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:0706.0761 · Physical Review B

Myriad phases of the checkerboard Hubbard model

Hong Yao, Wei-Feng Tsai, Steven Allan Kivelson

Published Oct 24, 2007• 52 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.76.161104

Related papers

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

Superconductivity in inhomogeneous Hubbard models

Wei-Feng Tsai, Steven Allan Kivelson

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 $\ensuremath{\mid}U\ensuremath{-}{U}_{c}\ensuremath{\mid}<\mathcal{O}({t}^{\ensuremath{'}})$.

arXiv:cond-mat/0601113•Physical Review B•Jun 9, 2006•62 citations

η -pairing superconductivity in the Hubbard chain with pair hopping

G. I. Japaridze, A. P. Kampf, Michael Sekania +2 more

The ground-state phase diagram of the one-dimensional Hubbard chain with pair-hopping interaction is studied. The analysis of the model is performed using the continuum-limit field theory approach and exact diagonalization studies. At half-filling the phase diagram is shown to consist of two superconducting states with Cooper-pair center-of-mass momentum $Q=0 (\mathrm{BCS}\ensuremath{-}{\ensuremath{\eta}}_{0}$ phase) and $Q=\ensuremath{\pi} ({\ensuremath{\eta}}_{\ensuremath{\pi}}$ phase) and four insulating phases corresponding to the Mott antiferromagnet, the Peierls dimerized phase, the charge-density-wave (CDW) insulator, and an unconventional insulating phase characterized by the coexistence of a CDW and a bond-located staggered magnetization. Away from half-filling the phase diagram consists of the superconducting $\mathrm{BCS}\ensuremath{-}{\ensuremath{\eta}}_{0}$ and ${\ensuremath{\eta}}_{\ensuremath{\pi}}$ phases and the metallic Luttinger-liquid phase. The $\mathrm{BCS}\ensuremath{-}{\ensuremath{\eta}}_{0}$ phase exhibits a smooth crossover from a weak-coupling BCS type to a strong-coupling local-pair regime. The ${\ensuremath{\eta}}_{\ensuremath{\pi}}$ phase shows the properties of the doublon (zero-size Cooper-pair) superconductor with Cooper-pair center-of-mass momentum $Q=\ensuremath{\pi}.$ The transition into the ${\ensuremath{\eta}}_{\ensuremath{\pi}}$-paired state corresponds to an abrupt change in the ground-state structure. After the transition the conduction band is completely destroyed and a new ${\ensuremath{\eta}}_{\ensuremath{\pi}}$-pair band, corresponding to the strongly correlated doublon motion, is created.

arXiv:cond-mat/0006401•Physical review. B, Condensed matter•Dec 13, 2001•64 citations

Ground-state phase diagram of the one-dimensional half-filled extended Hubbard model

Masahisa Tsuchiizu, Akira Furusaki

We revisit the ground-state phase diagram of the one-dimensional half-filled extended Hubbard model with on-site (U) and nearest-neighbor (V) repulsive interactions. In the first half of the paper, using the weak-coupling renormalization-group approach $(g$-ology) including second-order corrections to the coupling constants, we show that bond-charge-density-wave (BCDW) phase exists for $U\ensuremath{\approx}2V$ in between charge-density-wave (CDW) and spin-density-wave (SDW) phases. We find that the umklapp scattering of parallel-spin electrons disfavors the BCDW state and leads to a bicritical point where the CDW-BCDW and SDW-BCDW continuous-transition lines merge into the CDW-SDW first-order transition line. In the second half of the paper, we investigate the phase diagram of the extended Hubbard model with either additional staggered site potential $\ensuremath{\Delta}$ or bond alternation $\ensuremath{\delta}.$ Although the alternating site potential $\ensuremath{\Delta}$ strongly favors the CDW state (that is, a band insulator), the BCDW state is not destroyed completely and occupies a finite region in the phase diagram. Our result is a natural generalization of the work by Fabrizio, Gogolin, and Nersesyan [Phys. Rev. Lett. $83,$ 2014 (1999)], who predicted the existence of a spontaneously dimerized insulating state between a band insulator and a Mott insulator in the phase diagram of the ionic Hubbard model. The bond alternation $\ensuremath{\delta}$ destroys the SDW state and changes it into the BCDW state (or Peierls insulating state). As a result the phase diagram of the model with $\ensuremath{\delta}$ contains only a single critical line separating the Peierls insulator phase and the CDW phase. The addition of $\ensuremath{\Delta}$ or $\ensuremath{\delta}$ changes the universality class of the CDW-BCDW transition from the Gaussian transition into the Ising transition.

