In a 1/S loop expansion of the two-dimensional Hubbard model, doping leads to incommensurate twist spin-density-wave (SDW) states at the saddle point. Further doping reduces the on-site magnetization and increases the SDW pitch, conspiring to close the effective Mott-Hubbard gap at \ensuremath{\delta}\ensuremath{\sim}0.07--0.30 for U/t\ensuremath{\sim}2.5--10. At the one-loop level, the two massless fluctuations map to the nonlinear \ensuremath{\sigma} model at zero doping while for the twist SDW states, three massless spin-wave modes arise which are strongly coupled to the charge fluctuations, therefore implying nontrivial charge-spin response functions for the metal.
Using a path-integral formalism and decomposing the Hubbard interaction into charge and spin order parameters, we develop a 1/S loop expansion of the two-dimensional Hubbard model. At the saddle-point level, we present a detailed study of the energetics of twisted antiferromagnetic order. Doping the antiferromagnetic state leads to a rich mean-field phase diagram at zero temperature, which includes spiral phases, a column phase, and ferromagnetism. The on-site magnetization, the density of states, the plasma frequency, and the quasiparticle weights are evaluated for the twisted spin-density-wave (SDW) states as a function of doping and U/t. Doping-induced frustration remarkably leads to a closure of the Mott-Hubbard band gap for any U/t. At one-loop order, we delineate the collective spin and charge fluctuations of the mean-field SDW state. The action for the two transverse spin modes at half-filling maps to a nonlinear \ensuremath{\sigma} model, which tends to disorder upon doping. Twisted magnetic ground states exhibit a third Goldstone mode due to the complete breaking of rotational invariance. For the commensurate SDW, the charge fluctuations couple only to longitudinal spin fluctuations, whereas, for the incommensurate metallic SDW states, the charge modes are coupled nontrivially to the massless spin waves. This suggests a metallic state quite unlike traditional Fermi liquids.
Determining the properties of the two-dimensional Hubbard model is an outstanding problem in physics. Applying recent advances in constrained path auxiliary-field quantum Monte Carlo techniques and simulating large rectangular supercells, we characterize the magnetic and charge properties in the ground state as a function of doping. At intermediate interaction strengths, an incommensurate spin density wave (SDW) state is found, with antiferromagnetic order and essentially homogeneous charge correlation. The wavelength of the collective mode decreases with doping, as does its magnitude. The SDW order vanishes beyond a critical doping. As the interaction is increased, the holes go from a wavelike (delocalized) to a particlelike (localized) state, and charge ordering develops which eventually evolves into stripelike states.
We study the Hubbard model on the square lattice coupled in addition to the optical Su-Schrieffer-Heeger (SSH) phonons, using the singular-mode functional renormalization group method. At half filling and in the absence of the Hubbard interaction
U, we find the degenerate spin-density-wave--charge-density-wave--
s-wave-superconductivity state at smaller electron-phonon coupling strength $\ensuremath{\lambda}$ and higher phonon frequency $\ensuremath{\omega}$, and the valence bond solid (VBS) state at larger $\ensuremath{\lambda}$ and lower $\ensuremath{\omega}$. After switching on a positive
U, the VBS state is suppressed, while the spin-density wave (SDW) state is enhanced. At finite doping, the SSH phonon is found to favor
s-wave superconductivity (sSC) at
U=0. With increasing positive
U, we find
d-wave superconductivity (dSC) and incommensurate SDW states. In a narrow window of moderate
U and $\ensuremath{\lambda}$, we also find the incommensurate VBS state. The sSC and dSC here can be naturally related to the charge-density wave (CDW) and SDW fluctuations, both of which can be triggered by the SSH phonons. In contrast, the repulsive interaction
U enhances SDW but suppresses CDW fluctuations. Our results at half filling are consistent with quantum Monte Carlo (QMC) and provide insights at finite doping where QMC may suffer from the minus sign problem.
We study the two-dimensional one-band Hubbard model on the square lattice through the spin rotation invariant Kotliar-Ruckenstein slave-boson method, with an emphasis on the electron-doped regime. The phase diagram determined within mean field approximation reveals three different magnetically ordered phases separated by first-order transitions, and a wide instability region as indicated by a negative compressibility. We show that this conundrum of the unstable mean field domain is resolved by spatially mixed phases of neighboring stable states. We also find charge density wave type instabilities in the antiferromagnetically ordered (AFM) state at wave vectors commensurate with the doping. The corresponding charge and spin density response functions are calculated from a Gaussian fluctuations formalism about a magnetic saddle point, for which we provide a detailed derivation. Further, we present density and spin excitation spectra in the AFM phase, allowing for a quantitative calculation of the spin wave parameters, the gap in the longitudinal spin excitation spectrum, and the density excitations in the upper Hubbard band.
A path-integral field-theoretic derivation of electromagnetic linear response for the two-dimensional Hubbard model is given. We find, remarkably, that changes in the Fermi-surface topology associated with incommensurate planar spin-density-wave saddle points induce a change in sign of the Hall coefficient at dopings ${\mathrm{\ensuremath{\delta}}}_{\mathit{H}}$=0.02--0.5 for U/t=2--10. The change in sign is not affected by short-range magnetic domains. We delinate from first principles an anomalous temperature dependence of the Hall carrier density at dopings close to ${\mathrm{\ensuremath{\delta}}}_{\mathit{H}}$. An additional anisotropic component to the usual dc conductivity is nonvanishing for certain types of spirals. The paper extends the Bloch-Boltzmann theory to the case of untraditional Fermi liquids where the damping of the quasiparticles is \ensuremath{\Gamma}(\ensuremath{\epsilon})\ensuremath{\sim}max(
kBT,\ensuremath{\epsilon}).
We have applied mean-field theory to calculate the spin-density-wave (SDW) order parameter \ensuremath{\Delta}, the chemical potential \ensuremath{\mu}, and the ground-state energy at various band fillings for the Hubbard model. We have found that at zero temperature and in the large-U limit, the SDW order parameter and the ground-state energy can be written as \ensuremath{\Delta}=U(1-x)/2-O(J) and
Eg=-U(1-x
)2/4-0(J). The chemical potential obtained here for x=0 and 1 agrees well with the result obtained by the exact-diagonalization calculation on a 4\ifmmode\times\else\texttimes\fi{}4 sites cluster by Dagotto et al. [Phys. Rev. B 45, 10 741 (1992)]. In the SDW phase, the mean-field theory gives an upward shift in the chemical potential as one dopes away from half-filling, which is in contrast with the exact-diagonalization calculations. This result indicates that the Mott-Hubbard gap is depressed much more rapidly in the mean-field theory than the exact-diagonalization calculation. The doping dependence of ground-state energy calculated here is in good agreement with the above numerical calculation for doping up to 0.5. Here, U is the intrasite Coulomb repulsion, x is doping concentration, J=4
t2/U is the superexchange interaction parameter, and t is the hopping integral for the nearest neighbors.