By break you mean trash, since breaking in netrunner has a different meaning, breaking subroutines on ice, which gets you past but doesn't remove the ice. The answer is yes, there are ways to do this.
- Cuttlery Suite - Forked, Knifed and Spooned - these each destroy one ice you pass on the run you make using them, each one works on a different one of the three major ice types.
- En Passant - This event trashes an ice you passed that the corp didn't rez on the run before you play it.
- Forged Activation Orders - Forces the corp to pay the cost to rez ice now, or trash it, particularly useful early game if the corp has low funds, the ice may be a bluff they can't yet afford.
- Immolation Script - If you've already trashed a piece of ice, the corp discarded it, or it entered archives some other way, this card lets you trash a copy of that ice on the board.
- Kraken - Gives the corp the choice, but you pick the server, they pick the ice and it gets trashed.
- Parasite - Parasite weakens ice every turn and eventually that ice hits 0 strength, due to parasite's other ability once the ice has 0 strength it is trashed. Players speed this up by using Datasucker and later Şifr.
- Prey - It's an expensive option, but if you have alot of cards on the field, you can use this to trash ice you pass by trashing your own cards, equal to the strength. Another good use for strength reducing tricks.
- Run Amok - Trashes one piece of ice that was rezzed in the run you use this, either they dont' rez and let you in free, or they lose ice.
- Traps - Trap ice, like It's A Trap! trash themselves once a runner lets them fire, sometimes it's worth taking the hit early to get it off the board.
In addition to trashing there are other ways to handle ice - one deck was built on the combination of these cards - False Echo and DDoS - preventing the corp from rezzing the outermost piece of ice, then returning ice that was not rezzed to the corp's hand, making the next ice the outermost, so they can't rez it either.
Derezzing ice is also an option, cards like Maxwell James, Rubicon Switch, or the bird breakers like Golden let you flip them back face down, forcing the corp to pay and flip them again.
Finally bypassing the ice is an option, Femme Fatale, Inside Job and Spear Phishing all let you skip one piece of ice.