arXiv:cond-mat/0308157•Physical Review B•Jan 15, 2004•78 citations

Phase diagram of the asymmetric Hubbard model

Pavol Farkašovský

The ground-state phase diagram of the asymmetric Hubbard model is studied in one and two dimensions by a well-controlled numerical method. The method allows us to calculate directly the probabilities of particular phases in the approximate ground state and thus to specify the stability domains corresponding to phases with the highest probabilities. Depending on the electron filling $n$ and the magnitude of the asymmetry ${t}_{f}∕{t}_{d}$ between the hopping integrals of $f$ and $d$ electrons two different scenarios in formation of ground states are observed. At low electron fillings $(n\ensuremath{\leqslant}1∕3)$ the ground states are always phase segregated in the limit of strong asymmetry $({t}_{d}⪢{t}_{f})$. With decreasing asymmetry the system undergoes a transition to the phase-separated state and then to the homogeneous state. For electron fillings $n>1∕3$ and weak Coulomb interactions the ground state is homogeneous for all values of asymmetry, while for intermediate and strong interactions the system exhibits the same sequence of phase transitions as for $n$ small. Moreover, it is shown that the segregated phase is significantly stabilized with increasing electron filling, while the separated phases disappear gradually from the ground-state phase diagrams.

arXiv:0805.3988•Physical Review B•Feb 12, 2008•21 citations

Anomalous superconductivity and its competition with antiferromagnetism in doped Mott insulators

S. S. Kancharla, Bumsoo Kyung, David Sénéchal +4 more

Proximity to a Mott insulating phase is likely to be an important physical ingredient of a theory that aims to describe high-temperature superconductivity in the cuprates. Quantum cluster methods are well suited to describe the Mott phase. Hence, as a step toward a quantitative theory of the competition between antiferromagnetism and $d$-wave superconductivity in the cuprates, we use cellular dynamical mean-field theory to compute zero-temperature properties of the two-dimensional square lattice Hubbard model. The $d$-wave order parameter is found to scale like the superexchange coupling $J$ for on-site interaction $U$ comparable to or larger than the bandwidth. The order parameter also assumes a dome shape as a function of doping, while, by contrast, the gap in the single-particle density of states decreases monotonically with increasing doping. In the presence of a finite second neighbor hopping ${t}^{\ensuremath{'}}$, the zero-temperature phase diagram displays the electron-hole asymmetric competition between antiferromagnetism and superconductivity that is observed experimentally in the cuprates. Adding realistic third neighbor hopping ${t}^{\ensuremath{''}}$ improves the overall agreement with the experimental phase diagram. Since band parameters can vary depending on the specific cuprate considered, the sensitivity of the theoretical phase diagram to band parameters challenges the commonly held assumption that the doping vs ${T}_{c}∕{T}_{c}^{\mathit{max}}$ phase diagram of the cuprates is universal. The calculated angle-resolved photoemission spectrum displays the observed electron-hole asymmetry. The tendency to homogeneous coexistence of the superconducting and antiferromagnetic order parameters is stronger than observed in most experiments but consistent with many theoretical results and with experiments in some layered high-temperature superconductors. Clearly, our calculations reproduce important features of $d$-wave superconductivity in the cuprates that would otherwise be considered anomalous from the point of view of the standard Bardeen--Cooper--Schrieffer approach. At strong coupling, $d$-wave superconductivity and antiferromagnetism naturally appear as two equally important competing instabilities of the normal phase of the same underlying Hamiltonian.

