This document presents direct searches for lepton flavour violation in Higgs boson decays, $H\rightarrow e\tau$ and $H\rightarrow\mu\tau$, performed using data collected with the ATLAS detector at the LHC. The searches are based on a data sample of proton--proton collisions at a centre-of-mass energy $\sqrt{s} = 13$\,TeV, corresponding to an integrated luminosity of 138~fb$^{-1}$.
Both leptonic ($\tau \to \ell \nu_\ell \nu_\tau$) and hadronic ($\tau \to \text{hadrons } \nu_\tau$) decays of the $\tau$-lepton are considered and two background estimation techniques are employed: the MC-template method, based on data-corrected simulation samples, and the data-driven Symmetry method, based on exploiting the symmetry between electrons and muons in the Standard Model backgrounds.
The observed (expected) upper limits set on the branching ratios at 95\% confidence level, $\mathcal{B}(H\to e\tau)<0.20\%$ ($0.12\%$) and $\mathcal{B}(H\to \mu\tau)<0.18\%$ ($0.09\%$), are obtained with the MC-template method from a simultaneous measurement of potential $H \rightarrow e\tau$ and $H \rightarrow\mu\tau$ signals. The best-fit branching ratio difference, $\mathcal{B}(H\to \mu\tau)- \mathcal{B}(H\to e\tau)$, measured with the Symmetry method in the channel where the $\tau$-lepton decays to leptons, is $(0.25 \pm 0.10)\%$, compatible with a value of zero within $2.5\sigma$.