An employee worked unauthorized overtime. Do we have to pay them time and a half for these hours?

Answer from Margaret, PHR, SHRM-CP: Yes. Any overtime worked by non-exempt employees must be compensated, regardless of whether the overtime was authorized. We recommend communicating with the employee about work expectations outside of their scheduled workday, making it clear that the company does not expect or permit employees to work unauthorized time and that working without permission is subject to disciplinary action.

Assuming you have a policy that requires overtime be authorized ahead of time, you can (and should) discipline employees for working unauthorized overtime, just as you would for any other violation of your policy. An oral or written warning may suffice for the first offense. Repeated offenses may warrant further corrective action, up to and including termination.