What are some of the perspectives about what connection (if any) remains between a man and woman who have divorced?

I understand that from a perspective of religious law, a legitimate Jewish divorce severs the formal connection between a man and a woman. But it seems a little odd to only consider a divorce in legal terms when Jewish marriage is supposed to have such profound spiritual implications.

Within the range of opinions of Jewish scholarship, is there a consensus about what the relationship is between a couple who divorced? Or a few opinions which are prominent? Are there arguments that even when divorced, some sort of connection remains, or is it held that after divorce the pair spiritually and legally return to a status like those of two strangers?