Topological Phases in Condensed Matter and Cold Atom Systems: towards quantum computations
- Eddy Ardonne (Stockholm University, Sweden)
- Didier Poilblanc ( CNRS - University Toulouse III, France)
- Matthias Troyer (Swiss Federal Institute of Technology Zurich (ETHZ), Switzerland)
- Nicolas Regnault (CNRS, ENS -Paris, France)
The study of topological phases of matter has recently experienced a tremendous intensification, with much progress on both the experimental as well as theoretical side. Most notably are the newly discovered topological insulators (or superconductors), which combines physics from the quantum Hall effect and graphene. Currently, most of the interesting physics in topological insulators emerges from combining non-interacting band theory with the notion of topology, which has led to some spectacular results.
Most of the developments in the field of topological insulators, has focussed on the effects of the topological properties, without taking the electron interactions into account. While giving rise to very interesting physics, combining the topological effects with electron interactions, will most certainly lead to many interesting discoveries. The fractional quantum Hall effect is a good example of where this interplay indeed has led to very exciting new physics.
The theoretical prediction of (non-abelian) Majorana particles in topological insulators and closely related systems by Kane and Mele  (see  for a review) has recently boosted the quest for the discovery of emergent non-abelian particles [3,4] beyond the realm of the quantum Hall effect . Recent experiments on strongly spin-orbit coupled nano-wires deposited on a superconductor have shown very strong indications that such non-abelian particles exist [5,6]. The experimental progress in establishing the existence of non-abelian particles is very exciting, but it is only the first step in utilizing topological phases of matter for (for instance) computing purposes.
In parallel to the developments in condensed matter physics, tremendous progress has been made in the field of cold atomic systems . Such systems are extremely versatile, because of their tunability, and there are several proposals to use exploit the properties cold atomic gasses offer. Amongst these are the realization of interesting model lattice systems, which are known to exhibit interesting topological phases, such as Kitaev's honeycomb model , to name an interesting example. In addition, there are several proposals to use atomic gasses, to emulate non-abelian gauge-potentials . Success in this direction, in particular in combination with `traditional' condensed matter physics, would open up a whole new realm of interesting topological phases of matter.
During the recent years, the field of topological phases has been boosted by the possible application to quantum computing . Topological quantum computation solves by construction, the problem of local decoherence. Implementing topological quantum computation  in realistic experimental systems is one of the grail of the community.
Numerical simulations with theoretical guidance have provided enormous insights on these complex many-body systems - quantum Monte Carlo simulations of cold atoms , density matrix renormalisation group  and tensor network studies  of topological spin liquids or exact diagonalizations of non-Abelian strongly interacting anyons  to cite only a few.
From the above, it should be clear that the field of topological phases in condensed matter physics is an active field, where theoretical (both analytic approaches and simulations) and experimental progress go hand in hand. It is therefor important to have a regular platform, where physicist with different backgrounds - numerical, theoretical or experimental - but with the common interest of topological phases of matter, can report and discuss the recent developments in the field.
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