Does refusal to live together break nikah?

My father believes and says that praying salat makes bad things happen to him. Bahishti Zewar book states that this is shirk and marriage to someone who does shirk is annulled. Does that mean my mother is no longer his wife? Another questions is, my mother refuses to live with my father because of his very bad conduct and denial of basic matters of deen. She goes to visit him only once a year in Eid and does not divorce him saying that when he is old and cannot torment her anymore, she wishes to serve him till the death (his or her). Since she does not sleep with him, does this break her nikah with him?