What if two pieces of graphene were stacked on top of each other? If the layers were aligned perfectly, two graphene layers would behave essentially the same as a single graphene sheet. But when one of the layers was twisted slightly compared to the other, the rotational misalignment of the two lattices produces a repeating “ moiré pattern ” stretching across many atoms.

“That’s where I started,” Dr. MacDonald said. “What if they were nearly aligned?”

Electrons could easily hop between the two sheets where their lattices lined up. But in places where they were misaligned, the flow would be more difficult. In 2011, Dr. MacDonald and Rafi Bistritzer , a postdoctoral researcher, calculated that at a small angle, the electronic structure would become “flat,” with the electrons jammed like cars trying to make their way across Times Square.

The slowly moving electrons would be more likely to interact with each other — “strongly correlated,” in the language of physics — and from experience, physicists knew that strongly correlated systems are often surprising ones.

“We threw out a few guesses,” Dr. MacDonald said.

The paper was intriguing but largely ignored. The equations, encompassing a multitude of particles at once, are generally far too complex to solve exactly. So Dr. MacDonald and Dr. Bistritzer had made some simplifications to come up with rough answers. Many scientists thought their results were an artifact of their approximations and not a likely description of what would actually be observed.

Philip Kim , a Harvard physicist who did many of the early graphene experiments — Dr. Efetov and Dr. Jarillo-Herrero both worked in his laboratory — thought the glossed-over details in the calculations would be important. “I was skeptical,” he said.

But Dr. Jarillo-Herrero decided to test the prediction. “There was good theoretical motivation to see what would happen,” he said.