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On the scaling properties of quasicrystalline potentials of eightfold rotational symmetry

A. Ya. Maltsev

TL;DR

This work studies the scaling properties of level lines for two-dimensional quasiperiodic potentials with eightfold rotational symmetry, motivated by photonic and ultracold-atom systems. It introduces an extended family of potentials built from a base fourfold-symmetric component and its rotated copies, analyzes open versus closed level lines (Novikov problem), and employs periodic approximations at magic angles to derive explicit bounds on the growth of closed level lines as $\epsilon \to 0$, notably $D(\epsilon) \leq C |\epsilon|^{-1}$. The key contributions include a detailed classification of A− and A+ regimes, the role of singular nets in potentials with exact rotational symmetry, and a rigorous Appendix result bounding the diameter of singular-net cells by $D \leq \sqrt{2}\,T_{n,m}$. The findings position these quasiperiodic potentials as models of random-planar behavior with long-range order, offering both qualitative and quantitative insights into percolation-like scaling and level-line topology.

Abstract

We consider a special class of quasi-periodic potentials arising in the physics of photonic systems and possessing rotational symmetry of the 8th order. We are interested in the ``scaling'' properties of such potentials, namely, the growth rate of their closed level lines near the percolation threshold. Estimates of the corresponding scaling indices allow, in particular, to carry out some comparison of such potentials with various models of random potentials on the plane.

On the scaling properties of quasicrystalline potentials of eightfold rotational symmetry

TL;DR

This work studies the scaling properties of level lines for two-dimensional quasiperiodic potentials with eightfold rotational symmetry, motivated by photonic and ultracold-atom systems. It introduces an extended family of potentials built from a base fourfold-symmetric component and its rotated copies, analyzes open versus closed level lines (Novikov problem), and employs periodic approximations at magic angles to derive explicit bounds on the growth of closed level lines as , notably . The key contributions include a detailed classification of A− and A+ regimes, the role of singular nets in potentials with exact rotational symmetry, and a rigorous Appendix result bounding the diameter of singular-net cells by . The findings position these quasiperiodic potentials as models of random-planar behavior with long-range order, offering both qualitative and quantitative insights into percolation-like scaling and level-line topology.

Abstract

We consider a special class of quasi-periodic potentials arising in the physics of photonic systems and possessing rotational symmetry of the 8th order. We are interested in the ``scaling'' properties of such potentials, namely, the growth rate of their closed level lines near the percolation threshold. Estimates of the corresponding scaling indices allow, in particular, to carry out some comparison of such potentials with various models of random potentials on the plane.

Paper Structure

This paper contains 5 sections, 88 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Extended family of quasiperiodic potentials
  3. Potential $\, V ({\bf r} , \, 45^{\circ} , \, {\bf 0}) \,$ and related potentials
  4. Appendix
  5. Conclusion

Figures (9)

  • Figure 1: "Topologically regular" level line of a quasiperiodic potential (schematically)
  • Figure 2: "Chaotic" level line of a quasiperiodic potential (schematically)
  • Figure 3: Situations of type $\, A_{-} \,$ and $\, A_{+} \,$ in the plane $\, \mathbb{R}^{2} \,$ (schematically)
  • Figure 4: Rotation from the vector $\, (n, \, - m) \,$ to the vector $\, (n, \, m) \,$, and also from the vector $\, (1/2) (n + m , \, n - m ) \,$ to the vector $\, (1/2) (n - m , \, n + m ) \,$ in the integer lattice.
  • Figure 5: Periodic open level lines of the potentials $\, V ({\bf r} , \, \bar{\alpha}_{n,m} , \, {\bf a}) \,$ (schematically).
  • ...and 4 more figures