A search for a heavy, CP-odd Higgs boson, $A$, decaying into a $Z$ boson and a 125 GeV Higgs boson, $h$, with the ATLAS detector at the LHC is presented. The search uses proton--proton collision data at a centre-of-mass energy of 8 TeV corresponding to an integrated luminosity of 20.3 fb$^{-1}$. Decays of CP-even $h$ bosons to $\tau\tau$ or $bb$ pairs with the $Z$ boson decaying to electron or muon pairs are considered, as well as $h \rightarrow bb$ decays with the $Z$ boson decaying to neutrinos. No evidence for the production of an $A$ boson in these channels is found and the 95% confidence level upper limits derived for \sigma (gg\rightarrow A) \times \mbox{BR}(A\rightarrow Zh) \times \mbox{BR}(h\rightarrow f\bar{f}) are 0.098--0.013 pb for $f=\tau$ and 0.57--0.014 pb for $f=b$ in a range of $m_A =$ 220--1000 GeV. The results are combined and interpreted in the context of two-Higgs-doublet models.

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