Tunneling Anomaly in a Superconductor above the Paramagnetic Limit

We study the tunneling density of states (DOS) in superconducting systems driven by Zeeman splitting $E_Z$ into the paramagnetic phase. We show that, even though the BCS gap disappears, superconducting fluctuations cause a strong DOS singularity in the vicinity of energies ${-E}^{*}$ for electrons polarized along the magnetic field and $E^{*}$ for the opposite polarization. The position of the singularity $E^{*}=\frac{1}{2}(E_Z+\sqrt{E_Z^2-{\Delta }^2})$ (where $\Delta$ is the BCS gap at $E_Z=0\mathrm{}$) is universal. We find analytically the shape of the DOS for different dimensionalities of the system. For ultrasmall grains the singularity has the form of the hard gap, while in higher dimensions it appears as a significant though finite dip.