(a) System of two FMs parallel to the x–y-plane separated by a distance l along the z-axis. The magnified area depicts the interactions between the spins of the different FMs. (b) Phase diagram showing the orientation of the spins in FM A and B depending on the distance l/a and the dipole–dipole interaction strength D=Kz . For small distances or small dipole–dipole interactions, spins in both FM point out of plane forming a ferromagnetic ordering between the FMs (green, OOP FM). For large distances and large dipole–dipole interactions, spins in both FM lie in-plane forming an antiferromagnetic ordering between the FMs (yellow, IP AFM). The border between both phases (black solid line) is given by Eq. (7). The black dashed line shows the phases depending on the distance for D 1⁄4 0:5Kz. The orientation of the spins inside each FM is depicted by the arrows with the figure-plane being the x–z-plane.

Dipole–dipole-interaction-induced entanglement between two-dimensional ferromagnets

We investigate the viability of dipole–dipole interaction as a means of entangling two distant ferromagnets. To this end, we make use of the Bogoliubov transformation as a symplectic transformation. We show that the coupling of the uniform magnon modes can be expressed using four squeezing parameters, which we interpret in terms of hybridization, one-mode, and two-mode squeezing. We utilize the expansion in terms of the squeezing parameters to obtain an analytic formula for the entanglement in the magnon ground state using the logarithmic nega- tivity as entanglement measure. Our investigation predicts that for infinitely large two-dimensional ferromagnets, the dipole–dipole interac- tion does not lead to significant long-range entanglement. However, in the case of finite ferromagnets, finite entanglement can be expected.

D. Wuhrer, N. Rohling and W. Belzig
Appl. Phys. Lett. 125, 2 (2024)