# Topology in Condensed Matter

## Abstract

Fundamental Physics focuses its efforts on finding out the smallest building blocks of matter. Following this purpose along the centuries, the elements were discovered in the 19th century and Chemistry achieved unprecedented significance. Throughout the 20th century, Physics has endorsed a feverishly search for elementary particles. On the contrary, Condensed Matter Physics plays around with the same atoms and electrons which have been studied for many centuries. Discoveries in this field are not new elements but the emergence of new phases using the same elements. Emergence describes the properties of a material by how the electrons and atoms are organised [1]. This implies that the wide range of materials does not come from the variety of the components. As a natural consequence, the main interest in this area is to find out how they combine together forming new states of matter. These very well-known components form a whole plethora of new states of matter: crystalline solids, magnets and superconductors are some representative examples. The effective field theory known as Landau-Ginzburg theory [2] gave a universal description for quantum phases of matter based on the dimensionality and the symmetries of local order parameters. The great success of the last century was to classify phases of matter following the principle of spontaneous symmetry breaking [3]. What Landau’s theory is lacking is precisely the quantum effects, since it is developed for systems at finite temperatures.

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