What is the ruling of Islamic law regarding this estrangement between spouses? Is it obligatory to divorce the wife? And in the event of her divorce, is she entitled to a consolatory gift (mut'ah)?

If a wife cuts off her husband and refuses to communicate without a valid excuse, this is a forbidden act of defiance (nushooz). The husband is advised to try to communicate with her and remind her of the obligation to obey him within what is right and reasonable. If he asks her to move in and live with him, she does not have the right to refuse unless she stipulated this condition in the marriage contract. If she refuses intimacy (mu'asharah) in a reasonable manner without a valid excuse, he is permitted to restrict her until she seeks a divorce (khul'). If he divorces her unconditionally, she is entitled to all the rights of a divorced wife with whom consummation has occurred, including the dowry (mahr), maintenance during the waiting period (iddah), and compensatory gift (mut'ah).