A319 Captain reported loss of aircraft control during pushback from gate.

Narrative

During pushback; the ground crew stated breakaway; breakaway on the intercom. I used pedal brakes to slowly bring the aircraft to a complete stop. Once the aircraft stopped; I set the parking brake and informed the ground crew that the parking brake was set and pressure was normal. The ground crew then reported that the aircraft had detached from the tug due to an improperly installed pin and confirmed that the aircraft did not sustain any damage.Shortly after; we lost communication with the ground crew due to a faulty headset. With the First Officer; we referenced the FOM and FM (Flight Manual) procedures and notified ramp control and operations. A maintenance defect report was sent via ACARS; and we requested a gate return for maintenance to inspect the aircraft.Once communication was re-established with the ground crew and the tow-in checklist was completed; we were towed back to the gate. Flight attendants and passengers were informed of the situation prior to going back to the gate. Upon arrival at the gate; maintenance personnel inspected the aircraft; and I contacted dispatch to coordinate the new departure.