I'm not a lawyer or Realtor, but I've been there.
When a lease is no longer in effect the time period for notice is determined by the original payment terms, i.e. weekly, biweekly, monthly, etc.
Having said that you are not required to give notice and can leave at any time as long as you aren't leaving without paying any rent due. The landlord cannot hold any funds from your deposit for rent owed or as a penalty for leaving without notice. IF they do sue them in small claims.
On the flip side the landlord would be required to give you notice based on your payment terms, again weekly, biweekly, monthly, etc. The only exception is an eviction which can be as little as 7 days. To meet the qualifications of a 7 day eviction however the rent has to be in arrears and they must show that their property is in imminent danger by the renter.
P.S. In regards to a security deposit they must send you notice by certified mail within 15 days of vacating informing you of any claim against the deposit. If there is no claim the deposit must be returned within 15 days. Again if this is an issue sue them in small claims. The one time I had to do it the filing fee was about $175.00, and I'm sure it's gone up, but the landlord will be ordered to pay that back to you if they lose.