Yes, it is possible to run anything that runs on electricity off a solar grid. The more power you need, the more solar cells, storage batteries, and bigger inverter you need. To run a heat pump, you'll need a hell of a lot of batteries and quite a big inverter. It would also help if your house was located a lot closer to the sun.
If you care to post your zip code, I could run some rough numbers to let you know what size of a solar system we're talking about to run your heat pump effectively when the outdoor temp is 45 or greater (as per your spec).