Lattice Unitarity: Saturated Collisional Resistivity of Strongly Interacting Metals

Abstract

We investigate the interaction-induced resistivity of ultracold fermions in a three-dimensional optical lattice. In situ observations of transport dynamics enable the determination of real and imaginary conductivity (or resistivity). In the strongly interacting metallic regime, we observe a striking saturation of the current-dissipation rate to a value independent of the interaction strength. This behavior is quantitatively captured by a dissipation model that uses a renormalized two-body scattering matrix. The highest observed dissipation rates approach, but do not reach, the unitarity bound on two-body scattering in the lattice, owing to momentum dispersion. We further measure the temperature dependence of resistivity in the strongly interacting limit and compare it to the predicted asymptotic behaviors. These results provide a clear microscopic understanding of bounded resistivity of low-density metals, thus providing a useful benchmark for studies of strongly correlated atomic and electronic systems.

Figures (6)

• Figure 1: Physical regime of measurements. The current dissipation rate, normalized as $\hbar \Gamma/nt \equiv C_\Gamma$, is indicated by color throughout various regimes of temperature and on-site interaction strength, for filling $n=0.1$. At low $T$, dissipation is described by the Fermi liquid picture. For $U^2 \lesssim t^2$, the first Born approximation of the scattering $\mathcal{T}$-matrix is valid. The measurements we report (indicated as "Fig. 3" and "Fig. 4") explores the regime beyond either of these approximations, and in the crossover to a saturated regime in which $C_\Gamma$ is independent of $U$ and $T$.
• Figure 2: Example of saturation. The real and imaginary conductivity, scaled by $\sigma_0 = a_L^2 N/\hbar$, are shown versus drive frequency. Lines show a fit to a perturbative quantum model, described in the text, which yield a best-fit width $\tau_Q^{-1}$. Spectra are shown at $U^2/t^2=1.18(6)$, $\tau_Q^{-1} = 37(14)$ s$^{-1}$ (panels a,b), $U^2/t^2=16.2(8)$, $\tau_Q^{-1} = 110(27)$ s$^{-1}$ (panels c,d), and $U^2/t^2=34(2)$, $\tau_Q^{-1} = 116(19)$ s$^{-1}$ (panels e,f). Despite the strong increase in interactions, the current dissipation rate given by $\tau_Q^{-1}$ saturates, remaining constant between c,d and e,f.
• Figure 3: Resistivity saturation.(a) The real and imaginary components of $\rho$ are shown versus drive frequency for $U^2/t^2=16.2(8)$. While $\Im \rho$ depends strongly on $\omega$, $\Re \rho$ shows no clear trend. The weighted average of $\Re\rho$ across the plotted range is indicated with a horizontal line, with the standard error shown by a shaded band. (b) Frequency-averaged $\Re \rho$ (filled circles) is shown versus interaction strength $U^2/t^2$. The solid line shows a calculation of $\Gamma m^*/N$ using the full $\mathcal{T}$-matrix; the dashed line shows the first Born approximation, $\mathcal{T} \approx U$, for which $\rho$ would be proportional to $U^2$. Instead, $\Re\rho$ shows a clear saturation effect in $U$. These data are taken at $T/t =2.04(31)$ and peak filling $n = 0.088(18)$. The shaded uncertainty bands show $\rho$ calculated for a wider parameter range, corresponding to experimental fluctuations in $T$ and $n$ for data sets at each $U$. Inset: For the same data and horizontal axis, $\hbar/(\tau_Q t)$ is determined through a fit to $\sigma(\omega)$ as in Fig. \ref{['fig:saturation']}, and compared to the calculated $\hbar \Gamma/t$ (solid line and shaded bands).
• Figure 4: Temperature dependence of interaction-saturated resistivity. Measured (filled circles) and calculated (solid line) resistivity are both determined as in Fig. \ref{['fig:Uscan']}, but for $n = 0.095(20)$, fixed $U/t=5.82(16)$, and variable $T$. The range of $T/t$ is expanded for comparison to the $T$-linear regime (dashed line). The shaded bands in both parts include variation in $n$ at each $T$. Inset: Filled circles show $\hbar/(\tau_Q t)$ for the same data set and horizontal axis, as in Fig. \ref{['fig:saturation']}, but here compared to calculations of $\hbar \Gamma/t$ either with all events (solid line) or with only umklapp events (dotted line).
• Figure S1: In situ atomic distribution. Example in situ digitized images at $T/t = 1.19(4)$ (left) and $T/t = 2.35(16)$ (right). The orange atomic clouds represent an average of $\sim$ 280 digitized images, smoothed by a Gaussian filter with a binning size of 1.5 lattice sites. The purple atoms show a single digitized image randomly selected from the set of images used in the averaged distributions.