International atomic clock comparison achieving $5\times 10^{-18}$ uncertainty
Two single-ion optical clocks, located in the United Kingdom and in Germany, are compared remotely via a 2220\,km long optical fiber link. Both clocks are based on the ${}^2S_{1/2} (F=0)\leftrightarrow {}^2F_{7/2} (F=3)$ electric octupole (E3) transition in $^{171}\text{Yb}^+$. Using more than $10^6$ seconds of measurement data, we determine a fractional frequency difference of $1.0(4.8)\times 10^{-18}$, demonstrating agreement of the independently developed systems. To our knowledge, this constitutes the first international clock comparison reaching a fractional uncertainty below $5\times 10^{-18}$, an important milestone towards an anticipated optical redefinition of the SI second. The measurement uncertainty is dominated by that of the relativistic redshift correction between the two sites, which includes time-varying gravity potential differences. We exploit the large baseline between the two clocks to search for violations of Lorentz invariance. In the absence of statistically significant modulations, we improve constraints on the Robertson-Mansouri-Sexl parameter $α$ by approximately a factor of five in the uncertainty, and provide limits on corresponding parameters within the Standard-Model Extension framework.