AC Fingerprints of 2D Electron Hydrodynamics: Superdiffusion and Drude Weight Suppression

Abstract

Clean two-dimensional Fermi liquids are now known to exhibit an intermediate tomographic regime, between ballistic and Navier--Stokes transport, caused by the anomalously slow relaxation of parity-odd multipolar deformations of the Fermi surface. Here we show that this anomaly extends to the dynamical realm. Starting from a microscopic numerical evaluation of the linearized electron--electron collision operator, we find that the finite-frequency nonlocal conductivity is controlled at low frequency by a single hydrodynamic pole, $σ(q,ω)=\mathcal{D}(q)/(iω+η_\star q^z)$, with dynamical exponent $z=4/3$ and superdiffusive viscosity $η_\star$. Remarkably, the pole residue itself is scale dependent and obeys $\mathcal{D}(q)\sim q^{-α}$ with $α=1/3$, so the dynamical properties are described by two separate exponents rather than one. We interpret the residue suppression using a Krylov-chain description of current relaxation: as $q$ increases, the longest-lived quasinormal mode ceases to be a nearly pure current excitation and spreads over higher odd angular harmonics. Finally, we show that AC transport in narrow channels provides a direct route to measuring the exponents $z$ and $α$ separately.

Figures (8)

• Figure 1: Spectrum of the linearized interparticle collision operator in the cylindrical-harmonic basis, shown as the decay rates $\gamma_{m,\text{even}}$ in blue and $\gamma_{m,\text{odd}}$ in orange at $T/T_F = 0.005$. One can observe the different behavior of the even and odd rates for $m \leq m_\text{max} \approx 12$ as discussed around Eqs. \ref{['eq:hofmann-even']} and \ref{['eq:hofmann-odd']}.
• Figure 1: Semi-log plot of the even spectra with the low temperature analytical model shown in green. As temperature is lowered, the lowest even modes collapse onto the model spectrum over a region which grows as $\sim 1 / \sqrt{T/T_F}$.
• Figure 2: Finite-frequency nonlocal conductivity $\sigma(q,\omega)$ at $T/T_F=0.005$. The low-frequency line shape is well fit by a Lorentzian, $\sigma(q,\omega)=\cmcal{D}(q)/(i\omega+\gamma(q))$, from which we extract the decay rate $\gamma(q)$. Rescaling the vertical axis by $\gamma(q)$ and the horizontal axis by $1/\gamma(q)$ would collapse the curves if the residue were constant. This collapse is observed for the two curves with $q\ll q_\mathrm{min}$, while the two curves with $q\gtrsim q_\mathrm{min}$ are progressively suppressed, indicating a decreasing residue.
• Figure 2: Spectrum of $L_q$ for $T/T_F = 0.005$ at three values of $q$ in the tomographic regime. The red dot identifies the eigenvalue with minimal real part; i.e., the late-time decay rate of the Krylov wavefunction.
• Figure 3: (Left column) Extracted $\gamma(q)$, $\cmcal{D}(q)$, and $\sigma(q,\omega=0)$ for a range of temperatures. (Middle column) Local exponents calculated by logarithmic derivatives, i.e., $z(q)\equiv d\ln\gamma/d\ln q$ and $\alpha \equiv d\ln\cmcal{D}/d\ln q$, plotted versus $q/q_\mathrm{min}$. (Right column) Local exponents shown over a larger range of $q/q_\mathrm{min}$, for $T/T_F = 10^{-6}$, demonstrating the slow approach to their asymptotic values.