[leni536]

Are there any efficient electric heating systems for a household based on a thermodynamic cycle with the environment?

I think the most electric heating systems I have seen are using the rather bruteforce and inefficient (only near 100%) method. The basic principle that they heat a resistor and something quickly transfers the heat away so the resistor doesn't burn out and/or your house doesn't catch on fire.

An other fun method to increase the "efficiency" of electric heating would be heating with bitcoin miners. It wouldn't make the heating more efficient in the sense it would cost the same energy but at least you could get some of your money back spent on heating.

[abelsson]

I'm not quite sure what you mean your first question, but I think heat pumps using air, ground or water as heat reservoirs would qualify. They are in common household use at least where I am located, and have a input-power-to-heat efficiency of up to 400% (the Carnot cycle imposes a limit of roughly 7-10x depending on which exact scenario you calculate)

Here's an example of one manufacturer and products: http://www.nibe.eu/Domestic-heatingcooling/Airwater-heat-pum...