What acts mandate a husband and wife to separate and enter into a new marriage contract? How long a separation is required?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

It seems that the question is about an irrevocable divorce which becomes effective in six situations:

First, Khul' where the woman pays her husband a financial compensation in exchange for his divorcing her.

Second, apostasy when one of the spouses gives up Islam; i.e., becomes apostate.

Third, when the husband divorces his wife thrice at once.

Fourth, when the husband divorces his wife before having sexual intercourse with her.

Fifth, when the divorce is imposed by the judge.

Sixth, when the ‘Iddah (waiting period for the divorced woman before she could marry again) of a revocable divorce expires.

As for the period the couple should remain separated before they could remarry, it differs from one situation to another.

After the Khul’, he cannot take her back to the conjugal life as long as her ‘Iddah has not expired. But he can marry her with a new marriage contract. The same ruling applies to the divorce imposed by the judge unless the cause of the divorce is the husband's failure to spend on his wife or in the case of Eelaa’ (the oath taken by a husband that he would not have sexual relations with his wife for a certain period). In these two situations the marriage can be resumed as soon as the husband promises to spend on his wife in the future and when the husband terminates the Eelaa’. The wife who has been divorced three times cannot return to her former husband unless she gets married to another man and has sexual intercourse with the new husband and provided the last man divorces her and her ‘Iddah from that divorce expires.

The ruling concerning apostasy is the following: as soon as the apostate spouse repents, then the marriage contract can be renewed. The same ruling applies to the divorce done before having sex with one's new wife, and to the wife, whose ‘Iddah (from a revocable divorce) has expired. Her former husband can return her with a new marriage contract.

Allah knows best.