Can a senator give up his home in one state and move to Washington dc?

Yes, a senator can give up his home in one state and move to Washington, D.C., while still representing the state. Senators represent their entire state, regardless of where they reside.

As long as they retain their legal residency in the state they represent and continue to fulfill their duties as a senator, including regularly visiting the state and communicating with constituents, they are eligible for election or reelection.