We study the two-dimensional attractive Hubbard model using the mapping onto the half-filled repulsive Hubbard model in a uniform magnetic field coupled to the fermion spins. The low-energy effective action for charge and pairing fluctuations is obtained in the hydrodynamic regime. We recover the action of a Bose superfluid where half the fermion density is identified as the conjugate variable of the phase of the superconducting order parameter. By integrating out charge fluctuations, we obtain a phase-only action. In the zero-temperature superconducting state, this action describes a collective phase mode smoothly evolving from the Anderson-Bogoliubov mode at weak coupling to the Bogoliubov mode of a Bose superfluid at strong coupling. At finite temperature, the phase-only action can be used to extract an effective $XY$ model and thus obtain the Berezinskii-Kosterlitz-Thouless (BKT) phase transition temperature. We also identify a renormalized classical regime of superconducting fluctuations above the BKT phase transition, and a regime of incoherent pairs at higher temperature. Special care is devoted to the nearly half-filled case where the symmetry of the order parameter is enlarged to SO(3) due to strong $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$ charge fluctuations. The low-energy effective action is then an SO(3) nonlinear $\ensuremath{\sigma}$ model with a (symmetry breaking) magnetic field proportional to the doping. In the strong-coupling limit, the attractive Hubbard model can be mapped onto the Heisenberg model in a magnetic field, which reduces to the quantum $XY$ model (except for a weak magnetic field, i.e., in the low-density limit of the attractive model). In the low-density limit, the Heisenberg model allows one to recover the action of a Bose superfluid, including the ${(\mathbf{\ensuremath{\nabla}}\ensuremath{\rho})}^{2}$ term (with $\ensuremath{\rho}$ the density), and in turn the Gross-Pitaevskii equation.We study the two-dimensional attractive Hubbard model using the mapping onto the half-filled repulsive Hubbard model in a uniform magnetic field coupled to the fermion spins. The low-energy effective action for charge and pairing fluctuations is obtained in the hydrodynamic regime. We recover the action of a Bose superfluid where half the fermion density is identified as the conjugate variable of the phase of the superconducting order parameter. By integrating out charge fluctuations, we obtain a phase-only action. In the zero-temperature superconducting state, this action describes a collective phase mode smoothly evolving from the Anderson-Bogoliubov mode at weak coupling to the Bogoliubov mode of a Bose superfluid at strong coupling. At finite temperature, the phase-only action can be used to extract an effective $XY$ model and thus obtain the Berezinskii-Kosterlitz-Thouless (BKT) phase transition temperature. We also identify a renormalized classical regime of superconducting fluctuations above the BKT phase transition, and a regime of incoherent pairs at higher temperature. Special care is devoted to the nearly half-filled case where the symmetry of the order parameter is enlarged to SO(3) due to strong $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$ charge fluctuations. The low-energy effective action is then an SO(3) nonlinear $\ensuremath{\sigma}$ model with a (symmetry breaking) magnetic field proportional to the doping. In the strong-coupling limit, the attractive Hubbard model can be mapped onto the Heisenberg model in a magnetic field, which reduces to the quantum $XY$ model (except for a weak magnetic field, i.e., in the low-density limit of the attractive model). In the low-density limit, the Heisenberg model allows one to recover the action of a Bose superfluid, including the ${(\mathbf{\ensuremath{\nabla}}\ensuremath{\rho})}^{2}$ term (with $\ensuremath{\rho}$ the density), and in turn the Gross-Pitaevskii equation.

arXiv:cond-mat/0311374Physical Review BOct 13, 200425 citations

Strong Coupling Limit of the Holstein-Hubbard Model

Zhaoyu Han, Steven Allan Kivelson, Hong Yao

We analyze the quantum phase diagram of the Holstein-Hubbard model using an asymptotically exact strong coupling expansion. We find all sorts of interesting phases including a pair-density wave, a charge 4e (and even a charge 6e) superconductor, regimes of phase separation, and a variety of distinct charge-density-wave, spin-density-wave, and superconducting regimes. We chart the crossovers that occur as a function of the degree of retardation, i.e., the ratio of characteristic phonon frequency to the strength of interactions.We analyze the quantum phase diagram of the Holstein-Hubbard model using an asymptotically exact strong coupling expansion. We find all sorts of interesting phases including a pair-density wave, a charge 4e (and even a charge 6e) superconductor, regimes of phase separation, and a variety of distinct charge-density-wave, spin-density-wave, and superconducting regimes. We chart the crossovers that occur as a function of the degree of retardation, i.e., the ratio of characteristic phonon frequency to the strength of interactions.

arXiv:2006.16253Physical Review LettersOct 13, 202053 citations

High correlation regime of the Hubbard model

Miguel Lagos, Miguel Kiwi, R. Ramirez

Solid State CommunicationsAug 1, 19880 citations

Magnetic susceptibility and the Landau Fermi-liquid parameter of the Hubbard model and the three-band CuO2 model

