Optical conductivity of the two-dimensional Hubbard model: Vertex corrections, emergent Galilean invariance, and the accuracy of the single-site dynamical mean field approximation

We compute the frequency-dependent conductivity of the two-dimensional square lattice Hubbard model at zero temperature as a function of density to second order in the interaction strength, and compare the results to the predictions of single-site dynamical mean field theory computed at the same order. We find that despite the neglect of vertex corrections, the single-site dynamical mean field approximation produces semiquantitatively accurate results for most carrier concentrations, but fails qualitatively for the nearly empty or nearly filled band cases where the model exhibits an emergent Galilean invariance. The DMFT approximation also becomes qualitatively inaccurate very near half filling if nesting is important.

Figure 1

Main panel: Total spectral weight correction as a function of density obtained by numerical integration of Eqs. (22) and (23). The upper left inset shows the low-density dependence of total spectral weight correction as a function of $n^2$ and the inset in the middle shows the renormalized difference $\mathrm{\Delta }$ between two curves, which is defined as $\mathrm{\Delta }=(\delta K_{perturb}-\delta K_{DMFT})/\delta K_{perturb}$.

Figure 2

Optical conductivity for different chemical potential for the perturbative (solid blue) and DMFT (dashed yellow) cases ($\omega >0$). The inset in (a) shows the conductivity for half filled case multiplied by frequency.

Figure 3

(a) When the chemical potential $\mu <-2t$, there is no Umklapp scattering at low frequency. (b) When $-2t<\mu <2t$ Umklapp scattering is allowed at low frequency.

Figure 4

Optical conductivity for the same carrier densities as in Fig. 2 but with $t^{\prime }=0.1$ for the perturbative (solid blue) and DMFT (dashed yellow) cases ($\omega >0$).

Figure 5

Main panel: Drude weight correction as a function of density. The upper left inset shows the low-density dependence of Drude weight correction as a function of $n^2$ and the upper-right inset shows the renormalized difference between two curves $\mathrm{\Delta }=(\delta K_{D_{perturb}}-\delta K_{D_{DMFT}})/\delta K_{D_{perturb}}$. Note that in the exact calculation $-\delta K_{D_{perturb}}/U^2$ diverges as $n\to 1$.

Figure 6

Convolution of two bubbles.