The Drude weight of the Hubbard model on the two-dimensional square lattice is studied by the exact diagonalizations applied to clusters up to 20 sites. We carefully examine finite-size effects by consideration of the appropriate shapes of clusters and the appropriate boundary condition beyond the imitation of employing only the simple periodic boundary condition. We successfully capture the behavior of the Drude weight that is proportional to the squared hole doping concentration. Our present result gives a consistent understanding of the transition between the Mott insulator and doped metals. We also find, in the frequency dependence of the optical conductivity, that the mid-gap incoherent part emerges more quickly than the coherent part and rather insensitive to the doping concentration in accordance with the scaling of the Drude weight.
The Drude weight of the Hubbard model on the two-dimensional square lattice is studied by the exact diagonalizations applied to clusters up to 20 sites. We carefully examine finite-size effects by consideration of the appropriate shapes of clusters and the appropriate boundary condition beyond the imitation of employing only the simple periodic boundary condition. We successfully capture the behavior of the Drude weight that is proportional to the squared hole doping concentration. Our present result gives a consistent understanding of the transition between the Mott insulator and doped metals. We also find, in the frequency dependence of the optical conductivity, that the mid-gap incoherent part emerges more quickly than the coherent part and rather insensitive to the doping concentration in accordance with the scaling of the Drude weight.
The optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ in the two-dimensional Hubbard model is examined by applying the exact diagonalization technique to small square clusters with periodic boundary conditions up to $\sqrt{20}\ifmmode\times\else\texttimes\fi{}\sqrt{20}$ sites. Spectral-weight distributions at half-filling and their doping dependence in the 20-site cluster are found to be similar to those in a $\sqrt{18}\ifmmode\times\else\texttimes\fi{}\sqrt{18}$ cluster, but different from $4\ifmmode\times\else\texttimes\fi{}4$ results. The results for the 20-site cluster enable us to perform a systematic study of the doping dependence of the spectral-weight transfer from the region of the Mott-gap excitation to lower-energy regions. We discuss the dependence of the Drude weight and the effective carrier number on the electron density at a large on-site Coulomb interaction.
We study Mott transition in the two-dimensional Hubbard model on an anisotropic triangular lattice. We use the Lanczos exact diagonalization of finite-size clusters up to eighteen sites, and calculate Drude weight, charge gap, double occupancy and spin structure factor. We average these physical quantities over twisted boundary conditions in order to reduce finite-size effects. We find a signature of the Mott transition in the dependence of the Drude weight and/or charge gap on the system size. We also examine the possibility of antiferromagnetic order from the spin structure factor. Combining these information, we propose a ground-state phase diagram which has a nonmagnetic insulating phase between a metallic phase and an insulating phase with antiferromagnetic order. Finally, we compare our results with those reported in the previous theoretical studies, and discuss the possibility of an unconventional insulating state.
We study the Drude weight, optical conductivity, and flux-periodicity properties of one-dimensional Hubbard chains using Bethe-ansatz, Lanczos, and exact-diagonalization techniques. We find that the Drude weight D is unexpectedly negative for half-filled Hubbard rings with N=4n sites reflecting a strong paramagnetic response, while it is positive (diamagnetic) for half-filled rings of N=4n+2 sites. In both cases at half filling, D vanishes exponentially with N on a length scale set by the inverse of the gap for small U/t. Near half filling, we find that D approaches a constant, positive, diamagnetic value as N increases, indicating metallic behavior. Combining Bethe-ansatz results near half filling with the known U=0 and U=\ensuremath{\infty} limits and Lanczos finite-size extrapolations, we obtain a general picture of D in the thermodynamic limit as a function of band filling and the ratio U/t. We note similarities with the low-frequency integrated spectral weight of certain hole-doped and electron-doped high-Tc compounds. We investigate the finite-frequency optical conductivity, finding structure at \ensuremath{\omega}\ensuremath{\approxeq}U and, for one hole off half filling, at \ensuremath{\omega}\ensuremath{\propto}t2/U. We find that minima in the energy of a Hubbard ring enclosing a flux \ensuremath{\Phi} occur with changes of half a flux quantum rather than a full flux quantum when the Coulomb repulsion is turned on. Lastly, we discuss the relationship of the Drude weight computed from open chains to that obtained from rings, and also the effect of arbitrary phases at the ring boundary.
