Rudner Photonic Topological Insulators in the Language of the Zhegalkin Operators

Abstract

A topological insulator is a material that exhibits the properties of a conductor on the surface and of an insulator in the bulk. The Rudner game is a simplified model of a topological insulator implemented on a two-dimensional photonic lattice of resonators, which is described in the language of tricolor four-cycle two-dimensional Wolfram cellular automata. It is a case of a regular two-dimensional lattice, in which each cell is colored in one of three colors (for a photonic topological insulator, these colors mean the presence of a photon in a resonator, the absence of a photon, and a topological insulator boundary). By setting the transformation rule for each cell, depending on the state of the nearest neighbors and the cell itself, for equal discrete time intervals we obtain a cellular automaton. In this study, the Rudner game is rewritten equivalently in terms of operators in the Zhegalkin polynomial ring with coefficients in a field consisting of three elements.