In a mountain range, there are trails that can be contracted to a point if you walk through a flat valley. But there are also loop routes that cannot be shrunk without crossing a ridge. Physicists studying quantum phases have found that their phase diagram is structured the same way: it contains closed paths that cannot be contracted to a point without crossing a region where the energy gap vanishes.

Such paths are called non-contractible loops. They show that you cannot smoothly transition between two phases without hitting a critical point. Previously, these loops were studied in isolation: either when symmetry is preserved or when it is completely broken. But in real systems, symmetries interact, and one may be broken while another is not.

The authors used the method of duality, which connects the phase diagrams of two systems, much like a mirror swaps left and right. For example, the Kramers-Wannier duality exchanges domain walls and charges. This allowed them to transfer the classification of loops from one side to the other and to construct explicit unitary evolutions that generate non-contractible loops.

The results were unexpected. For systems with anomalous Z₂×Z₂ symmetry, loops are classified by the group Z₄, because the domain wall of one symmetry carries half the charge of the other. For dihedral groups D₂ₙ, the classification depends on the parity of n: for odd n it equals D₂ₙ, and for even n it is D₈ or Z₂×Z₂. For non-invertible symmetries, the classification of loops depends on whether the base point is trivial, which contrasts with ordinary phases.

These results provide a tool for finding inevitable gapless points in quantum systems. If there is a non-contractible loop in the phase diagram, then somewhere on that loop there must be a point where the gap closes. This is like knowing that on a loop route you will inevitably encounter a pass, even if you cannot see it from the start. The work deepens our understanding of the topological structure of phase diagrams and opens the way to classifying phases with exotic symmetries.