Physical realization of the four color problem in quantum systems

Details Physical realization of the four color problem in quantum systems Physical realization of the four color problem in quantum systems

2005-10-07

arxiv.org/abs/cond-mat/0510161v1

Physics

Condensed Matter

Strongly Correlated Electrons

5 pages, 3 figures

Scientific paper

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.

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