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Common mathematical theory can have profound applications in understandingreal materials. The intrinsic connection between aperiodic orders observed inthe Fibonacci sequence, Penrose tiling, and quasicrystals is a well-knownexample. Another example is the self-similarity in fractals and dendrites. Fromtransmission electron microscopy experiments, we found that FexTaS2 crystalswith x=1/4 and 1/3 exhibit complicated antiphase and chiral domain structuresrelated to ordering of intercalated Fe ions with 2a*2a and sqrt3a*sqrt3asuperstructures, respectively. These complex domain patterns are found to bedeeply related with the four color theorem, stating that four colors aresufficient to identify the countries on a planar map with proper coloring, andits variations for two-step proper coloring. Furthermore, the domain topologyis closely relevant to their magnetic properties. Our discovery unveils theimportance of understanding the global topology of domain configurations infunctional materials.

Color theorems, chiral domain topology and magnetic properties of FexTaS2