When we tear paper does it's particles also get separated into two?
Any atom splitting event would lead to some amounts of radiation, and considering the amount of papers that are torn around the globe, I’d say there is no radiation at all. So atoms are not being split.
What is being split then? An intuitive guess would be bonds between atoms. This might be wrong though. Paper us usually made out of wood. Meaning that it has plant cells in it. So splitting atomic bonds would mean breaking molecules, which again does not happen.
What happens is that you break intermolecular bonds. (In a very general sense since this depends on the paper’s chemistry)