Using modern mathematics – specifically

**group cohomology theory and group super-cohomology theory – the researchers have constructed and classified the symmetry-protected phases in any number of dimensions and for any symmetries.**Their new classification system will provide insight about these quantum phases of matter, which may in turn increase our ability to design states of matter for use in superconductors or quantum computers. Examples of symmetry-protected phases include some topological superconductors and topological insulators, which are of widespread immediate interest because they show promise for use in the coming first generation of quantum electronics.

To understand symmetry, imagine flying through liquid water in an impossibly tiny ship: the atoms would swirl randomly around you and every direction – whether up, down, or sideways – would be the same. The technical term for this is "symmetry" – and liquids are highly symmetric. Crystal ice, another phase of water, is less symmetric. If you flew through ice in the same way, you would see the straight rows of crystalline structures passing as regularly as the girders of an unfinished skyscraper. Certain angles would give you different views. Certain paths would be blocked, others wide open. Ice has many symmetries – every "floor" and every "room" would look the same, for instance – but physicists would say that the high symmetry of liquid water is broken.

Classifying the phases of matter by describing their symmetries and where and how those symmetries break is known as the Landau paradigm. More than simply a way of arranging the phases of matter into a chart, Landau’s theory is a powerful tool which both guides scientists in discovering new phases of matter and helps them grapple with the behaviours of the known phases. Physicists were so pleased with Landau’s theory that for a long time they believed that all phases of matter could be described by symmetries. That’s why it was an eye-opening experience when they discovered a handful of phases that Landau couldn’t describe.

New states contain a new kind of order: topological order. Topological order is a quantum mechanical phenomenon: it is not related to the symmetry of the ground state, but instead to the global properties of the ground state’s wave function. Therefore, it transcends the Landau paradigm, which is based on classical physics concepts.

*String net theory of light and electrons*

Science - etry-Protected Topological Orders in Interacting Bosonic Systems

Order Parameters, Broken Symmetry, and Topology - Online introduction to the theoretical framework used to study the bewildering variety of phases in condensed-matter physics. They emphasize the importance of the breaking of symmetries, and develop the idea of an order parameter through several examples. They discuss elementary excitations and the topological theory of defects.