Is the spouse responsible for half of the rent if we are married, we rented the home and spouse’s name is on the lease?

You don't say if you are living together or apart. Legally, if you are both on the lease, you are both responsible for the entire amount of the lease payments. If you are separated, then the spouse living there should pay. If the spouse not residing in the apartment pays, they may be entitled to recover a credit in a divorce case, unless the payment is considered spousal support.