Physical realization of the four color problem in quantum systems

TL;DR

This paper constructs a quantum lattice model whose ground state demonstrates a spontaneous coloring order related to the four color problem, revealing a topological phase transition and hidden lattice topology.

Contribution

It introduces a multi-component electron model linking the four color problem to quantum ground states, utilizing topological and graph-theoretic concepts.

Findings

Ground state exhibits a coloring order consistent with the four color problem.

Identification of a topological order in the coloring phase.

Discovery of a hidden-topological structure transition in the system.

Abstract

A multi-component electron model on a lattice is constructed whose ground state exhibits a spontaneous ordering which follows the rule of map-coloring used in the solution of the four color problem. The number of components is determined by the Euler characteristics of a certain surface into which the lattice is embedded. Combining the concept of chromatic polynomials with the Heawood-Ringel-Youngs theorem, we derive an index theorem relating the degeneracy of the ground state with a hidden topology of the lattice. The system exhibits coloring transition and hidden-topological structure transition. The coloring phase exhibits a topological order.