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:1005.0978 · Physical Review B

Contractor-renormalization study of Hubbard plaquette clusters

Shirit Baruch, Dror Orgad

Published Oct 29, 2010• 21 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.82.134537

Related papers

Optimal inhomogeneity for superconductivity: Finite-size studies

Wei-Feng Tsai, Hong Yao, Andreas Martin Läuchli +1 more

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.

arXiv:0803.0933•Physical Review B•Jun 4, 2008•58 citations

Possible cluster pairing correlation in the checkerboard Hubbard model: a quantum Monte Carlo study

Yongzheng Wu, Shi-Chao Fang, Guangkun Liu +1 more

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.

Journal of Physics Condensed Matter•May 30, 2019•5 citations

Enhanced pairing in the checkerboard Hubbard ladder

George Karakonstantakis, Erez Berg, Steven R. White +1 more

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.

arXiv:1008.3908•Physical Review B•Feb 22, 2011•30 citations

A pure Hubbard model with demonstrable pairing adjacent to the Mott-insulating phase

James Champion, M W Long

Abstract. We introduce a Hubbard model on a particular class of geometries, and consider the effect of doping the highly spin-degenerate Mott-insulating state with a microscopic number of holes in the extreme strong-coupling limit. The geometry is quite general, with pairs of atomic sites at each superlattice vertex, and a highly frustrated inter-atomic connectivity: the one dimensional realization is a chain of edge-sharing tetrahedra. The sole model parameter is the ratio of intrapair to inter-pair hopping matrix elements. If the intra-pair hopping is negligible then introducing a microscopic number of holes results in a ferromagnetic Nagaoka groundstate. Conversely, if the intra-pair hopping is comparable with the inter-pair hopping then the groundstate is low spin with short-ranged spin correlations. We exactly solve the correlated motion of a pair of holes in such a state and find that, in 1-d and 2-d, they form a bound pair on a length scale that increases with diminishing binding energy. This result is pertinent to the long-standing problem of hole-motion in the CuO2 planes of the high-temperature superconductors: we have rigorously shown that, on our frustrated geometry, the holes pair up and a short-ranged low-spin state is generated by hole motion alone. Submitted to: J. Phys. A: Math. Gen. 1.

•Jan 1, 2003•1 citation

Exact study of charge-spin separation, pairing fluctuations, and pseudogaps in four-site Hubbard clusters

ARMEN N. KOCHARIAN, Gayanath Fernando, Kalum Palandage +1 more

An exact study of charge-spin separation, pairing fluctuations, and pseudogaps is carried out by combining the analytical eigenvalues of the four-site Hubbard clusters with the grand canonical and canonical ensemble approaches in a multidimensional parameter space of temperature $(T)$, magnetic field $(h)$, on-site interaction $(U)$, and chemical potential $(\ensuremath{\mu})$. Our results, near the average number of electrons $⟨N⟩\ensuremath{\approx}3$, strongly suggest the existence of a critical parameter ${U}_{c}(T)$ for the localization of electrons and a particle-hole binding (positive) gap ${\ensuremath{\Delta}}^{e\text{\ensuremath{-}}h}(T)>0$ at $U>{U}_{c}(T)$, with a zero temperature critical value, ${U}_{c}(0)=4.584$. For $U<{U}_{c}(T)$, the particle-particle pair binding is found with a (positive) pairing gap ${\ensuremath{\Delta}}^{P}(T)>0$. The ground state degeneracy is lifted at $U>{U}_{c}(T)$ and the cluster becomes a Mott-Hubbard like insulator due to the presence of energy gaps at all (allowed) integer numbers $(1\ensuremath{\le}N\ensuremath{\le}8)$ of electrons. In contrast, for $U\ensuremath{\le}{U}_{c}(T)$, we find an electron pair binding instability at finite temperature near $⟨N⟩\ensuremath{\approx}3$, which manifests a possible pairing mechanism, a precursor to superconductivity in small clusters. Rigorous criteria for the existence of many-body Mott-Hubbard like particle-hole and particle-particle pairings, spin pairing, (spin) pseudogap, and (spin) antiferromagnetic critical crossover temperatures, at which the corresponding pseudogaps disappear, are also formulated. In particular, the resulting phase diagram consisting of charge and spin pseudogaps, antiferromagnetic correlations, hole pairing with competing hole-rich $(⟨N⟩=2)$, hole-poor $(⟨N⟩=4)$, and magnetic $(⟨N⟩=3)$ regions in the ensemble of clusters near 1/8 filling closely resembles the phase diagrams and inhomogeneous phase separation recently found in the family of doped high-${\mathrm{T}}_{\mathrm{c}}$ cuprates.

arXiv:cond-mat/0606165•Physical Review B•Jul 17, 2006•53 citations

A pure Hubbard model with demonstrable pairing adjacent to the Mott-insulating phase

J D Champion, M W Long

We introduce a Hubbard model on a particular class of geometries, and consider the effect of doping the highly spin-degenerate Mott-insulating state with a microscopic number of holes in the extreme strong-coupling limit. The geometry is quite general, with pairs of atomic sites at each superlattice vertex, and a highly frustrated inter-atomic connectivity: the one-dimensional realization is a chain of edge-sharing tetrahedra. The sole model parameter is the ratio of intra-pair to inter-pair hopping matrix elements. If the intra-pair hopping is negligible then introducing a microscopic number of holes results in a ferromagnetic Nagaoka groundstate. Conversely, if the intra-pair hopping is comparable with the inter-pair hopping then the groundstate is low spin with short-ranged spin correlations. We exactly solve the correlated motion of a pair of holes in such a state and find that, in 1d and 2d, they form a bound pair on a length scale that increases with diminishing binding energy. This result is pertinent to the long-standing problem of hole motion in the CuO 2 planes of the high-temperature superconductors: we have rigorously shown that, on our frustrated geometry, the holes pair up and a short-ranged low-spin state is generated by hole motion alone.