Hiroyuki Kaga, Taiichirou Saikawa, Chihiro Ohmori

Magnetic susceptibilities of interacting (d) electrons in the Hubbard model and the three-band ${\mathrm{CuO}}_{2}$ model are worked out within the Gaussian fluctuations of the functional integral in the slave boson representation. The normalized Landau Fermi-liquid parameter | ${\mathit{F}}_{0}^{\mathit{a}}$ |/\ensuremath{\alpha} (\ensuremath{\alpha}\ensuremath{\equiv} ${\mathit{W}}_{0}$ ${\mathit{N}}_{\mathit{F}}^{0}$ , ${\mathit{N}}_{\mathit{F}}^{0}$ the density of states at the Fermi level, and ${\mathit{W}}_{0}$ the bare bandwidth) defined from the susceptibility expression is investigated for the two models as functions of Coulomb repulsion U, charge-transfer energy \ensuremath{\Delta}, and doping concentration \ensuremath{\delta}. It is shown that the Landau parameter for the three-band model is larger and exhibits different renormalization behaviors than that for the Hubbard model in both the weak- and strong-coupling regimes except in the Mott-Hubbard weak-coupling regime, where it takes the same random-phase approximation form U/ ${\mathit{W}}_{0}$ as the latter. The reasons for these results are analyzed. The slave-boson Landau parameter for the Hubbard model exactly reproduces the T-matrix expression | ${\mathit{F}}_{0}^{\mathit{a}}$ |/\ensuremath{\alpha}=U/ ${\mathit{W}}_{0}$ /(1+U/ ${\mathit{W}}_{0}$ ) in the low-density limit, whose renormalization has the different origin from that in the dilutely doped regime. \textcopyright{} 1996 The American Physical Society.Magnetic susceptibilities of interacting (d) electrons in the Hubbard model and the three-band ${\mathrm{CuO}}_{2}$ model are worked out within the Gaussian fluctuations of the functional integral in the slave boson representation. The normalized Landau Fermi-liquid parameter |${\mathit{F}}_{0}^{\mathit{a}}$|/\ensuremath{\alpha} (\ensuremath{\alpha}\ensuremath{\equiv}${\mathit{W}}_{0}$${\mathit{N}}_{\mathit{F}}^{0}$, ${\mathit{N}}_{\mathit{F}}^{0}$ the density of states at the Fermi level, and ${\mathit{W}}_{0}$ the bare bandwidth) defined from the susceptibility expression is investigated for the two models as functions of Coulomb repulsion U, charge-transfer energy \ensuremath{\Delta}, and doping concentration \ensuremath{\delta}. It is shown that the Landau parameter for the three-band model is larger and exhibits different renormalization behaviors than that for the Hubbard model in both the weak- and strong-coupling regimes except in the Mott-Hubbard weak-coupling regime, where it takes the same random-phase approximation form U/${\mathit{W}}_{0}$ as the latter. The reasons for these results are analyzed. The slave-boson Landau parameter for the Hubbard model exactly reproduces the T-matrix expression |${\mathit{F}}_{0}^{\mathit{a}}$|/\ensuremath{\alpha}=U/${\mathit{W}}_{0}$/(1+U/${\mathit{W}}_{0}$) in the low-density limit, whose renormalization has the different origin from that in the dilutely doped regime. \textcopyright{} 1996 The American Physical Society.

Physical review. B, Condensed matterJul 1, 19961 citation

XY

model and thus obtain the Berezinskii-Kosterlitz-Thouless (BKT) phase transition temperature. We also identify a renormalized classical regime of superconducting fluctuations above the BKT phase transition, and a regime of incoherent pairs at higher temperature. Special care is devoted to the nearly half-filled case where the symmetry of the order parameter is enlarged to SO(3) due to strong $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$ charge fluctuations. The low-energy effective action is then an SO(3) nonlinear $\ensuremath{\sigma}$ model with a (symmetry breaking) magnetic field proportional to the doping. In the strong-coupling limit, the attractive Hubbard model can be mapped onto the Heisenberg model in a magnetic field, which reduces to the quantum

XY

model (except for a weak magnetic field, i.e., in the low-density limit of the attractive model). In the low-density limit, the Heisenberg model allows one to recover the action of a Bose superfluid, including the ${(\mathbf{\ensuremath{\nabla}}\ensuremath{\rho})}^{2}$ term (with $\ensuremath{\rho}$ the density), and in turn the Gross-Pitaevskii equation.

Magnetic susceptibilities of interacting (d) electrons in the Hubbard model and the three-band

{\mathrm{CuO}}_{2}

model are worked out within the Gaussian fluctuations of the functional integral in the slave boson representation. The normalized Landau Fermi-liquid parameter |

{\mathit{F}}_{0}^{\mathit{a}}

|/\ensuremath{\alpha} (\ensuremath{\alpha}\ensuremath{\equiv}

{\mathit{W}}_{0}

{\mathit{N}}_{\mathit{F}}^{0}

{\mathit{N}}_{\mathit{F}}^{0}

the density of states at the Fermi level, and

{\mathit{W}}_{0}

the bare bandwidth) defined from the susceptibility expression is investigated for the two models as functions of Coulomb repulsion U, charge-transfer energy \ensuremath{\Delta}, and doping concentration \ensuremath{\delta}. It is shown that the Landau parameter for the three-band model is larger and exhibits different renormalization behaviors than that for the Hubbard model in both the weak- and strong-coupling regimes except in the Mott-Hubbard weak-coupling regime, where it takes the same random-phase approximation form U/

{\mathit{W}}_{0}

as the latter. The reasons for these results are analyzed. The slave-boson Landau parameter for the Hubbard model exactly reproduces the T-matrix expression |

{\mathit{F}}_{0}^{\mathit{a}}

|/\ensuremath{\alpha}=U/

{\mathit{W}}_{0}

/(1+U/

{\mathit{W}}_{0}

) in the low-density limit, whose renormalization has the different origin from that in the dilutely doped regime. \textcopyright{} 1996 The American Physical Society.