Generally, the Landlord must do their best to re-rent the unit as soon as possible.
How early the lease was broken, and how much damage there is to repair will determine how quickly this can be done.
If a new tenant is found before the end of the original lease period, then there could be a period of time that the departed tenant would not be liable for the rent. Unless they paid rent in advance, they would not get any back.
Of course, if your lease includes an Early Termination Fee - or you strike any other deal with a tenant who wants to break the lease - make sure the terms are very clear and abide by whatever written agreement you have.
If you are referring to security deposit: That should be refunded according to state law. If there is any left over after damages, you might call that an "unused portion".
Follow the link in the upper left area of this page to find your state laws.
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