Dynamical response functions of the Hubbard model on an infinite dimensional lattice (D=\ensuremath{\infty}) are investigated by the self-consistent second-order perturbation theory with respect to the Coulomb interaction U. The vertex corrections are properly considered so that the conservation laws are satisfied, i.e., the f-sum rule is satisfied. The explicit expression for the f-sum rule is obtained in the case of infinite dimensions. It is found that even in the second-order perturbation theory the electronic states show many-body features at finite temperature and/or frequency regions: the system crosses over from the Fermi-liquid state at the low temperature and low-frequency region to the incoherent regime at high temperatures.
Dynamical response functions of the Hubbard model on an infinite dimensional lattice (D=\ensuremath{\infty}) are investigated by the self-consistent second-order perturbation theory with respect to the Coulomb interaction U. The vertex corrections are properly considered so that the conservation laws are satisfied, i.e., the f-sum rule is satisfied. The explicit expression for the f-sum rule is obtained in the case of infinite dimensions. It is found that even in the second-order perturbation theory the electronic states show many-body features at finite temperature and/or frequency regions: the system crosses over from the Fermi-liquid state at the low temperature and low-frequency region to the incoherent regime at high temperatures.
A model for nonequilibrium dynamical mean-field theory is constructed for the infinite-dimensional Hubbard lattice. We impose nonequilibrium by expressing the physical orbital as a superposition of a left (L)-moving and right (R)-moving electronic state with the respective chemical potentials ${\ensuremath{\mu}}_{L}$ and ${\ensuremath{\mu}}_{R}$. Using the second-order iterative perturbation theory we calculate the quasiparticle properties as a function of the chemical potential bias between the L and R movers, i.e., $\ensuremath{\Phi}={\ensuremath{\mu}}_{L}\ensuremath{-}{\ensuremath{\mu}}_{R}$. The evolution of the nonequilibrium quasiparticle spectrum is mapped out as a function of the bias and temperature. The quasiparticle states with the renormalized Fermi-energy scale ${\ensuremath{\epsilon}}_{QP}^{0}$ disappear at $\ensuremath{\Phi}\ensuremath{\sim}{\ensuremath{\epsilon}}_{QP}^{0}$ in the low-temperature limit. The second-order perturbation theory predicts that in the vicinity of the Mott-insulator transition at the Coulomb-parameter U=Uc, there exists another critical Coulomb-parameter Ud(<Uc) such that, for Ud<U<Uc, quasiparticle states are destroyed abruptly when ${({\ensuremath{\epsilon}}_{QP}^{0})}^{2}\ensuremath{\sim}a{(\ensuremath{\pi}{k}_{B}{T}_{c})}^{2}+b{\ensuremath{\Phi}}_{c}^{2}$ with the critical temperature Tc, the critical bias ${\ensuremath{\Phi}}_{c}$, and the numerical constants a and b on the order of unity.
The NMR relaxation rate in the Hubbard model in high dimensions is investigated. First, by an analysis of the response functions, it is shown that, in the infinite-dimensional models, the intersite correlation does not directly affect the local response of the system. A simple approximation to consider the effects of the finite dimensions is then presented, and effects of the antiferromagnetic (AF) correlation on the NMR relaxation rate in the finite-dimensional Hubbard model is studied by treating the interaction by the self-consistent second-order perturbation theory. It is found that at low temperatures the AF correlation seriously affects the temperature dependence of the NMR relaxation rate in contrast to the case in the infinite dimensions.
We study the Hubbard model in dimension d= infinity , within an approximation that simultaneously reproduces the atomic limit and the weak-coupling limit up to second order in the interaction strength U. This method, which is a slightly modified version of an approximation suggested by Edwards and Hertz, is applied to the calculation of the self-energy and of the spectral function in the paramagnetic regime for different U, band fillings n, and temperatures T. We find Fermi-liquid behaviour for small U and (for half-filling n=0.5) a metal-insulator (Mott) transition at a critical value U cr , but also a metallic non-Fermi-liquid phase for intermediate values U<U cr . We compare our results with results obtained within the second-order perturbation theory in U and (at finite temperature) with the (essentially exact) quantum Monte Carlo (QMC) results. Qualitatively, the approximation reproduces the correct tendencies obtained by QMC, but in detail there are still discrepancies, in particular for intermediate values of U.
We have investigated the attractive Hubbard model in the low-density limit for the 2D square lattice using the ladder approximation for the vertex function in a self-consistent, conserving formulation. In the parameter region where the on-site attraction is of the order of the bandwidth, we found no evidence of a pseudo-gap. Furthermore, we have observed that the suppression of the Fermi surface known to destroy superconductivity in one and two dimensions, when these systems are treated using a non-self-consistent theory (Schmitt-Rink, Varma C M and Ruckenstein A E 1989 Phys. Rev. Lett. 63 445), does not occur when pair-pair interactions are included. However, we do find a quasi-particle lifetime that varies linearly with temperature, which is similar to the findings from many experiments. Thus, although this system has a Fermi surface, it shows non-Fermi-liquid-type behaviour over a wide temperature range. We stress that our work uses thermal Green's functions along the real-time axis, and thus allows for a more accurate determination of the dynamical properties of a model than theories that require extrapolations from the imaginary-frequency axis.
I present a numerical treatment for the retarded self-energy in two dimensions (2D), which is formally exact for the frequency and the momentum sums occurring in perturbation theory with respect to the Coulomb interaction U. In this frequency- and momentum-renormalization-group technique, which allows the sequential elimination of regions of high frequencies and momenta, I focus on the 2D fluctuation-exchange approximation for the Hubbard model. I present results for the self-energy and the spectral density down to temperatures t=${10}^{\mathrm{\ensuremath{-}}4}$/t (the next neighbor hopping amplitude). The obtained result shows evidence for a Fermi-liquid phase for bandfillings n near half filling for U/t=2 to U/t=8 for T\ensuremath{\rightarrow}0.
We investigate the Hubbard model on a two-dimensional square lattice by the perturbation expansion to the fourth order in the on-site Coulomb repulsion U. Numerically calculating all diagrams up to the fourth order in self-energy, we examine the convergence of perturbation series in the lattice system. We indicate that the coefficient of each order term rapidly decreases as in the impurity Anderson model for T > 0.1t in the half-filled case, but it holds in the doped case even at lower temperatures. Thus, we can expect that the convergence of perturbation expansion in U is very good in a wide parameter region also in the lattice system, except for T < 0.1t in the half-filled case. We next calculate the density of states in the fourth-order perturbation. In the half-filled case, the shape in a moderate correlation regime is quite different from the three peak structure in the second-order perturbation. Remarkable upper and lower Hubbard bands locate at w = +(-)U/2, and a pseudogap appears at the Fermi level w=0. This is considered as the precursor of the Mott-Hubbard antiferromagnetic structure. In the doped case, quasiparticles with very heavy mass are formed at the Fermi level. Thus, we conclude that the fourth-order perturbation theory overall well explain the asymptotic behaviors in a strong correlation regime.