Fig. 4: Dipole–dipole interaction in three contexts.

The central dipole (open arrow) can interact either ferroelectrically (in blue) or anti-ferroelectrically with dipoles (in red). a In an ordinary ionic crystal, the interaction is ferroelectric along one orientation and anti-ferroelectric along the perpendicular one, impeding the emergence of long-range ferroelectricity. b In a highly polarizable lattice, the interaction is ferroelectric over a length scale, r\({}_{{\rm{c}}}\), longer than the interdopant distance, \({l}_{{\rm{dd}}}\). This allows the formation of large ferroelectric cluster composed of many dipoles. The typical size of these clusters is set by r\({}_{c}\).^49,50,51 c In the presence of a Fermi sea, ferroelectric and anti-ferroelectric coupling between dipoles alternate radially. According to our observations, anti-ferroelectric coupling between dipoles destroys the ferroelectric order in Sr\({}_{1-x}\)Ca\({}_{x}\)TiO\({}_{3-\delta }\) at a carrier density of \({n}^{* }(x)\), which corresponds to \(2{k}_{{\rm{F}}}{l}_{{\rm{dd}}}=\pi\).