My understanding is that heat pumps can be over 100% efficient because they're actually moving heat from A to B where one side is the outside environment. It doesn't violate thermodynamics if you view the Earth as a closed system. The heat pump itself is not a closed system.
Thanks for highlighting this because it's a common misconception.
"The coefficient of performance or COP (sometimes CP or CoP) of a heat pump, refrigerator or air conditioning system is a ratio of useful heating or cooling provided to work (energy) required.[1][2] Higher COPs equate to higher efficiency, lower energy (power) consumption and thus lower operating costs. The COP usually exceeds 1, especially in heat pumps, because, instead of just converting work to heat (which, if 100% efficient, would be a COP of 1), it pumps additional heat from a heat source to where the heat is required. Most air conditioners have a COP of 2.3 to 3.5. Less work is required to move heat than for conversion into heat, and because of this, heat pumps, air conditioners and refrigeration systems can have a coefficient of performance greater than one. However, this does not mean that they are more than 100% efficient, in other words, no heat engine can have a thermal efficiency of 100% or greater. For complete systems, COP calculations should include energy consumption of all power consuming auxiliaries. The COP is highly dependent on operating conditions, especially absolute temperature and relative temperature between sink and system, and is often graphed or averaged against expected conditions."
In short, a heat pump is more efficient when compared to using the energy to directly generate heat because it's more efficient to move heat than generate it.
That means you need somewhere to move the heat to, and the thermoynamics of heat capacity are not forgiving if you are trying to make something lightweight. You need atoms and lots of entropic states to store heat :)
