On a regular square lattice, the extended Hubbard model produces d-wave superconducting pairing as the dominant instability. The sign of the order parameter alternates across the Fermi surface—positive in one direction, negative at ninety degrees—and this sign structure is the defining feature of d-wave symmetry.
Canyellas, Katsnelson, and Bagrov place the same model on a Sierpinski carpet—a fractal lattice with self-similar holes at every scale. The d-wave channel, dominant on the regular lattice, is destabilized. Extended s-wave pairing is strongly enhanced instead.
The mechanism: the fractal boundary geometrically frustrates sign-changing order parameters. If your pairing symmetry requires alternating signs across the lattice, and the lattice has holes punched through it at every scale, those sign changes become topologically unsustainable. The boundaries interrupt the coherent pattern before it can establish itself.
On the triangular Sierpinski gasket, the geometry is different enough that hybrid s+d+id states emerge, with critical temperatures comparable to pure s-wave. The fractal doesn't suppress superconductivity—it selects which symmetry survives.
This reframes what material removal does. Punching holes in a superconductor sounds destructive. But the holes act as a filter on pairing symmetry, promoting channels that are topologically compatible with the fractal structure while killing those that aren't. The structure that removes material selects for robustness rather than reducing capability. A lattice made worse in every local sense—fewer sites, more boundaries—becomes a tool for symmetry engineering at the global level.