Critical scaling of the ac conductivity and momentum dissipation

The scaling of the AC conductivity in quantum critical holographic theories at finite density, finite temperature, and in the presence of momentum dissipation is considered. It is shown that there is generically an intermediate window of frequencies in which the IR scaling of the AC conductivity is clearly visible.

Figure 1

The parameter landscape of the DBI Theory and the regimes for the DC conductivity. Figure reused from [60] under license https://creativecommons.org/licenses/by/4.0/.

Figure 2

Frequency dependence of the conductivity at $\tau =\kappa =0$. The real part (left) shows the scaling behavior $\mathrm{Re}\left[\sigma \right]\sim w^2$. On the contrary, the imaginary part (right) shows a $\mathrm{Im}\left[\sigma \right]\sim w^{-1}$ divergence, indicating the presence of a $\delta \left(w\right)$ in the real part, as required by momentum conservation. The black dashed line corresponds to the fitting $\mathrm{Re}\left[\sigma \right]=w^2/3$ and $\mathrm{Im}\left[\sigma \right]=1/\left(2\sqrt{3}w\right)$.

Figure 3

Absolute value (left) and argument of the conductivity (right) at zero temperature and zero momentum breaking parameter ($\tau =\kappa =0$). The dashed line shows a fitting with $\left|\sigma \right|=\frac{1}{2\sqrt{3}w}$ and $arg\left[\sigma \right]=\frac{\pi }{2}$, respectively.

Figure 4

Finite temperature conductivity at zero momentum relaxation parameter ($\kappa =0$). In the left plot we show the real part for several values of the temperature $\tau$. The $\delta \left(w\right)$ in the real part is not drawn. In the right plot the imaginary part is shown. The dots represent numerical data, while the continuous lines are given by (53). The black lines correspond to $\text{Re}\left[\sigma \right]=\frac{1}{3}{w}^2$ and $\mathrm{Im}\left[\sigma \right]=1/\left(2\sqrt{3}w\right)$.

Figure 5

The absolute value and argument of the conductivity at $\kappa =0$ and at various value of $\tau$. The dots show the numerical data, while the continuous lines are given by (53). The black line shows a fitting with the zero temperature result $\left|\sigma \right|=\frac{1}{2\sqrt{3}w}$ and $arg\left[\sigma \right]=\frac{\pi }{2}-\frac{2}{\sqrt{3}}{w}^3$.

Figure 6

The real and imaginary parts of the conductivity at $\tau =2\pi \times {10}^{-3}$ and for various values of $\kappa$. The dots show the numerical data, while the continuous lines are given by (62). Note that the peaks that appear in the real part as $w\to 0$ are not visible in this plot for most values of $\kappa$. As we further decrease the frequency, the lines cross and the peaks appear in reverse order.

Figure 7

The absolute value and argument of the conductivity at $\tau =2\pi \times {10}^{-3}$ and for various values of $\kappa$. The dots show the numerical data, while the continuous lines are given by (62).

Figure 8

Real part of the conductivity as a function of the frequency at $\tau =2\pi \times {10}^{-3}$. Left plot ($\kappa ={10}^{-7}$) shows the conductivity for the case in which $\lambda \ll \tau \ll 1$. The Drude peak turns into the “flat” temperature-dominated behavior, which gives its turn to the scaling behavior $\sim w^2$, before reaching the UV at $w>1$. The vertical lines correspond to $\frac{\kappa }{\left(2\tau \right)}$ and $\tau$ from left to right. On the contrary the right plot ($\kappa ={10}^{-4}$) corresponds to regime $\tau \ll \lambda \ll 1$. The Drude peak shows a transitions directly to the $\sim w^2$ scaling behavior. The dashes show the numerical data, while the continuous line is given by (63). The vertical line corresponds to $\lambda =\sqrt{\frac{\kappa }{2}}$.