Physical realization of the four color problem in quantum systems

Masanori Yamanaka; Akinori Tanaka

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Physical realization of the four color problem in quantum systems

Masanori Yamanaka·Akinori Tanaka

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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.

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Masanori Yamanaka, Akinori Tanaka. 2005-10-07. Physical realization of the four color problem in quantum systems. https://arxiv.org/abs/cond-mat/0510161

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