Non-Drude universal scaling laws for the optical response of local Fermi liquids

We investigate the frequency and temperature dependence of the low-energy electron dynamics in a Landau Fermi liquid with a local self-energy. We show that the frequency and temperature dependencies of the optical conductivity obey universal scaling forms, for which explicit analytical expressions are obtained. For the optical conductivity and the associated memory function, we obtain a number of surprising features that differ qualitatively from the Drude model and are universal characteristics of a Fermi liquid. Different physical regimes of scaling are identified, with marked non-Drude features in the regime where $\hslash \omega \sim k_{\text{B}}T$. These analytical results for the optical conductivity are compared to numerical calculations for the doped Hubbard model within dynamical mean-field theory. For the “universal” low-energy electrodynamics, we obtain perfect agreement between numerical calculations and analytical scaling laws. Both results show that the optical conductivity displays a non-Drude “foot,” which could be easily mistaken as a signature of breakdown of the Fermi liquid, while it actually is a striking signature of its applicability. The aforementioned scaling laws provide a quantitative tool for the experimental identification and analysis of the Fermi-liquid state using optical spectroscopy, and a powerful method for the identification of alternative states of matter, when applicable.

Figure 1

Fermi liquid scattering rate $\hslash {\tau }_{\text{qp}}^{-1}$. (a) At zero energy, $\hslash {\tau }_{\text{qp}}^{-1}$ increases quadratically with temperature (solid line). The dashed line indicates $2\pi k_{\text{B}}T$. The intercept defines a temperature scale $T_0$ and a frequency scale $\hslash {\omega }_0=2\pi k_{\text{B}}{T}_0$. (b) At finite $T$, $\hslash {\tau }_{\text{qp}}^{-1}\left(\omega \right)$ increases quadratically with the frequency $\omega$. The applicability of the frequency-independent (Drude) approximation is limited to low frequencies.

Figure 2

Fermi-liquid conductivity (13) at low temperature (solid lines), below the temperature $T_1$ defined in Fig. 3. The blue (red) lines show the real (imaginary) part of the conductivity. The dashed lines show the low-frequency Drude-like behavior given by Eq. (15). The dotted lines show Eq. (18). The characteristic frequency scales ${\omega }_{\text{L}}$ and ${\omega }_{\text{H}}$ correspond to those defined in Fig. 3 and Eq. (20).

Figure 3

Regimes of conduction. The Drude regime applies if $\omega <2\pi k_{\text{B}}T/\hslash$, and the thermal regime if $\omega \gtrsim 2\pi k_{\text{B}}T/\hslash$. The conductivity along the two horizontal cuts at fixed temperature is displayed in Figs. 2 and 6.

Figure 4

ac resistivity $1/{\sigma }_1(\omega ,T)$ as a function of $T$ for several frequencies $\omega$. These curves correspond to vertical slices in Fig. 3, and the green bars indicate the temperature at which the dome is crossed for each frequency. The temperatures are measured in units of $T_0$, and the frequencies in units of ${\omega }_0=2\pi k_{\text{B}}{T}_0/\hslash$. The normalization of the resistivity is ${\sigma }_{\text{dc}}{(T/T_0)}^2=\pi \hslash Z\Phi \left(0\right)/\left(24k_{\text{B}}{T}_0\right)$.

Figure 5

Imaginary part of the memory function in a Fermi liquid at $T=0.02T_0$ (thick red lines). (a) At frequencies $\omega <{\tau }_{\text{qp}}^{-1}$, $M_2\left(\omega \right)$ increases as ${\omega }^2$ (thin blue line), with a temperature-dependent curvature given by Eq. (E2). (b) In the thermal regime, $M_2\left(\omega \right)/M_2\left(0\right)$ scales as ${(\pi /3)}^2(1+x^2)$ with $x=\hslash \omega /\left(2\pi k_{\text{B}}T\right)$ (dashed line).

Figure 6

Fermi-liquid conductivity Eq. (13) at high temperature (solid lines), above the temperature $T_1$ defined in Fig. 3. The blue (red) lines show the real (imaginary) part of the conductivity. The dashed lines show the low-frequency Drude-like behavior given by Eq. (15). The dotted lines show Eq. (18).

Figure 7

DMFT self-energy scaling plot. At low temperatures, the curves collapse to a parabola. By comparing with Eq. (11) and taking into account that one has $Z=0.22$ for a doping $\delta =0.2$, one can determine $k_{\text{B}}{T}_0\approx 0.1D\approx 10k_{\text{B}}{T}_{\text{FL}}$.

Figure 8

Comparison of the optical conductivity at $k_{\text{B}}T/D=0.0025D$ ($T=T_{\text{FL}}/4$) for the doped Hubbard model calculated within DMFT (solid lines) to the universal FL scaling form (dashed lines). The real (blue) and imaginary (red) parts are plotted on a lin-lin (a) and log-log (b) scale. (Inset) Same data as in (a), showing the Drude-like response in the low-frequency region.

Figure 9

Memory function at $k_{\text{B}}T/D=0.0025$ ($T=T_{\text{FL}}/4$). (a)–(d) Real (solid blue) and imaginary (solid red) parts of the memory function, compared with the FL scaling forms (dashed). The data are shown for several frequency windows ranging from a very broad one (a), to a very narrow one (d). (e) and (f) Frequency derivative of $M_1$ (solid lines) compared with the FL expressions (dashed lines).

Figure 10

Comparison of the real part of the optical conductivity in the Hubbard model from DMFT (solid lines) with the analytical FL scaling expressions (dashed lines). Inverse temperatures are $D/k_{\text{B}}T=400,200,100,50,20,10$.

Figure 11

(a) Temperature dependence of the imaginary part of the memory function in DMFT. The temperatures and the color code are the same as in Fig. 10. (b) Scaling plot. The $\hslash \omega /\left(2\pi k_{\text{B}}T\right)$ scaling applies at low temperatures and intermediate frequencies. The dashed line shows the parabolic behavior corresponding to Eq. (21).

Figure 12

Microwave optical response of UPd${}_2$Al${}_3$ at $T=2.75$ K. The thin lines show the measurements of Ref. 47. The solid lines show a fit to Eqs. (13) with the parameters $T_0=350$ K and $Z\Phi \left(0\right)={\epsilon }_0{(1.6\times {10}^{14}{\text{s}}^{-1})}^2$. (Inset) Same data on a wider frequency range. The dashed lines show the Drude model.

Figure 13

Real part (top) and imaginary part (bottom) of the optical conductivity normalized by the dc value, as a function of frequency and temperature. $T_0$ is defined in Eq. (11), and ${\omega }_0=2\pi k_{\text{B}}{T}_0/\hslash$. The straight orange lines indicate $\hslash \omega =2\pi k_{\text{B}}T$, and the green lines correspond to the dome in Fig. 3.

Figure 14

Comparison of the Fermi-liquid conductivity in the Drude regime, Eq. (15), with the Drude model (1). The solid blue and red lines show the real and imaginary parts of Eq. (15), respectively. The dashed lines in (a) show the Drude model assuming ${\tau }_{\text{D}}={\tau }_{\text{qp}}$. The dotted lines in (b) show the Drude model assuming ${\tau }_{\text{D}}=({\pi }^2/12){\tau }_{\text{qp}}$ (see text).