Critical Zeeman Splitting of a Unitary Fermi Superfluid

We determine the critical Zeeman energy splitting of a homogeneous Fermi superfluid at unitary in terms of the Fermi energy $\epsilon_{\text F}$ according to recent experimental results in LKB-Lhomond. Based on the universal equations of state for the superfluid and normal phases, we show that there exist two critical fields $H_{c1}$ and $H_{c2}$, between which a superfluid-normal mixed phase is energetically favored. Universal formulae for the critical fields and the critical population imbalance $P_c$ are derived. We have found a universal relation between the critical fields and the critical imbalances: $H_{c1}=\gamma\xi\epsilon_{\text F}$ and $H_{c2}=(1+\gamma P_c)^{2/3}H_{c1}$ where $\xi$ is the universal constant and $\gamma$ is the critical value of the chemical potential imbalance in the grand canonical ensemble. Since $\xi$, $\gamma$ and $P_c$ have been measured in the experiments, we can determine the critical Zeeman fields without the detail information of the equation of state for the polarized normal phase. Using the experimental data from LKB-Lhomond, we have found $H_{c1}\simeq0.37\epsilon_{\text F}$ and $H_{c2}\simeq0.44\epsilon_{\text F}$. Our result of the polarization $P$ as a function of the Zeeman field $H/\epsilon_{\text F}$ is in good agreement with the data extracted from the experiments. We also give an estimation of the critical magnetic field for dilute neutron matter at which the matter gets spin polarized, assuming the properties of the dilute neutron matter are close to those of the unitary Fermi gas.