At least one species of ant sleeps. In the July 2009 edition of the Journal of Insect Behaviour, Deby L. Cassill, Skye Brown, Devon Swick and George Yanev published an article entitled 'Polyphasic Wake/Sleep Episodes in the Fire Ant, Solenopsis Invicta'.
In that article, Cassill et al. found that fire ants do sleep, although the amount of sleep an individual ant had varied by caste. Workers had an average of 253 'sleep episodes' a day, but each lasted an average of only 1.1 minutes, for a total of approximately 4.8 hours of sleep overall. By contrast, queens only experienced 92 sleep episodes, but each lasted an average of 6 minutes, for a total of 9.4 hours of sleep overall.
In addition, the study found that fire ant sleep was unaffected by light or darkness, and that queens experienced periods of 'rapid antenna movement' sleep. The authors suggested this 'RAM' sleep might be equivalent to vertebrate REM (rapid eye movement) sleep.