Many-body effects on graphene conductivity: Quantum Monte Carlo calculations

Optical conductivity of graphene is studied using quantum Monte Carlo calculations. We start from a Euclidean current-current correlator and extract $\sigma \left(\omega \right)$ from Green-Kubo relations using the Backus-Gilbert method. Calculations were performed both for long-range interactions and taking into account only the contact term. In both cases we vary interaction strength and study its influence on optical conductivity. We compare our results with previous theoretical calculations choosing $\omega \approx \kappa$, thus working in the region of the plateau in $\sigma \left(\omega \right)$ which corresponds to optical conductivity of Dirac quasiparticles. No dependence of optical conductivity on interaction strength is observed unless we approach the antiferromagnetic phase transition in the case of an artificially enhanced contact term. Our results strongly support previous theoretical studies that claimed very weak regularization of graphene conductivity.

Figure 1

The ratio of a current-current correlator calculated in the presence of interaction and for free fermions $\left(G\left(\tau \right)/G_0\left(\tau \right)\right)$ as a function of Euclidean time $\tau$ for different bare masses. Temperature is equal to $T=0.5$ eV and lattice size is ${24}^2\times 20$. Interaction strength corresponds to suspended graphene.

Figure 2

The ratio of a current-current correlator calculated in the presence of interaction $\left(G\left(\tau \right)\right)$ and the one for free fermions $\left(G_0\left(\tau \right)\right)$ as a function of Euclidean time $\tau$ for different interaction strengths. Temperature is equal to $T=0.5$ eV and lattice size is ${24}^2\times 20$. Both plots show the data for zero bare mass. (a) The plot corresponds to the model with long-range interaction. (b) The plot corresponds to the model with pure on-site interaction.

Figure 3

Results of test analytical continuation for free fermions: $\overline{\sigma }\left(\omega \right)$ and corresponding resolution functions. Regularization is absent: $\lambda =1$. Bare mass term is introduced: $m=0.05$ eV. $\kappa =2.7$ eV is the hopping between nearest neighbors. (a) $\overline{\sigma }\left(\omega \right)$ extracted from Green-Kubo relations. Vertical error bar is statistical uncertainty. Horizontal error bar shows the width of the corresponding resolution function. It is calculated as a width of the peak at half of its maximum height. A central point is placed at ${\omega }_c$: the maximum of the corresponding resolution function. (b) Resolution functions $\delta ({\omega }_0,\omega )$.

Figure 4

Real calculation for suspended graphene (long-range interaction with $\epsilon =1)$. In this case regularization is imposed: $\lambda =1-4\times {10}^{-5}$. Bare mass term is introduced: $m=0.05$ eV. $\kappa =2.7$ eV is the hopping between nearest neighbors. (a) $\overline{\sigma }\left(\omega \right)$ extracted from Green-Kubo relations. Vertical error bar is statistical uncertainty. Horizontal error bar shows the width of the corresponding resolution function. It is calculated as a width of the peak at half of its maximum height. A central point is placed at ${\omega }_c$: the maximum of the corresponding resolution function. (b) Resolution functions $\delta ({\omega }_0,\omega )$.

Figure 5

Dependence of $\overline{\sigma }\left({\omega }_c\right){|}_{{\omega }_c=1.06\kappa }$ on bare mass for $\lambda =1-4\times {10}^{-5}$. Inset: Dependence of conductivity on regularization constant $\lambda$ for fixed mass $m=0.05$ eV. Both plots are for long-range interactions with $\epsilon =1$ (suspended graphene).

Figure 6

Dependence of $\overline{\sigma }\left({\omega }_c\right){|}_{{\omega }_c=1.06\kappa }$ on interaction strength in the case of long-range interactions. All calculations were performed for $\lambda =1-4\times {10}^{-5}$ and $m=0.05$ eV. Two variants of analytical predictions for graphene conductivity renormalization [see Eq. (2)] are plotted for reference.

Figure 7

Left scale: dependence of conductivity $\tilde{\sigma }$ [calculated from the middle point of the current-current correlator, see Eq. (16)] on the interaction strength both for long-range and short-range interactions. Phase transition into an antiferromagnetic state is situated at $\varepsilon =0.6...0.7$ in the case of long-range interactions (see Ref. [12]) and at $\varepsilon =0.91$ (see Ref. [30]) in the case of pure on-site interaction.