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:cond-mat/0601113 · Physical Review B

Superconductivity in inhomogeneous Hubbard models

Wei-Feng Tsai, Steven Allan Kivelson

Published Jun 9, 2006• 62 citations
arXivHTMLPDF

Abstract

We present a controlled perturbative approach to the low-temperature phase diagram of highly inhomogeneous Hubbard models in the limit of small coupling, ${t}^{\ensuremath{'}}$, between clusters. We apply this to the dimerized and checkerboard models. The dimerized model is found to behave like a doped semiconductor, with a Fermi-liquid groundstate with parameters (e.g., the effective mass) which are smooth functions of the Hubbard interaction, $U$. By contrast, the checkerboard model has a Fermi liquid phase at large $U>{U}_{c}=4.58$, a $d$-wave superconducting state with a full gap for ${U}_{c}>U>0$, and a narrow strip of an intermediate $d$-wave superconducting phase with gapless ``nodal'' quasiparticles for $\ensuremath{\mid}U\ensuremath{-}{U}_{c}\ensuremath{\mid}<\mathcal{O}({t}^{\ensuremath{'}})$.

DOI
10.1103/physrevb.73.214510

Related papers

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

Myriad phases of the checkerboard Hubbard model

Hong Yao, Wei-Feng Tsai, Steven Allan Kivelson

The zero-temperature phase diagram of the checkerboard Hubbard model is obtained in the solvable limit in which it consists of weakly coupled square plaquettes. As a function of the on-site Coulomb repulsion $U$ and the density of holes per site, $x$, we demonstrate the existence of at least 16 distinct phases. For instance, at zero doping, the ground state is a novel $d$-wave Mott insulator ($d$-Mott), which is not adiabatically continuable to a band insulator; by doping the $d$-Mott state with holes, depending on the magnitude of $U$, it gives way to a $d$-wave superconducting state, a two-flavor spin-$1∕2$ Fermi liquid (FL), or a spin-$3∕2$ FL.

arXiv:0706.0761•Physical Review B•Oct 24, 2007•52 citations

Fidelity study of the superconducting phase diagram in the two-dimensional single-band Hubbard model

Chunjing Jia, Brian Moritz, Cheng-Chien Chen +2 more

Extensive numerical studies have demonstrated that the two-dimensional single-band Hubbard model contains much of the key physics in cuprate high-temperature superconductors. However, there is no definitive proof that the Hubbard model truly possesses a superconducting ground state or, if it does, of how it depends on model parameters. To answer these longstanding questions, we study an extension of the Hubbard model including an infinite-range $d$-wave pair field term, which precipitates a superconducting state in the $d$-wave channel. Using exact diagonalization on 16-site square clusters, we study the evolution of the ground state as a function of the strength of the pairing term. This is achieved by monitoring the fidelity metric of the ground state, as well as determining the ratio between the two largest eigenvalues of the $d$-wave pair/spin/charge-density matrices. The calculations show a $d$-wave superconducting ground state in doped clusters bracketed by a strong antiferromagnetic state at half filling controlled by the Coulomb repulsion $U$ and a weak short-range checkerboard charge ordered state at larger hole doping controlled by the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$. We also demonstrate that negative ${t}^{\ensuremath{'}}$ plays an important role in facilitating $d$-wave superconductivity.

arXiv:1012.4013•Physical Review B•Sep 9, 2011•24 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

d -Wave Superconductivity in the Hubbard Model

Thomas Maier, Mark Jarrell, Thomas Pruschke +1 more

The superconducting instabilities of the doped repulsive 2D Hubbard model are studied in the intermediate to strong coupling regime with the help of the dynamical cluster approximation. To solve the effective cluster problem we employ an extended noncrossing approximation, which allows for a transition to the broken symmetry state. At sufficiently low temperatures we find stable d-wave solutions with off-diagonal long-range order. The maximal T(c) approximately 150 K occurs for a doping delta approximately 20% and the doping dependence of the transition temperatures agrees well with the generic high- T(c) phase diagram.

