We numerically study optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ near the ``antiferromagnetic'' phase transition in the square-lattice Hubbard model at half filling. We use a cluster dynamical mean field theory and calculate conductivity including vertex corrections and, to this end, we have reformulated the vertex corrections in the antiferromagnetic phase. We find that the vertex corrections change various important details in temperature and $\ensuremath{\omega}$ dependencies of conductivity in the square lattice, and this contrasts sharply the case of the Mott transition in the frustrated triangular lattice. Generally, the vertex corrections enhance variations in the $\ensuremath{\omega}$ dependence, and sharpen the Drude peak and a high-$\ensuremath{\omega}$ incoherent peak in the paramagnetic phase. They also enhance the dip in $\ensuremath{\sigma}(\ensuremath{\omega})$ at $\ensuremath{\omega}=0$ in the antiferromagnetic phase. Therefore, the dc conductivity is enhanced in the paramagnetic phase and suppressed in the antiferromagnetic phase, but this change occurs slightly below the transition temperature. We also find a temperature region above the transition temperature in which the dc conductivity shows an insulating behavior but $\ensuremath{\sigma}(\ensuremath{\omega})$ retains the Drude peak, and this region is stabilized by the vertex corrections. We also investigate which fluctuations are important in the vertex corrections and analyze momentum dependence of the vertex function in detail.
We numerically study optical conductivity
σ(ω) near the "antiferromagnetic" phase transition in the square-lattice Hubbard model at half filling. We use a cluster dynamical mean field theory and calculate conductivity including vertex corrections, and to this end, we have reformulated the vertex corrections in the antiferromagnetic phase. We find that the vertex corrections change various important details in temperature- and
ω-dependencies of conductivity in the square lattice, and this contrasts sharply the case of the Mott transition in the frustrated triangular lattice. Generally, the vertex corrections enhance variations in the
ω-dependence, and sharpen the Drude peak and a high-
ω incoherent peak in the paramagnetic phase. They also enhance the dip in
σ(ω) at
ω=0 in the antiferromagnetic phase. Therefore, the dc conductivity is enhanced in the paramagnetic phase and suppressed in the antiferromagnetic phase, but this change occurs slightly below the transition temperature. We also find a temperature region above the transition temperature in which the dc conductivity shows an insulating behavior but
σ(ω) retains the Drude peak, and this region is stabilized by the vertex corrections. We also investigate which fluctuations are important in the vertex corrections and analyze momentum dependence of the vertex function in detail.
The qualitative reliability of the dynamical mean field theory (DMFT) is investigated for systems in which either the actual carrier density or the effective carrier density is low, by comparing the exact perturbative and dynamical mean field expressions of electron scattering rates and optical conductivities. We study two interacting systems: tight binding models in which the chemical potential is near a band edge and Dirac systems in which the chemical potential is near the Dirac point. In both systems it is found that DMFT underestimates the low frequency, near-Fermi surface single particle scattering rate by a factor proportional to the particle density. The quasiparticle effective mass is qualitatively incorrect for the low density tight binding model but not necessarily for Dirac systems. The dissipative part of the optical conductivity is more subtle: in the exact calculation vertex corrections, typically neglected in DMFT calculations, suppress the low frequency optical absorption, compensating for some of the DMFT underestimate of the scattering rate. The role of vertex corrections in calculating the conductivity for Dirac systems is clarified and a systematic discussion is given of the approach to the Galilean/Lorentz invariant low density limit. Relevance to recent calculations related to Weyl metals is discussed.
The optical conductivity of the weakly doped two-dimensional repulsive Hubbard model on the square lattice with the nearest and next-nearest hoppings is calculated within the generalized dynamical mean-field $(\mathrm{DMFT}+{\ensuremath{\sum}}_{\mathbf{p}})$ approach, which includes correlation length scale $\ensuremath{\xi}$ into the standard DMFT equations via the momentum dependent self-energy ${\ensuremath{\sum}}_{\mathbf{p}}$, with a full account of appropriate vertex corrections. This approach takes into consideration the nonlocal dynamical correlations induced, e.g., by short-ranged collective spin-density-wavelike antiferromagnetic spin fluctuations, which (at high enough temperatures) can be viewed as a quenched Gaussian random field with finite correlation length $\ensuremath{\xi}$. The DMFT effective single-impurity problem is solved by numerical renormalization group. We consider both the case of correlated metal with the bandwidth $W\ensuremath{\lesssim}U$ and that of doped Mott insulator with
U⪢W (
U---the value of local Hubbard interaction). The optical conductivity calculated within $\mathrm{DMFT}+{\ensuremath{\sum}}_{\mathbf{p}}$ demonstrates typical pseudogap behavior within the quasiparticle band, in qualitative agreement with experiments in copper oxide superconductors. For large values of
U, pseudogap anomalies are effectively suppressed.
The optical conductivity for the Hubbard and t-J models is calculated numerically by use of a Lanczos algorithm. One- and two-dimensional systems of up to 16 sites with periodic and open boundary conditions are examined for the t-J model, as well as Hubbard-model systems of up to 9 sites. In the one-dimensional case, the low-frequency part of the conductivity away from half filling is shown to consist almost entirely of a single \ensuremath{\delta} function (Drude peak) even for J of the order of t. This is a direct consequence of the decoupling of charge and spin in one dimension. In two dimensions there is an additional broad absorption band above the Drude peak, consistent with the experimentally observed midinfrared band in the copper oxide superconductors. The importance of vertex corrections is demonstrated by comparing the exact conductivity with the convolution integral of the exact single-particle Green's function.
The optical conductivity of the one-band Hubbard model is calculated using the ``dynamical cluster approximation'' implementation of dynamical mean-field theory for parameters appropriate to high-temperature copper-oxide superconductors. The calculation includes vertex corrections and the result demonstrates their importance. At densities of one electron per site, an insulating state is found with gap value and above-gap absorption consistent with measurements. As carriers are added the above-gap conductivity rapidly weakens and a three component structure emerges, with a low-frequency ``Drude'' peak, a mid-infrared absorption, and a remnant of the insulating gap. The mid-infrared feature obtained at intermediate dopings is shown to arise from a pseudogap structure in the density of states. On further doping the conductivity evolves to the Drude peak plus weakly frequency dependent tail structure expected for less strongly correlated metals.