We show that interacting bosons on a ring which are driven periodically by a rotating potential can support discrete time crystals whose absolute stability can be proven. The absolute stability is demonstrated by an exact mapping of discrete-time-crystal states to low-lying eigenstates of a time-independent model that reveals spontaneous breaking of space translation symmetry. The mapping ensures that there are no residual time-dependent terms that could lead to heating of the system and destruction of discrete time crystals. We also analyze periodically kicked bosons where the mapping is approximate only and cannot guarantee the absolute stability of discrete time crystals. Besides illustrating potential sources of instability, the kicked bosons model demonstrates a rich field for investigating the interplay between different time and space symmetry breaking, as well as the stability of time crystal behavior in contact with a thermal reservoir.

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• Cold atoms & matter waves
• Crystal phenomena
• Time crystals