  A measurement of observables sensitive to spin correlations in $t\bar{t}$
production is presented, using 36.1 fb$^{-1}$ of $pp$ collision data at
$\sqrt{s} = 13$ TeV recorded with the ATLAS detector at the Large Hadron
Collider. Differential cross-sections are measured in events with exactly one
electron and one muon with opposite-sign electric charge as a function of the
azimuthal opening angle and the absolute difference in pseudorapidity between
the electron and muon candidates in the laboratory frame. The azimuthal opening
angle is also measured as a function of the invariant mass of the $t\bar{t}$
system. The measured differential cross-sections are compared to predictions by
several NLO Monte Carlo generators and fixed-order calculations. The observed
degree of spin correlation is somewhat higher than predicted by the generators
used. The data are consistent with the prediction of one of the fixed-order
calculations at NLO, but agree less well with higher-order predictions. Using
these leptonic observables, a search is performed for pair production of
supersymmetric top squarks decaying into Standard Model top quarks and light
neutralinos. Top squark masses between 170 and 230 GeV are largely excluded at
the 95% confidence level for kinematically allowed values of the neutralino
mass.
