In a conventional superconductor, electrons team up into pairs that stay put: their total momentum is zero. But sometimes it's different: pairs gain a shared momentum and start moving, like a school of fish swimming against the current. This state is called a pair density wave, and it's very finicky: it needs special conditions.
In this work, they considered a crystal with attraction between electrons and added two kinds of magnetic patterns. The first pattern, resembling a diagonal stripe, helped the pair density wave emerge when there were few electrons. The second pattern, like a checkerboard, only worked at high electron counts, and even then only in a narrow range.
The takeaway: a magnetic pattern can either help or hinder the pair density wave. Knowing this, it's easier to search for materials where such exotic superconductivity appears on its own.
The pair density wave in superconductors was predicted back in the 1960s, but experimentally it has been observed in only a few materials, and each time it was a big deal.