If renters move out without telling us until after they already moved, are they in violation of breaking the lease?

It depends:

1) If their tenancy had not yet terminated or expired, then they would presumaby be in breach. If there was a written lease for a defined period of time (e.g. a one-year lease), they'd be liable for rent for the remainder of the term of the lease. If they were month-to-month tenants, they'd owe an extra month rent, for the month of notice they should have given.

2) If the lease required some notice before tenants move out, they would have breached that term; refer to the lease to see whether they breached.

But if no notice was required by the lease and their tenancy had terminated or expired, they would not have breached. So, say their lease ends December 31st, 2012, they could move out the 31st (or earlier; see below) without penalty.

The real issue is NOT, however, occupancy but rent. So say the tenants paid rent through to the end of their lease term--then if they moved out early, their is no liability. Their obligation to you is to you rent; it doesn't matter legally whether they reside there or not. So when above I write that if they move out before the lease ends, they would be liable for rent due until the end of the lease, that's not quite accurate, since so long as they paid all the rent they were supposed to pay, there is no violation. It's moving out *and* ceasing to pay rent early that is a breach.