Vuilleumier cycle heat pump
 

The Solar-AC FAQ: Vuilleumier cycle (similar to duplex Stirling)

It uses heat, not mechanical energy, to do the work.

Correction. It uses heat to create the mechanical motion to pump the heat.

Edit: Let me elaborate. Some of the energy goes to move the pistons and gasses around. The heat pumping is largely done by fluctuating gas pressures in the different areas.