Renormalization flow and universality for ultracold fermionic atoms

A functional renormalization-group study for the Bose-Einstein condensate (BEC)-BCS crossover for ultracold gases of fermionic atoms is presented. We discuss the fixed point which is at the origin of universality for broad Feshbach resonances. All macroscopic quantities depend only on one relevant parameter, the concentration $a{k}_{\mathrm{F}}$, besides their dependence on the temperature in units of the Fermi energy. In particular, we compute the universal ratio between molecular and atomic scattering length in vacuum. We also present an estimate to which level of accuracy universality holds for gases of Li and K atoms.

Figure 1

Location of fixed points (A)–(D) projected onto the plane which is spanned by the couplings ${\tilde{\lambda }}_{\psi }∕\left(8\pi \right)$ and ${\tilde{h}}_{\phi }^2∕\left(32\pi \right)$. The arrows characterize the flow of the couplings towards the infrared.

Figure 2

Sensitivity of the renormalized Yukawa coupling $h_{\phi }$ to the two-body Yukawa coupling ${\widehat{h}}_{\phi ,0}$ at fixed ${\widehat{\lambda }}_{\psi ,0}$: (a) uv and (b) ir flow of the Yukawa coupling. We use $\widehat{k}=k_{\mathrm{F}}=1\mathrm{eV}$, $a_{\mathrm{bg}}\widehat{k}={\widehat{\lambda }}_{\psi ,0}∕\left(8\pi \right)=0.38$ as appropriate for ${}^6\mathrm{Li}$ and $\widehat{T}=0.5$, $c^{-1}=1$. The different curves correspond to the two-body value of the Yukawa coupling ${\widehat{h}}_{\phi ,0}^2=3.72\times ({10}^3,{10}^4,{10}^5)$ (from top to bottom), where the last value corresponds to ${}^6\mathrm{Li}$ while the first one is comparable to that of ${}^{40}\mathrm{K}$. Universality with respect to the value of the two-body Yukawa coupling is very strong even for a comparatively large (fixed) ${\widehat{\lambda }}_{\psi }$.

Figure 3

Sensitivity of the renormalized Yukawa coupling $h_{\phi }$ to the two-body background scattering length ${\widehat{\lambda }}_{\psi }$ for fixed two-body Yukawa coupling ${\widehat{h}}_{\phi ,0}=3.72\times {10}^5$, $\widehat{k}=k_{\mathrm{F}}=1\mathrm{eV}$, as appropriate for ${}^6\mathrm{Li}$: (a) uv and (b) ir flow of the Yukawa coupling. We plot curves for $a_{\mathrm{bg}}\widehat{k}={\widehat{\lambda }}_{\psi }∕\left(8\pi \right)=-0.38\times ({10}^{-3},{10}^{-2},{10}^{-1},1)$ (the last value corresponds to ${}^6\mathrm{Li}$) and $\tilde{T}=0.5$, $c^{-1}=1$. The uppermost line in (b) has the largest $a_{\mathrm{bg}}$ and starts in (a) as the lowest line.