Frequency-temperature crossover in the conductivity of disordered Luttinger liquids

The temperature $\left(T\right)$ and frequency $\left(\omega \right)$ dependent conductivity of weakly disordered Luttinger liquids is calculated in a systematic way both by perturbation theory and from a finite temperature renormalization group (RG) treatment to leading order in the disorder strength. Whereas perturbation theory results in $\omega ∕T$ scaling of the conductivity, such scaling is violated in the RG treatment. We also determine the nonlinear field dependence of the conductivity, whose power law scaling is different from that of temperature and frequency dependence.

Figure 1

(Color online) Plots for the self-energy and conductivity calculated using the finite temperature RG result [Eq. (11)]. (a) Imaginary part $s_i=\mathrm{Im}(\Sigma ∕\omega )$ of the self-energy for several values of $K_0$ (shown in the plots) at fixed $u_0^2={10}^{-4}$ and $t_0={10}^{-3}$. (b) Double logarithmic plot of the real part of the conductivity [Eq. (2)] for the same parameters, rescaled to $1$ at $\omega =0$.

Figure 2

(Color online) Temperature dependence of the dc conductivity in log-log representation for several values of $K_0$ and $u_0^2={10}^{-4}$. Lines: Results obtained by truncation of the zero temperature RG equations (9a, 9b) at the thermal de Broglie wavelength (cf. Ref. 10). Symbols: Conductivity calculated from finite temperature RG and the renormalized self-energy [Eq. (13)].

Figure 3

(Color online) Double logarithmic plot of the imaginary part $s_i=\mathrm{Im}(\Sigma ∕\omega )$ of the self-energy for $K=0.8$ and different temperatures $t_0$. As $K$ varies with frequency, curves for different temperatures do not collapse and $\omega ∕T$ scaling is violated.

Figure 4

(Color online) Field dependence of the nonlinear conductivity ${\sigma }_{\mathrm{nl}}\sim E^{(2-2K)∕(2K-1)}$ for different values of $K$. The straight lines represent the power law.