the system undergoes a first-order transition from the SC to the normal state in the underdoped regime. In the overdoped regime, the change in the value of the SC order parameter correlates with the height of the lowest-energy peak in the spectral weight of antiferromagnetic spin fluctuations, confirming the connection between antiferromagnetic fluctuations and ddd-wave SC found in earlier papers on the homogeneous case. Our results are benchmarked by comparisons with numerically exact results on the checkerboard Hubbard ladder.