arXiv:cond-mat/0302039•Journal of Physics A Mathematical and General•Aug 20, 2003•1 citation
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′{t}^{\ensuremath{'}}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′{t}^{\ensuremath{'}}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′<tmax\ensuremath′{t}^{\ensuremath{'}}<{t}_{\text{max}}^{\ensuremath{'}}t\ensuremath′<tmax\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′{t}^{\ensuremath{'}}t\ensuremath′ is increased beyond tmax\ensuremath′{t}_{\text{max}}^{\ensuremath{'}}tmax\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′=tmax\ensuremath′{t}^{\ensuremath{'}}={t}_{\text{max}}^{\ensuremath{'}}t\ensuremath′=tmax\ensuremath′​ is a factor of 2--6 smaller than the pairing temperature.
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.
Abstract. We introduce a Hubbard model on a particular class of geometries, and consider the effect of doping the highly spin-degenerate Mott-insulating state with a microscopic number of holes in the extreme strong-coupling limit. The geometry is quite general, with pairs of atomic sites at each superlattice vertex, and a highly frustrated inter-atomic connectivity: the one dimensional realization is a chain of edge-sharing tetrahedra. The sole model parameter is the ratio of intrapair to inter-pair hopping matrix elements. If the intra-pair hopping is negligible then introducing a microscopic number of holes results in a ferromagnetic Nagaoka groundstate. Conversely, if the intra-pair hopping is comparable with the inter-pair hopping then the groundstate is low spin with short-ranged spin correlations. We exactly solve the correlated motion of a pair of holes in such a state and find that, in 1-d and 2-d, they form a bound pair on a length scale that increases with diminishing binding energy. This result is pertinent to the long-standing problem of hole-motion in the CuO2 planes of the high-temperature superconductors: we have rigorously shown that, on our frustrated geometry, the holes pair up and a short-ranged low-spin state is generated by hole motion alone. Submitted to: J. Phys. A: Math. Gen. 1.
An exact study of charge-spin separation, pairing fluctuations, and pseudogaps is carried out by combining the analytical eigenvalues of the four-site Hubbard clusters with the grand canonical and canonical ensemble approaches in a multidimensional parameter space of temperature (T)(T)(T), magnetic field (h)(h)(h), on-site interaction (U)(U)(U), and chemical potential $(\ensuremath{\mu})$. Our results, near the average number of electrons $⟨N⟩\ensuremath{\approx}3$, strongly suggest the existence of a critical parameter Uc(T){U}_{c}(T)Uc​(T) for the localization of electrons and a particle-hole binding (positive) gap ${\ensuremath{\Delta}}^{e\text{\ensuremath{-}}h}(T)>0$ at U>Uc(T)U>{U}_{c}(T)U>Uc​(T), with a zero temperature critical value, Uc(0)=4.584{U}_{c}(0)=4.584Uc​(0)=4.584. For U<Uc(T)U<{U}_{c}(T)U<Uc​(T), the particle-particle pair binding is found with a (positive) pairing gap ${\ensuremath{\Delta}}^{P}(T)>0$. The ground state degeneracy is lifted at U>Uc(T)U>{U}_{c}(T)U>Uc​(T) and the cluster becomes a Mott-Hubbard like insulator due to the presence of energy gaps at all (allowed) integer numbers $(1\ensuremath{\le}N\ensuremath{\le}8)$ of electrons. In contrast, for $U\ensuremath{\le}{U}_{c}(T)$, we find an electron pair binding instability at finite temperature near $⟨N⟩\ensuremath{\approx}3$, which manifests a possible pairing mechanism, a precursor to superconductivity in small clusters. Rigorous criteria for the existence of many-body Mott-Hubbard like particle-hole and particle-particle pairings, spin pairing, (spin) pseudogap, and (spin) antiferromagnetic critical crossover temperatures, at which the corresponding pseudogaps disappear, are also formulated. In particular, the resulting phase diagram consisting of charge and spin pseudogaps, antiferromagnetic correlations, hole pairing with competing hole-rich (⟨N⟩=2)(⟨N⟩=2)(⟨N⟩=2), hole-poor (⟨N⟩=4)(⟨N⟩=4)(⟨N⟩=4), and magnetic (⟨N⟩=3)(⟨N⟩=3)(⟨N⟩=3) regions in the ensemble of clusters near 1/8 filling closely resembles the phase diagrams and inhomogeneous phase separation recently found in the family of doped high-Tc{\mathrm{T}}_{\mathrm{c}}Tc​ cuprates.
We introduce a Hubbard model on a particular class of geometries, and consider the effect of doping the highly spin-degenerate Mott-insulating state with a microscopic number of holes in the extreme strong-coupling limit. The geometry is quite general, with pairs of atomic sites at each superlattice vertex, and a highly frustrated inter-atomic connectivity: the one-dimensional realization is a chain of edge-sharing tetrahedra. The sole model parameter is the ratio of intra-pair to inter-pair hopping matrix elements. If the intra-pair hopping is negligible then introducing a microscopic number of holes results in a ferromagnetic Nagaoka groundstate. Conversely, if the intra-pair hopping is comparable with the inter-pair hopping then the groundstate is low spin with short-ranged spin correlations. We exactly solve the correlated motion of a pair of holes in such a state and find that, in 1d and 2d, they form a bound pair on a length scale that increases with diminishing binding energy. This result is pertinent to the long-standing problem of hole motion in the CuO 2 planes of the high-temperature superconductors: we have rigorously shown that, on our frustrated geometry, the holes pair up and a short-ranged low-spin state is generated by hole motion alone.