It's not specifically stated, it's an implicit prohibition. Under the Constitution, States are not allowed to, for example, change their borders by even an inch without Congressional approval. It logically follows that they can't unilaterally secede, either.
But, in any event, the Southern states never even bothered to discuss the issue of secession within a Constitutional or legal process. They never even contemplated whether they were on the hook for any portion of the Federal government's debt. They raised an army, started a war by firing on a US fort and ended up losing.
Implication is NOT specficially in the US Constitution: I think the 10th Amendement Contradicts your logic: since NOTHING prohibits states from leaving the USA, and since they entered into the Union on their own accord (when they individually ratifed the current Constitution), then it is not only implied, but SPECFICALLY stated that that the states “reserve” the power (right) to seceed~!