thermodynamics - Temperature below absolute zero?

I saw this Nature article today, which cites e.g. arXiv:1211.0545.

And it makes no sense to me. The temperature of a collection of particles is the average kinetic energy of those particles. Kinetic energy cannot be less than zero (as far as I'm aware), so I don't understand what this article is trying to say, unless they're playing around with the conventional definition of "temperature".

The only thing I can thing of is if you have something like:

$$\frac{1}{kT} ~=~ \left(\frac{\partial S}{\partial E}\right)_{N,V}$$

And they've created a situation where entropy decreases with increasing energy.