Where to check for neighbors stealing power?
Understand how much power you are using
4000 KWH per month... 720 hours per month... that's 5600 watts on average all the time.
This is the equivalent of a dryer running continuously... 24 x 7...
If this was heating water, that is enough energy to heat nearly 1 gallon per minute continuously - nearly a low-flow showerhead.  A pretty big leak in the hot water system would do that.  A big leak!
If someone tapped an outdoor circuit, the most they could draw is 1800 watts or about 1/3 of your draw. That wouldn't explain this.
Since you don't have gas, the #1 probability is your heat. Resistive strip electric heat is CHEEEEAP to install and expensive to run.  (However in many locations where electric heat is popular, the power company often offers special electric tariff rates to make electric heat beneficial - so talk to your power company about rate plans).
To give you an idea how much heat this is, this is 19,000 BTU/hr continuously on average.   That's about right for heating a house.