Figure 1: Uniaxial electric dipoles on a triangular lattice in BaFe₁₂O₁₉.

(a) Crystal structure of the M-type hexaferrite BaFe₁₂O₁₉. It consists of alternate stacks of S and R blocks along the c axis. The asterisk symbols indicate that the corresponding blocks rotate about the c axis by 180°. The Fe³⁺ ions occupy three different sites: octahedral, tetrahedral and bipyramidal (blue) sites. A mirror plane (m) bisects equally the bipyramids in the R/R^* blocks. (b) The 2D triangular lattice of FeO₅ bipyramids in each R/R* block. (c) Illustration of Fe³⁺ off-equator displacements in the FeO₅ bypyramid. The upward or downward displacements at two 4e sites give rise to small electric dipoles along the c axis. Quantum fluctuations between two 4e sites persist to T=0 K. (d) Frustrated electric dipoles on a triangular lattice. Each site contains an Ising-type electric dipole (red arrow), while the neighbouring interactions favour anti-alignment. Quantum dimers (marked by green ovals) with either short-range or long-range entanglement continue to fluctuate and result in a QEL.