arXiv:cond-mat/0002352•Physical Review Letters•Aug 14, 2000•156 citations

Suppression of d -wave superconductivity in the checkerboard Hubbard model

D. G. S. P. Doluweera, Alexandru Macridin, Thomas Maier +2 more

Using a dynamical cluster quantum Monte Carlo approximation, we investigate the $d$-wave superconducting transition temperature ${T}_{c}$ in the doped two-dimensional repulsive Hubbard model with a weak inhomogeneity. The inhomogeneity is introduced in the hoppings ${t}^{\ensuremath{'}}$ and $t$ in the form of a checkerboard pattern where $t$ 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 ${T}_{c}$. The characteristic spin excitation energy and the strength of $d$-wave pairing interaction decrease with decreasing ${T}_{c}$, suggesting a strong correlation between these quantities.

arXiv:0806.1504•Physical Review B•Jul 9, 2008•19 citations
We present a controlled perturbative approach to the low-temperature phase diagram of highly inhomogeneous Hubbard models in the limit of small coupling, t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′, between clusters. We apply this to the dimerized and checkerboard models. The dimerized model is found to behave like a doped semiconductor, with a Fermi-liquid groundstate with parameters (e.g., the effective mass) which are smooth functions of the Hubbard interaction, UUU. By contrast, the checkerboard model has a Fermi liquid phase at large U>Uc=4.58U>{U}_{c}=4.58U>Uc​=4.58, a ddd-wave superconducting state with a full gap for Uc>U>0{U}_{c}>U>0Uc​>U>0, and a narrow strip of an intermediate ddd-wave superconducting phase with gapless ``nodal'' quasiparticles for \ensuremath∣U\ensuremath−Uc\ensuremath∣<O(t\ensuremath′)\ensuremath{\mid}U\ensuremath{-}{U}_{c}\ensuremath{\mid}<\mathcal{O}({t}^{\ensuremath{'}})\ensuremath∣U\ensuremath−Uc​\ensuremath∣<O(t\ensuremath′).
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 zero-temperature phase diagram of the checkerboard Hubbard model is obtained in the solvable limit in which it consists of weakly coupled square plaquettes. As a function of the on-site Coulomb repulsion UUU and the density of holes per site, xxx, we demonstrate the existence of at least 16 distinct phases. For instance, at zero doping, the ground state is a novel ddd-wave Mott insulator (ddd-Mott), which is not adiabatically continuable to a band insulator; by doping the ddd-Mott state with holes, depending on the magnitude of UUU, it gives way to a ddd-wave superconducting state, a two-flavor spin-1∕21∕21∕2 Fermi liquid (FL), or a spin-3∕23∕23∕2 FL.
Extensive numerical studies have demonstrated that the two-dimensional single-band Hubbard model contains much of the key physics in cuprate high-temperature superconductors. However, there is no definitive proof that the Hubbard model truly possesses a superconducting ground state or, if it does, of how it depends on model parameters. To answer these longstanding questions, we study an extension of the Hubbard model including an infinite-range ddd-wave pair field term, which precipitates a superconducting state in the ddd-wave channel. Using exact diagonalization on 16-site square clusters, we study the evolution of the ground state as a function of the strength of the pairing term. This is achieved by monitoring the fidelity metric of the ground state, as well as determining the ratio between the two largest eigenvalues of the ddd-wave pair/spin/charge-density matrices. The calculations show a ddd-wave superconducting ground state in doped clusters bracketed by a strong antiferromagnetic state at half filling controlled by the Coulomb repulsion UUU and a weak short-range checkerboard charge ordered state at larger hole doping controlled by the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$. We also demonstrate that negative ${t}^{\ensuremath{'}}$ plays an important role in facilitating ddd-wave superconductivity.
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.
The superconducting instabilities of the doped repulsive 2D Hubbard model are studied in the intermediate to strong coupling regime with the help of the dynamical cluster approximation. To solve the effective cluster problem we employ an extended noncrossing approximation, which allows for a transition to the broken symmetry state. At sufficiently low temperatures we find stable d-wave solutions with off-diagonal long-range order. The maximal T(c) approximately 150 K occurs for a doping delta approximately 20% and the doping dependence of the transition temperatures agrees well with the generic high- T(c) phase diagram.
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.