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Sub-diffusive critical states in 2D/3D quasiperiodic structures

Prove whether charge-carrier critical states in two- and three-dimensional quasiperiodic structures can generically exhibit sub-diffusive propagation with exponent β < 0.5 beyond a few very specific energies.

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Background

The authors survey theoretical expectations that critical states arising from quasiperiodicity might lead to sub-diffusive transport (β < 0.5), but emphasize that rigorous demonstrations in 2D and 3D have been limited to isolated energies.

Establishing the general existence (or nonexistence) of β < 0.5 behavior would clarify whether insulating-like transport can be attributed to quasiperiodicity alone without disorder, and delineate the energy ranges and structural conditions where such behavior occurs.

References

Many authors consider that critical states could lead to \beta < 0.5 but it has not yet been shown in 2D or 3D quasiperiodic structures (except for some very specific energies).

— Electronic structure and transport in materials with flat bands: 2D materials and quasicrystals (2506.06721 - Laissardière et al., 7 Jun 2025) in Subsection 2.2.2 (Sub-diffusive states in 2D quasiperiodic tilings)