Well .... who would have thought it!
They do !
This is from Antcam.com
Ants do seem to work very hard, but they also rest and sleep. The
reason why you think they are always working is because there are
always ants working at every hour. All of the workers look the
same, so you think they are working 24 hours a day. --Ant and UMI
Some ants are sleeping like behavior have been observed: some
workers of a colony laying around in a so-called sleeping
position (Australian Carpenter Ants, Camponotus perthiana). When
they are "awakened", they show a sluggishly moving. --E.O. Wilson
However, this behavior can also be seen as this: As workers have
to respond to the colony needs it could also be a reducing of
resources. It means that they have nothing to do and have to
reduce the need of food, so they fall in a sleep-like idleness
with reduced biological functions. When the colony needs them
again, they start acting normal with a side effect -- they need
time to get to normal functioning (resulting in a sluggish
movement). --Jinei
Other comments from my observations from Camponotus. Actually,
a friend noticed this watching them on my TV. It seems that at
least my carpenters will go into a kind of meditation or relaxed
state. One can see their legs waving slightly. Many times a whole
chamber will do this for a few minutes at a time until another
worker comes in and wakes them up -- or one of them just
spontaneously wakes up.
Read more here