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We report results from a deep search for redshifted HI 21cm absorption fromeight damped Lyman-$\alpha$ absorbers (DLAs) detected in our earlier opticalsurvey for DLAs towards radio-loud quasars. HI 21cm absorption was detectedfrom the $z = 2.192$ DLA towards TXS2039+187, only the sixth case of such adetection at $z > 2$, while upper limits on the HI 21cm optical depth wereobtained in six other DLAs at $z > 2$. Our detection of HI 21cm absorption inthe eighth system, at $z = 2.289$ towards TXS0311+430, has been reportedearlier. We also present high spatial resolution images of the backgroundquasars at frequencies close to the redshifted HI 21cm line frequency, allowingus to estimate the covering factor of each DLA, and then determine its spintemperature $T_s$. For three non-detections of HI 21cm absorption, we obtainstrong lower limits on the spin temperature, $T_s \gtrsim 790$ K, similar tothe bulk of the high-$z$ DLA population; three other DLAs yield weak lowerlimits, $T_s > 140-400$ K. However, for the two DLAs with detections of HI 21cmabsorption, the derived spin temperatures are both low $T_s = (160 \pm 35)\times (f/0.35)$ K for the $z = 2.192$ DLA towards TXS2039+187 and $T_s = (72\pm 18) \times (f/0.52)$ K for the $z = 2.289$ DLA towards TXS0311+430. Theseare the first two DLAs at $z > 1$ with $T_s$ values comparable to thoseobtained in local spiral galaxies. Based on the observed correlations betweenspin temperature and metallicity and velocity spread and metallicity in DLAs,we suggest that the hosts of the two absorbers are likely to be massive,high-metallicity galaxies.

A search for HI 21cm absorption towards a radio-selected quasar sample II: a new low spin temperature DLA at high redshift