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## Topological Order: Anyons and Fractons

#### Location: Ecole de physique des Houches, France

#### Organisers

Topological order (TO) was introduced by Xiao-Gang Wen in the 1990’s in order to explain chiral spin liquids and the fractional quantum Hall effect (see [1] for a review) and as an alternative to Landau's theory of phases of matter. More generally, TO is supposed to describe phases of matter that are neither distinguished by a local order parameter nor by spontaneous symmetry breaking. Later on, in the 2000’s, Alexei Kitaev proposed topologically-protected quantum computation using anyons [2]. Famous toy models such as the toric code [2] and string nets (Levin & Wen [3]) were also introduced around that time and led to key conceptual progress. It was then understood that the mathematics needed to handle TO was that of fusion categories. A lot of theoretical, numerical and now experimental work is being devoted to non-abelian anyons such as Majorana zero modes with the aim of realizing protected qu-bits for topological quantum computation. Very recently, the ideas of generalized symmetry breaking (such as higher-form symmetries or categorical symmetries) and of TO also in three spatial dimensions (with fractons) have emerged. There was also an experimental breakthrough with the discovery of anyonic statistics in 2021 in the fractional quantum Hall effect (in the two different groups of Gwendal Fève [4] and Michael Manfra [5]). On the theoretical and numerical sides, a lot of progress has been made recently through the developement of tensor network methods (such as matrix product states MPS and projected entangled pair PEPS) that rely on understanding the quantum entanglement content of groundstates of many-body systems featuring (or not) topological order (see [6] for review). It is therefore highly timely to organize a school in this field.

## References

**Austria**

Norbert Schuch (University of Vienna) - Organiser

**Belgium**

Frank Verstraete (University of Gent) - Organiser

**France**

Jean-Noël Fuchs (Sorbonne Université and CNRS) - Organiser

Julien Vidal (Sorbonne Université and CNRS) - Organiser