Izbrane teme sodobne fizike in matematike
V tem članku je opisana matematična obravnava usmerjenih snovi, natančneje nematske faze tekočih kristalov. Obravnavana je s klasično teorijo polja ter teorijo homotopij. Topološki koncepti so pojasnjeni na fizikalen način z zgledi ter fizikalnimi primeri. Obravnavane so homotopije krivulj in ploskev ter preko njih uvedeni homotopski razredi, ki sestavljajo homotopske grupe. Z njimi so obravnavani črtasti in točkasti topološki defekti ter topološke invariante, ki jih določajo. Prikazani so še črtasti solitoni. Pri obravnavi je poudarek na direktorskem polju nematika z upoštevanjem njegove simetrije. Prikazane so rešitve z več defekti ter razložena njihova dinamika.
In this article, the mathematical formulation of directed matter, specifically the nematic phase of liquid crystals, is presented. It is described using classical field theory and homotopy theory. Topological concepts are explained using physical examples. Homotopies between curves and surfaces are covered, with which homotopy classes, which constitute homotopy groups, are defined. Line and point defects are described using topological invariants. Line solitons are also covered. The focus is on the director field of nematics, taking its symmetries into account. Multidefect solutions are covered and their dynamics described.