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Phases of interacting many-body systems: from classical systems to non-abelian anyons

Charlotte Gils

Published Jan 1, 2008• 0 citations
Publisher
DOI
10.3929/ethz-a-005773779

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•Apr 1, 2011•22 citations
Non-Abelian anyons can exist as pointlike particles in two-dimensional systems and have particle exchange statistics which are neither bosonic nor fermionic. Like in spin systems, the role of fusion (Heisenberg-like) interactions between anyons has been well studied. However, unlike our understanding of the role of bosonic and fermionic statistics in the formation of different quantum phases of matter, little is known concerning the effect of non-Abelian braid statistics. We explore this physics using an anyonic Hubbard model on a two-legged ladder which includes braiding and nearest-neighbor Heisenberg interactions among anyons. We study two of the most prominent non-Abelian anyon models: the Fibonacci and Ising types. We discover rich phase diagrams for both anyon models and show the different roles of their fusion and braid statistics.
A set of localized, non-Abelian anyons—such as vortices in a p x +i p y superconductor or quasiholes in certain quantum Hall states—gives rise to a macroscopic degeneracy. Such a degeneracy is split in the presence of interactions between the anyons. Here, we show that in two spatial dimensions this splitting selects a unique collective state as ground state of the interacting many-body system. This collective state can be a novel gapped quantum liquid nucleated inside the original parent liquid (of which the anyons are excitations). This physics is of relevance for any quantum Hall plateau realizing a non-Abelian quantum Hall state when moving off the center of the plateau.
A set of localized, non-Abelian anyons—such as vortices in a px+ipy superconductor or quasiholes in certain quantum Hall states—gives rise to a macroscopic degeneracy. Such a degeneracy is split in the presence of interactions between the anyons. Here, we show that in two spatial dimensions this splitting selects a unique collective state as ground state of the interacting many-body system. This collective state can be a novel gapped quantum liquid nucleated inside the original parent liquid (of which the anyons are excitations). This physics is of relevance for any quantum Hall plateau realizing a non-Abelian quantum Hall state when moving off the center of the plateau.