"have broken up" is a correct and usable part of a sentence in written English.
It is used to denote an ended romantic relationship. Example: After four years together, John and Jane have broken up.
"It seems they really have broken up".
"The actions of Pakistan have broken up this network".
Small wonder, perhaps, that they have broken up a few days later.
He explained that some of those customers have broken up and now want privacy.
Police have broken up suspected terrorist cells in the Philippines, Cambodia, Thailand, and Indonesia.
We had to recover it quickly or it would have broken up and been lost forever.
In 1995, the electorate rejected a referendum that would have broken up Berlusconi's empire.
The show, which included lengthy nude scenes, was rumored to have broken up Mr. Tucci's marriage.
European officials trying to curb illegal migration have broken up several high-profile smuggling operations recently.
Still, their bond is steamy and perverse enough to have broken up Archie's marriage.
The Japanese police have broken up several theft rings shipping cars to Russia.
