A search for the pair-production of vector-like quarks optimized for decays into a $Z$ boson and a third-generation Standard Model quark is presented, using the full Run 2 dataset corresponding to 139 fb$^{-1}$ of $pp$ collisions at $\sqrt{s}=13$ TeV, collected in 2015-2018 with the ATLAS detector at the Large Hadron Collider. The targeted final state is characterized by the presence of a $Z$ boson with high transverse momentum, reconstructed from a pair of same-flavour leptons with opposite-sign charges, as well as by the presence of $b$-tagged jets and high-transverse-momentum large-radius jets reconstructed from calibrated smaller-radius jets. Events with exactly two or at least three leptons are used, which are further categorized by the presence of boosted $W$, $Z$, and Higgs bosons and top quarks. The categorization is performed using a neural-network-based boosted object tagger to enhance the sensitivity to signal relative to the background. No significant excess above the background expectation is observed and exclusion limits at 95% confidence level are set on the masses of the vector-like partners $T$ and $B$ of the top and bottom quarks, respectively. In the singlet model, the limits allow $m_T > 1.27$ TeV and $m_B > 1.20$ TeV. In the doublet model, allowed masses are $m_T > 1.46$ TeV and $m_B >1.32$ TeV. In the case of 100% branching ratio for $T\rightarrow Zt$ and 100% branching ratio for $B\rightarrow Zb$, the limits allow $m_T > 1.60$ TeV and $m_B > 1.42$ TeV, respectively.
Expected and observed lower limits on B masses at 95% CL in the BR plane from the combination of the two analysis channels for all BR configurations when assuming a total BR of 100% for H, W and Z.
Expected and observed lower limits on T masses at 95% CL in the BR plane from the combination of the two analysis channels for all BR configurations when assuming a total BR of 100% for H, W and Z.
Expected and observed combined limits at 95% CL on the production cross-section of vector-like T for 100% $T\rightarrow Zt$.
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