A mortgage is a legal contract between a lender (called a creditor) and the borrower or borrowers (called a debtor or debtors) and secured by real property. Like any good contract, a mortgage identifies the parties involved, the property, and the terms of the loan.
If two people agreed to be borrowers -- married or not -- they are both jointly and severally responsible for making the payments agreed. This means that if one person cannot pay, the other debtor on the mortgage must. Similarly, if the debtors stop paying the mortgage the creditor can sue both for breach of contract and get a judgment to collect on their damages from both debtors.
In no lease, vehicle financing, mortgage, loan, or co-signatory contract that I have seen is a "We broke up so I am not on the loan anymore" clause. The lender did not ask you for your relationship status when you signed the loan (you could have been platonic friends, for example) and it does not care now that you are no longer in a relationship with the cosignatory. You are bound by the contract, and if you choose not to honor the mortgage contracts terms you must be prepared for the consequences. (See Collections Advice to learn the rights of creditors and debtors in a collections context.)
To terminate a mortgage, the debtors must either pay-off the loan, sell the property (and thereby pay-off the loan), or refinance the mortgage. A mortgage refinance is where the debtor finds a new loan which he or she uses to repay the original loan.
To remove your name from the mortgage, your cosignatory must refinance the mortgage or sell the property. If your cosignatory can afford to continue to make the payments you may be tempted to leave the mortgage as-is. Do not do this for two reasons. First, if the cosignatory ever fails to make a payment the mortgage company will find you and demand that you make the payment. Second, when you apply for a loan in the future this mortgage will appear on your credit report and will alter your debt-to-income ratio significantly, and may make it impossible for you to get a loan.
To help your cosignatory get started shopping for a refinance, see the Bills.com refinance savings center.
You mentioned abandonment. In a real property context, abandonment of domicile refers to a tenant quitting a property before the lease term is over and stopping making lease payments with the intention of not returning. Abandonment of domicile doctrine does not apply in a mortgage situation, and would not apply here if this was a lease because one of the tenants is still in possession of the property.
I hope this information helps you Find. Learn & Save.