Thanks for the additional information, Jennifer.
Under CA law, the landlord is responsible for the repair or replacement of your appliance so long as the appliance was provided for you as part of the rental when you first rented the unit.
As the refrigerator is an important appliance for the preservation of food, it is not deemed an absolute necessity for the habitability of the rental and under the law the landlord has a reasonable about of time to replace or repair the appliance.
A reasonable amount of time is generally deemed within a week or so after you give notice of the fact that the appliance has quit working. Be sure to put the notice in writing and either mail or hand deliver the notice. While a phone call or email is a good idea, it does not constitute legal notice of the repair/replacement issue.
Because a refrigerator is not a required item to be included with a rental under CA law, the failure to repair or replace your broken unit would not violate the law. However, the failure to repair or replace would result in you no longer getting what you bargained for and you could sue the landlord to decrease the rent or to seek a rebate of the rent based on loss of the use of the appliance.
You may reply back to me using the Reply link and I will be happy to continue to assist you until I am able to address your concerns, to your satisfaction.
Please remember to rate my service to you so that I can be compensated for helping you. Thank you in advance.
I wish you and yours the best in 2015,