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A search for flavour-changing neutral current decays of a top quark into anup-type quark ($q=u, c$) and the Standard Model Higgs boson, $t\rightarrow Hq$,is presented. The search is based on a dataset of $pp$ collisions at$\sqrt{s}=13$ TeV recorded in 2015 and 2016 with the ATLAS detector at the CERNLarge Hadron Collider and corresponds to an integrated luminosity of 36.1fb$^{-1}$. Two complementary analyses are performed that search for top-quarkpair events in which one top quark decays into $Wb$ and the other top quarkdecays into $Hq$, and target the $H\rightarrow b\bar{b}$ and$H\rightarrow\tau^+\tau^-$ decay modes, respectively. The high multiplicity of$b$-quark jets, or the presence of hadronically decaying $\tau$-leptons, areexploited in the two analyses respectively. Multivariate techniques are used toseparate the signal from the background, which is dominated by top-quark pairproduction. No significant excess of events above the background expectation isfound, and 95% CL upper limits on the $t\rightarrow Hq$ branching ratios arederived. The combination of these searches with ATLAS searches in diphoton andmultilepton final states yields observed (expected) 95% CL upper limits on the$t\rightarrow Hc$ and $t\rightarrow Hu$ branching ratios of $1.1 \times10^{-3}$ ($8.3 \times 10^{-4}$) and $1.2 \times 10^{-3}$ ($8.3 \times10^{-4}$), respectively. The corresponding combined observed (expected) upperlimits on the $|\lambda_{tcH}|$ and $|\lambda_{tuH}|$ couplings are 0.064(0.055) and 0.066 (0.055), respectively.

Search for top-quark decays $t \rightarrow Hq$ with 36 fb$^{-1}$ of $pp$ collision data at $\sqrt{s}=13$ TeV with the ATLAS detector