The effect of doping in the two-dimensional Hubbard model is studied within finite-temperature exact diagonalization combined with cluster dynamical mean-field theory. By employing a mixed basis involving cluster sites and bath molecular orbitals for the projection of the lattice Green's function onto $2\ifmmode\times\else\texttimes\fi{}2$ clusters, a considerably more accurate description of the low-frequency properties of the self-energy is achieved than in a pure site picture. To evaluate the phase diagram, the transition from Fermi-liquid to non-Fermi-liquid behavior for decreasing hole doping is studied as a function of Coulomb energy, next-nearest-neighbor hopping, and temperature. The self-energy component ${\ensuremath{\Sigma}}_{X}$ associated with $X=(\ensuremath{\pi},0)$ is shown to develop a collective mode above EF, whose energy and strength exhibits a distinct dispersion with doping. This low-energy excitation gives rise to non-Fermi-liquid behavior as the hole doping decreases below a critical value ${\ensuremath{\delta}}_{c}$, and to an increasing particle-hole asymmetry, in agreement with recent photoemission data. This behavior is consistent with the removal of spectral weight from electron states above EF and the opening of a pseudogap, which increases with decreasing doping. The phase diagram reveals that ${\ensuremath{\delta}}_{c}\ensuremath{\approx}0.15\dots{}0.20$ for various system parameters. For electron doping, the collective mode of ${\ensuremath{\Sigma}}_{X}(\ensuremath{\omega})$ and the concomitant pseudogap are located below the Fermi energy, which is consistent with the removal of spectral weight from the hole states just below EF. The critical doping, which marks the onset of non-Fermi-liquid behavior, is systematically smaller than for hole doping.
We have examined the behavior of the compressibility, the dc conductivity, the single-particle gap, and the Drude weight as probes of the density-driven metal-insulator transition in the Hubbard model on a square lattice. These quantities have been obtained through determinantal quantum Monte Carlo simulations at finite temperatures on lattices up to $16\ifmmode\times\else\texttimes\fi{}16$ sites. While the compressibility, the dc conductivity, and the gap are known to suffer from ``closed-shell'' effects due to the presence of artificial gaps in the spectrum (caused by the finiteness of the lattices), we have established that the former tracks the average sign of the fermionic determinant ($\ensuremath{\langle}\text{sign}\ensuremath{\rangle}$), and that a shortcut often used to calculate the conductivity may neglect important corrections. Our systematic analyses also show that, by contrast, the Drude weight is not too sensitive to finite-size effects, being much more reliable as a probe to the insulating state. We have also investigated the influence of the discrete imaginary-time interval ($\ensuremath{\Delta}\ensuremath{\tau}$) on $\ensuremath{\langle}\text{sign}\ensuremath{\rangle}$, on the average density ($\ensuremath{\rho}$), and on the double occupancy (d): we have found that $\ensuremath{\langle}\text{sign}\ensuremath{\rangle}$ and $\ensuremath{\rho}$ are more strongly dependent on $\ensuremath{\Delta}\ensuremath{\tau}$ away from closed-shell configurations, but d follows the $\ensuremath{\Delta}{\ensuremath{\tau}}^{2}$ dependence in both closed- and open-shell cases.
We investigate the extended t-J model (which may include a small next-nearest-neighbor antiferromagnetic coupling or a small hole-hole nearest-neighbor repulsion) by exact-diagonalization techniques. The calculation is done for one and two holes on a 4\ifmmode\times\else\texttimes\fi{}4 torus with arbitrary twisted boundary conditions. The role of the boundary conditions on the ground-state energy, the band width, the charge stiffness, and the optical conductivity is extensively studied. In the two-dimensional space of the twist angles (to be viewed as fluxes through the holes of the torus) the ground-state energy surface exhibits multiple level crossings and may present, locally, a negative curvature (or stiffness) characteristic of a paramagnetic behavior. For two holes and sufficiently large J/t (J/t>0.7) the minima of the energy surface in the parameter space of the fluxes are separated by half a flux quantum. The optical conductivity is calculated by averaging over the boundary conditions in order to minimize finite-size effects and to converge more rapidly toward the thermodynamic limit. We find three different contributions to the optical conductivity; (i) a Drude peak at \ensuremath{\omega}=0, (ii) a broad absorption band in the range J2t, and (iii) a tail extending to larger frequencies (2t7t) associated to the diffusive character of the hole states at these energies.Typically, at low doping density (6%) for a wide range of J/t, and at intermediate doping (12%) for small J/t, these three components account for roughly 30%, 50%, and 20% of the total spectral weight, respectively. At small ratio J/t, both Drude weight and finite-frequency absorption scale almost proportionally with the doping density. However, with increasing J/t, the effective hole-hole attraction (for 12% doping) leads to a transfer of weight from finite frequency to zero frequency. In contrast to the quasiparticle mass, in the J/t\ensuremath{\rightarrow}0 limit, the optical mass defined through the weight of the Drude \ensuremath{\delta}-function peak does not increase like \ensuremath{\sim}t/J but, rather, approaches a limit around \ensuremath{\sim}5. We discuss our results in connection with the optical experiments in the high-Tc cuprates and suggest that the observed mid-infrared band is due to the strong coupling between the mobile holes and the spin excitations of the Mott insulator, as expected from a description in terms of the t-J Hamiltonian.