arXiv:cond-mat/0508205•Physical Review B•May 23, 2008•184 citations
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.
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, 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 $\ensuremath{\mid}U\ensuremath{-}{U}_{c}\ensuremath{\mid}<\mathcal{O}({t}^{\ensuremath{'}})$.
The ground-state phase diagram of the one-dimensional Hubbard chain with pair-hopping interaction is studied. The analysis of the model is performed using the continuum-limit field theory approach and exact diagonalization studies. At half-filling the phase diagram is shown to consist of two superconducting states with Cooper-pair center-of-mass momentum $Q=0 (\mathrm{BCS}\ensuremath{-}{\ensuremath{\eta}}_{0}$ phase) and $Q=\ensuremath{\pi} ({\ensuremath{\eta}}_{\ensuremath{\pi}}$ phase) and four insulating phases corresponding to the Mott antiferromagnet, the Peierls dimerized phase, the charge-density-wave (CDW) insulator, and an unconventional insulating phase characterized by the coexistence of a CDW and a bond-located staggered magnetization. Away from half-filling the phase diagram consists of the superconducting $\mathrm{BCS}\ensuremath{-}{\ensuremath{\eta}}_{0}$ and ${\ensuremath{\eta}}_{\ensuremath{\pi}}$ phases and the metallic Luttinger-liquid phase. The $\mathrm{BCS}\ensuremath{-}{\ensuremath{\eta}}_{0}$ phase exhibits a smooth crossover from a weak-coupling BCS type to a strong-coupling local-pair regime. The ${\ensuremath{\eta}}_{\ensuremath{\pi}}$ phase shows the properties of the doublon (zero-size Cooper-pair) superconductor with Cooper-pair center-of-mass momentum $Q=\ensuremath{\pi}.$ The transition into the ${\ensuremath{\eta}}_{\ensuremath{\pi}}$-paired state corresponds to an abrupt change in the ground-state structure. After the transition the conduction band is completely destroyed and a new ${\ensuremath{\eta}}_{\ensuremath{\pi}}$-pair band, corresponding to the strongly correlated doublon motion, is created.
We revisit the ground-state phase diagram of the one-dimensional half-filled extended Hubbard model with on-site (U) and nearest-neighbor (V) repulsive interactions. In the first half of the paper, using the weak-coupling renormalization-group approach (g(g(g-ology) including second-order corrections to the coupling constants, we show that bond-charge-density-wave (BCDW) phase exists for $U\ensuremath{\approx}2V$ in between charge-density-wave (CDW) and spin-density-wave (SDW) phases. We find that the umklapp scattering of parallel-spin electrons disfavors the BCDW state and leads to a bicritical point where the CDW-BCDW and SDW-BCDW continuous-transition lines merge into the CDW-SDW first-order transition line. In the second half of the paper, we investigate the phase diagram of the extended Hubbard model with either additional staggered site potential $\ensuremath{\Delta}$ or bond alternation $\ensuremath{\delta}.$ Although the alternating site potential $\ensuremath{\Delta}$ strongly favors the CDW state (that is, a band insulator), the BCDW state is not destroyed completely and occupies a finite region in the phase diagram. Our result is a natural generalization of the work by Fabrizio, Gogolin, and Nersesyan [Phys. Rev. Lett. 83,83,83, 2014 (1999)], who predicted the existence of a spontaneously dimerized insulating state between a band insulator and a Mott insulator in the phase diagram of the ionic Hubbard model. The bond alternation $\ensuremath{\delta}$ destroys the SDW state and changes it into the BCDW state (or Peierls insulating state). As a result the phase diagram of the model with $\ensuremath{\delta}$ contains only a single critical line separating the Peierls insulator phase and the CDW phase. The addition of $\ensuremath{\Delta}$ or $\ensuremath{\delta}$ changes the universality class of the CDW-BCDW transition from the Gaussian transition into the Ising transition.
The ground-state phase diagram of the asymmetric Hubbard model is studied in one and two dimensions by a well-controlled numerical method. The method allows us to calculate directly the probabilities of particular phases in the approximate ground state and thus to specify the stability domains corresponding to phases with the highest probabilities. Depending on the electron filling nnn and the magnitude of the asymmetry tf∕td{t}_{f}∕{t}_{d}tf​∕td​ between the hopping integrals of fff and ddd electrons two different scenarios in formation of ground states are observed. At low electron fillings $(n\ensuremath{\leqslant}1∕3)$ the ground states are always phase segregated in the limit of strong asymmetry (td⪢tf)({t}_{d}⪢{t}_{f})(td​⪢tf​). With decreasing asymmetry the system undergoes a transition to the phase-separated state and then to the homogeneous state. For electron fillings n>1∕3n>1∕3n>1∕3 and weak Coulomb interactions the ground state is homogeneous for all values of asymmetry, while for intermediate and strong interactions the system exhibits the same sequence of phase transitions as for nnn small. Moreover, it is shown that the segregated phase is significantly stabilized with increasing electron filling, while the separated phases disappear gradually from the ground-state phase diagrams.
Proximity to a Mott insulating phase is likely to be an important physical ingredient of a theory that aims to describe high-temperature superconductivity in the cuprates. Quantum cluster methods are well suited to describe the Mott phase. Hence, as a step toward a quantitative theory of the competition between antiferromagnetism and ddd-wave superconductivity in the cuprates, we use cellular dynamical mean-field theory to compute zero-temperature properties of the two-dimensional square lattice Hubbard model. The ddd-wave order parameter is found to scale like the superexchange coupling JJJ for on-site interaction UUU comparable to or larger than the bandwidth. The order parameter also assumes a dome shape as a function of doping, while, by contrast, the gap in the single-particle density of states decreases monotonically with increasing doping. In the presence of a finite second neighbor hopping ${t}^{\ensuremath{'}}$, the zero-temperature phase diagram displays the electron-hole asymmetric competition between antiferromagnetism and superconductivity that is observed experimentally in the cuprates. Adding realistic third neighbor hopping ${t}^{\ensuremath{''}}$ improves the overall agreement with the experimental phase diagram. Since band parameters can vary depending on the specific cuprate considered, the sensitivity of the theoretical phase diagram to band parameters challenges the commonly held assumption that the doping vs Tc∕Tcmax{T}_{c}∕{T}_{c}^{\mathit{max}}Tc​∕Tcmax​ phase diagram of the cuprates is universal. The calculated angle-resolved photoemission spectrum displays the observed electron-hole asymmetry. The tendency to homogeneous coexistence of the superconducting and antiferromagnetic order parameters is stronger than observed in most experiments but consistent with many theoretical results and with experiments in some layered high-temperature superconductors. Clearly, our calculations reproduce important features of ddd-wave superconductivity in the cuprates that would otherwise be considered anomalous from the point of view of the standard Bardeen--Cooper--Schrieffer approach. At strong coupling, ddd-wave superconductivity and antiferromagnetism naturally appear as two equally important competing instabilities of the normal phase of the same underlying Hamiltonian.