As a grad student how to approach a possible breakthrough research?
As a good researcher always be cautious with "breakthroughs". I would explain that you think it is a good idea worth pursuing. Coming off too strong may put new professors off. Unfortunately I have witnessed many times people believe they have come across breakthroughs when they are nothing of the sort. I think partly because unlike how science is portrayed in the media, "most" excellent science is a long grind, through years of experiments, thought, collaboration and trial and error. The road is often long and winding, and to approach someone well versed in the field with a "breakthrough" may be hubris. But, if it is a strong idea and worth pursuing with good reasoning/logic/science behind it, I cant see a problem convincing anybody in your field the importance of what you propose! Good luck!