On the Novikov problem for superposition of periodic potentials

TL;DR

This paper investigates the Novikov problem for quasiperiodic functions formed by superimposing periodic potentials with four quasiperiods, focusing on cases with identical rotational symmetry and special angles that produce periodic structures.

Contribution

It analyzes the geometric structure of level lines in superpositions of periodic potentials with four quasiperiods, highlighting special angles that yield periodic solutions.

Findings

Generic superpositions have chaotic open level lines.

Special rotation angles lead to periodic superpositions.

Potential applications in two-dimensional physics systems.

Abstract

We consider the Novikov problem, namely, the problem of describing the level lines of quasiperiodic functions on the plane, for a special class of potentials that have important applications in the physics of two-dimensional systems. Potentials of this type are given by a superposition of periodic potentials and represent quasiperiodic functions on a plane with four quasiperiods. Here we study an important special case when the periodic potentials have the same rotational symmetry. In the generic case, their superpositions have ``chaotic'' open level lines, which brings them close to random potentials. At the same time, the Novikov problem has interesting features also for ``magic'' rotation angles, which lead to the emergence of periodic superpositions.