Imagine a mountain landscape: valleys are different states of matter, and ridges are the transitions between them. Some circular routes can be shrunk to a point if you follow a flat valley. But there are routes that cannot be contracted without crossing a ridge. In physics, such routes are called non-contractible loops. They show that you cannot pass smoothly between two states without hitting a point where the energy gap vanishes.

Previously, scientists studied these loops only in simple cases: when the symmetry of the system is either fully preserved or fully broken. But in reality, symmetries can interact: one may be broken while another is intact. It is as if two different sets of trail rules applied simultaneously in the mountains. The new work shows that such interactions create more complex loops that had gone unnoticed.

The results were unexpected. For instance, for systems with anomalous Z₂×Z₂ symmetry, loops are classified by the group Z₄, because a domain wall of one symmetry carries half the charge of the other. For other symmetries, the classification depends on the parity of the number n. Thanks to this, one can now predict in advance where a critical point will inevitably appear in a quantum system, even if there are infinitely many parameters but only two need to be tuned.