Critical role of magnetic moments on lattice dynamics in YBa${}_2$Cu${}_3$O${}_6$

The role of lattice dynamics in unconventional high-temperature superconductivity is still vigorously debated. Theoretical insights into this problem have long been prevented by the absence of an accurate first-principles description of the combined electronic, magnetic, and lattice degrees of freedom. Utilizing the recently constructed r${}^2$SCAN density functional that stabilizes the antiferromagnetic (AFM) state of the pristine oxide YBa${}_2$Cu${}_3$O${}_6$, we faithfully reproduce the experimental dispersion of key phonon modes. We further find significant magnetoelastic coupling in numerous high energy Cu-O bond stretching optical branches, where the AFM results improve over the soft non-magnetic phonon bands.