A321 Captain reported the aircraft began to roll after parking at the gate when the towbar became disconnected.

Narrative

At the gate; after cleared to release the parking brake; the plane moved on its own which I only initially detected when the driver frantically said the plane was moving and to set brakes.His words kicked in my adrenaline instantly and I quickly glanced to my left and saw us rolling. Immediately; by instinct I grabbed the parking brake and we quickly; albeit abruptly stopped. I am not sure how far we rolled; maybe a couple feet but it didn't feel far. After querying the driver about what occurred; The driver was said the tow bar hadn't been properly connected and we slipped away from his tug and tow bar.After stepping back a bit to assess the situation; I asked my First Officer (FO) the question if had heard our prior clearance to release brakes per the tug driver request. He confirmed he had heard the same. Initially I was concerned I may have released the brakes in haste perhaps merely 'thinking' I heard that request given some sort of expectation bias.There had already been one attempt by the push crew to tell us they were ready for push prior to our XA:00 departure. At the same time we didn't have a valid release as Maintenance was wrapping up paperwork associated with an overhead bin deferral. I told the push back driver we would not be releasing brakes until our release was available.Thus I was doubly shocked unpacking how we moved on our own as intuitively it seemed minutes before their operation was ready for pushback.After the brake release I know I checked the brake pressure gauges then I was heads down looking at the Digital Radio and Audio Integration Management System (DRAIMS) as the next step was contacting ZZZ clearance for push. It must have been at that time the plane moved unintentionally. This extra time heads down checking over the DRAIMS setup may have caused me to be slower to detect the movement. I don't fly the NEO often; thus; all the little steps require a bit more attention see above quick check of the DRAIMS even as the pilot flying (PF) on the ground. Furthermore my FO is quite new to company and 121 operations which naturally led me to focus a bit more on monitoring. We both discussed the NEO and its differences as a threat during our briefing and we both consulted the NEO QRG during flight deck preparation and prior to pushback. After I had taken stock of the alarming situation; I told the FO I'd be off checking in with the flight attendants (FAs) to make sure they were OK given our abrupt stop. They were fine. I told the purser that was not a normal movement they should feel.Looking outside I could see the ramp crew swapping in a new bar and they were clearly examining it's connecting claw device where it hooks to the plane.I discussed with my FO; who is new to company and civilian aviation our options. I explained when a tow bar breaks under push a Maintenance inspection is required prior to continuing. In this case; I explained it wasn't a broken tow bar so much as it seemed to be ramp agent error or maybe an issue with the tow bar. I explained I was comfortable with us continuing given the situation while very unusual and very undesirable was quickly identified by the tug driver and his team.We pushed back again with the same driver and it was a standard pushback.