Does a prohibition apply to the wife if her husband says to her, "You are forbidden to me as a wife" after she went to her in-laws' house without his knowledge, and is there a sin or expiation incumbent upon the husband if he denies what he said and sexual intercourse occurs thereafter?

The preponderant opinion is that the husband's statement "you are forbidden to me" depends on his intention. If he intended divorce, then it is a divorce. If he intended ẓihār (a pre-Islamic form of divorce), then it is ẓihār. If he did not intend divorce or ẓihār, then he must offer expiation for an oath (kaffārat yamīn), and neither divorce nor ẓihār occurs.

As long as the husband denies uttering this prohibition, there is no sin upon the wife, and there is no sin or expiation upon him as long as he does not recall uttering this phrase and does not believe his wife in this regard.

Assuming he does believe her concerning his utterance of this phrase, then he must offer expiation for an oath, and neither divorce nor ẓihār is incumbent upon him.