We propose a novel ferroelectric switchable altermagnetism effect, the reversal of ferroelectric polarization is coupled to the switching of altermagnetic spin splitting. We demonstrate the design principles for the ferroelectric altermagnets and the additional symmetry constraints necessary for switching the altermagnetic spin splitting through flipping the electric polarization based on the state-of-the-art spin-group symmetry techniques. 22 ferroelectric altermagnets are found by screening through the 2001 experimental reported magnetic structures in the MAGNDATA database and 2 of them are identified as ferroelectric switchable altermagnets. Using the hybrid improper ferroelectric material [C(NH2)3]Cr(HCOO)3 as an example, we show how the altermagnetic spin splitting is tightly coupled to the ferroelectric polarization, providing an ideal platform for designing electric-field-controllable multiferroic devices. Finally, we find that such manipulation of altermagnetism can be detected by monitoring the physical quantities that are related to the non-vanishing Berry curvature dipole, such as the linearly polarized photogalvanic spin current.

7 pages, 4 figures