What constitute abandonment for the purposes of a divorce?

First of all, you leaving home and not returning at all would constitute abandonment and in most states, there is a time requirement to show or prove abandonment. As to him, his moving in with a girlfriend may be considered abandonment and adultery, all wrapped into one. I suggest if you plan on filing divorce, you hit first with the charges of adultery and abandonment. He cannot control who comes to your home because while he is on the deed, it seems you are, as well, and therefore you can have visitors. Just make sure you begin the process of separating your debts and showing that your income should now be considered your separate property and not a joint marital asset.