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On the level lines of two-layer symmetric potentials

A. Ya. Maltsev

TL;DR

The article analyzes level lines of two-dimensional quasiperiodic potentials arising from a four-quasiperiod Novikov problem, modeled as a bilayer system with $V(x,y)=V_{1}(x,y)+V_{2}(x,y)$ where $V_{2}$ is a rotated, shifted copy of $V_{1}$. It distinguishes between generic (non-magic) angles, where open level lines occur only at $c_{0}(\alpha)$, and magic angles where the potential is periodic and a singular net governs the threshold behavior. The main results show that near the threshold energy $c_{0}(\alpha)$, the sizes of closed level-line components obey bounds of the form $d(c) \lesssim |c-c_{0}(\alpha)|^{-1-\varepsilon}$ for almost all non-magic angles, with measure-zero exceptions that can exhibit cascade-like scalings. This work connects the specific structure of bilayer quasiperiodic potentials to models of random potentials, highlighting unique features due to symmetry, singular nets, and the finite-size context relevant to two-layer systems such as bilayer graphene.

Abstract

We consider the behavior of level lines of two-dimensional potentials, which play an important role in the physics of ``two-layer'' systems. Potentials of this type are quasiperiodic and, at the same time, can also be considered as a model of random potentials on a plane. The description of level lines of such potentials is a special case of the Novikov problem for potentials with four quasiperiods and uses many features that arise in the study of the general Novikov problem. At the same time, the potentials under consideration also have their own clearly expressed specificity, which makes them very interesting for research from a variety of points of view.

On the level lines of two-layer symmetric potentials

TL;DR

The article analyzes level lines of two-dimensional quasiperiodic potentials arising from a four-quasiperiod Novikov problem, modeled as a bilayer system with where is a rotated, shifted copy of . It distinguishes between generic (non-magic) angles, where open level lines occur only at , and magic angles where the potential is periodic and a singular net governs the threshold behavior. The main results show that near the threshold energy , the sizes of closed level-line components obey bounds of the form for almost all non-magic angles, with measure-zero exceptions that can exhibit cascade-like scalings. This work connects the specific structure of bilayer quasiperiodic potentials to models of random potentials, highlighting unique features due to symmetry, singular nets, and the finite-size context relevant to two-layer systems such as bilayer graphene.

Abstract

We consider the behavior of level lines of two-dimensional potentials, which play an important role in the physics of ``two-layer'' systems. Potentials of this type are quasiperiodic and, at the same time, can also be considered as a model of random potentials on a plane. The description of level lines of such potentials is a special case of the Novikov problem for potentials with four quasiperiods and uses many features that arise in the study of the general Novikov problem. At the same time, the potentials under consideration also have their own clearly expressed specificity, which makes them very interesting for research from a variety of points of view.
Paper Structure (5 sections, 101 equations, 10 figures)

This paper contains 5 sections, 101 equations, 10 figures.

Table of Contents

  1. Introduction
  2. General facts about level lines of quasiperiodic functions and potentials $\, V (x, y, \alpha, {\bf a}) \,$
  3. Features of periodic potentials $\, V (x, y, \alpha , {\bf a}) \,$
  4. Level lines of quasi-periodic potentials $\, V (x, y, \alpha, {\bf a}) \,$
  5. Conclusion

Figures (10)

  • Figure 1: The form of a "topologically regular" open level line of a quasi-periodic function $\, f (x, y) \,$
  • Figure 2: The form of a "chaotic" open level line of a quasiperiodic function $\, f (x, y) \,$ (schematically)
  • Figure 3: Situation of type $\, A_{-} \,$ ($c < c_{1} (\alpha)$) in the plane $\, \mathbb{R}^{2} \,$
  • Figure 4: Situation of type $\, A_{+} \,$ ($c > c_{2} (\alpha)$) in the plane $\, \mathbb{R}^{2} \,$
  • Figure 5: "Magic" rotation angles that translate the vector $m \, {\bf e}_{1} \, + \, n \, {\bf e}_{2}$ into the vector $n \, {\bf e}_{1} \, + \, m \, {\bf e}_{2}$, and the vector $\, (m + n) \, {\bf e}_{1}\, - \, n \, {\bf e}_{2} \,$ into the vector $\, (m + n) \, {\bf e}_{1}\, - \, m \, {\bf e}_{2} \,$ in the triangular lattice.
  • ...and 5 more figures