  A search for flavour-changing neutral-current processes in top-quark decays
is presented. Data collected with the ATLAS detector from proton-proton
collisions at the Large Hadron Collider at a centre-of-mass energy of
$\sqrt{s}=13$ TeV, corresponding to an integrated luminosity of 36.1 fb$^{-1}$,
are analysed. The search is performed using top-quark pair events, with one top
quark decaying through the $t\rightarrow qZ$ $(q = u, c)$ flavour-changing
neutral-current channel, and the other through the dominant Standard Model mode
$t\rightarrow bW$. Only $Z$ boson decays into charged leptons and leptonic $W$
boson decays are considered as signal. Consequently, the final-state topology
is characterized by the presence of three isolated charged leptons (electrons
or muons) and at least two jets, one of the jets originating from a $b$-quark,
and missing transverse momentum from the undetected neutrino. The data are
consistent with Standard Model background contributions, and at 95% confidence
level the search sets observed (expected) upper limits of $1.7\times10^{-4}$
($2.4\times10^{-4}$) on the $t\to uZ$ branching ratio and $2.4\times10^{-4}$
($3.2\times10^{-4}$) on the $t\to cZ$ branching ratio, constituting the most
stringent limits to date.
