Is it obligatory for a husband to pay an expiation for an oath (kaffarat yamin) if he swore to his wife, saying, "By Allah, if you don't do such-and-such, you will not sleep in this house today," and then, because she did not do what he asked, he turned her out of the house, and she remained outside for hours?

The wording of the questioner's oath is about sleeping, not about staying at home. So, if he intended the entire night, and his wife returned home before Fajr, he has broken his oath, and expiation is due. If he intended a specific duration, then his intention is what matters, as intention particularizes oaths.

The expiation for an oath is feeding ten needy persons, or clothing them, or freeing a slave. If one cannot find any of these, then it is fasting for three days, as Allah the Almighty says: "Allah will not impose blame upon you for what is unintentional in your oaths, but He will impose blame upon you for what you intended of oaths. So its expiation is the feeding of ten needy people from the average of that which you feed your [own] families or clothing them or the freeing of a slave. But whoever does not find [anything], then [it is] the fasting of three days. That is the expiation for oaths when you have sworn. And guard your oaths. Thus does Allah make clear to you His verses that you may be grateful." (Al-Ma'idah: 89).