Universal dynamic structure factor of a strongly correlated Fermi gas

Universality of strongly interacting fermions is a topic of great interest in diverse fields. Here we investigate theoretically the universal dynamic density response of resonantly interacting ultracold Fermi atoms in the limit of either high temperature or large frequency. At high temperature, we use quantum virial expansion to derive universal, nonperturbative expansion functions of the dynamic structure factor; at large momentum, we conjecture that the second-order expansion function gives the Wilson coefficient used in the operator product expansion method. The dynamic structure factor is therefore determined by its second-order expansion function with an overall normalization factor given by Tan's [S. Tan, Ann. Phys. (NY) 323, 2952 (2008); S. Tan, Ann. Phys. (NY) 323, 2971 (2008); S. Tan, Ann. Phys. (NY) 323, 2987 (2008)] contact parameter. We show that the spin-parallel and -antiparallel dynamic structure factors have, respectively, a tail of the form $\sim \pm {\omega }^{-5/2}$ for $\omega \to \infty$, decaying slower than the total dynamic structure factor found previously ($\sim {\omega }^{-7/2}$). Our predictions for the dynamic structure factor at high temperature or large frequency are testable using Bragg spectroscopy for ultracold atomic Fermi gases.

Figure 1

Universal second-order expansion function of the DSF at $\tilde{q}=1/3$, 1, and 3. The inset shows the rapid convergence of $\Delta {\tilde{S}}_2(\tilde{q},\tilde{\omega })$ at small $\hslash {\omega }_0/k_{B}T$ (thick lines), $\Delta {\tilde{S}}_{\uparrow \uparrow ,2}$ (thin solid line), and $\Delta {\tilde{S}}_{\uparrow \downarrow ,2}$ (thin dashed line) at $\tilde{q}=1$.

Figure 2

Diagrammatic contributions to the interaction dynamic susceptibility. The self-energy ($S$) and Maki-Thompson (MT) diagrams contribute to $\Delta {\chi }_{\uparrow \uparrow }(\mathbf{r},\tau )$ and $\Delta {\chi }_{\uparrow \downarrow }(\mathbf{r},\tau )$, respectively, while the Aslamazo-Larkin (AL) diagram contributes to both. The shadow in the diagrams represents the contact $\mathcal{I}$. The crossed-hatched part in the diagram (AL) is the vertex.

Figure 3

Function $f_{\sigma {\sigma }^{\prime }}=\sqrt{m\hslash }{\omega }^{3/2}(z^2/{\mathcal{I}}_2)\Delta S_{\sigma {\sigma }^{\prime },2}$ at $\tilde{q}=3$, 5, and 10. With increasing momentum and/or frequency, $f_{\sigma {\sigma }^{\prime }}$ approaches smoothly the result by Son and Thompson [18]: $f_{\uparrow \uparrow }=(f_S-f_{\text{AL}})/4{\pi }^2$ and $f_{\uparrow \downarrow }=(f_{\text{MT}}-f_{\text{AL}})/4{\pi }^2$.

Figure 4

(a) Universal total DSF and (b) spin-antiparallel DSF at $T/T_F=0.225$, $0.5$, 1, and 2, calculated using Eq. (14). We use a pair fluctuation theory to determine the fugacity $z$ and contact $\mathcal{I}$ [24].