We report the first investigation of cool-core properties of galaxy clusters selected via their Sunyaev--Zeldovich (SZ) effect. We use 13 galaxy clusters uniformly selected from 178 deg^2 observed with the South Pole Telescope (SPT) and followed up by the Chandra X-ray Observatory. They form an approximately mass-limited sample (> 3 x 10^14 M_sun h^-1_70) spanning redshifts 0.3 < z < 1.1. Using previously published X-ray-selected cluster samples, we compare two proxies of cool-core strength: surface brightness concentration (cSB) and cuspiness ({alpha}). We find that cSB is better constrained. We measure cSB for the SPT sample and find several new z > 0.5 cool-core clusters, including two strong cool cores. This rules out the hypothesis that there are no z > 0.5 clusters that qualify as strong cool cores at the 5.4{sigma} level. The fraction of strong cool-core clusters in the SPT sample in this redshift regime is between 7% and 56% (95% confidence). Although the SPT selection function is significantly different from the X-ray samples, the high-z cSB distribution for the SPT sample is statistically consistent with that of X-ray-selected samples at both low and high redshifts. The cool-core strength is inversely correlated with the offset between the brightest cluster galaxy and the X-ray centroid, providing evidence that the dynamical state affects the cool-core strength of the cluster. Larger SZ-selected samples will be crucial in understanding the evolution of cluster cool cores over cosmic time.
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