Is it permissible for a woman who has had her marriage contract (aqd qiran) solemnized to communicate with her husband despite her promise to her father not to communicate with him, and would she be considered to have broken her promise, incurring an expiation (kaffarah) if she communicates with him?

The marriage contract has been concluded, and you have become this man's wife. Therefore, your father has no right to seek your divorce, and you are not obligated to obey him in that, for obedience is only in what is right. If your husband requests consummation and has already paid you the advanced dowry, then neither you nor your father has the right to refuse. Not speaking with your husband is not necessary, and not answering his calls out of fear of your father is not a sin as long as you have not moved into his house. Honoring your father is obligatory, and you should strive to reconcile him with your husband, seeking help from wise people.