A study of the charge conjugation and parity ($CP$) properties of the interaction between the Higgs boson and $\tau$ leptons is presented. The study is based on measurement of $CP$-sensitive angular observables defined by the visible decay products of $\tau$ lepton decays, where at least one hadronic decay is required. The analysis uses 139 fb$^{-1}$ of proton–proton collision data recorded at a center-of-mass energy of $\sqrt{s}$= 13 TeV with the ATLAS detector at the Large Hadron Collider. Contributions from $CP$-violating interactions between the Higgs boson and $\tau$ leptons are described by a single mixing angle parameter $\phi_{\tau}$ in the generalized Yukawa interaction. Without assuming Standard Model hypothesis for the $H\rightarrow\tau\tau$ signal strength, the mixing angle $\phi_{\tau}$ is measured to be $9 \pm 16^{\circ}$, with an expected value of $0 \pm 28^{\circ}$ at the 68\% confidence level. The pure $CP$-odd hypothesis is disfavoured at 3.4 standard deviations. The results are compatible with the predictions for the Higgs boson in the Standard Model as well as $CP$-violating scenarios.

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