# Non Abelian Topological Systems

A fascinating property of many topological phases are collective fractionalized excitations which are highly non-local in nature. These carry fractions of the quantum numbers of the underlying microscopic degrees of freedom; e.g., they can carry fractions of the electron charge. In two spatial dimensions, such emergent excitations can have exotic exchange rules, which are different from those of either bosons or fermions. Most intriguingly, certain topological systems can support “non-Abelian” excitations, whose exchange rules are described by non-commuting matrices. In the presence of such excitations, the many-body ground state is multiply degenerate; this degeneracy is topological, in the sense that it cannot be lifted by any local perturbation. The degenerate ground states encode quantum information in a non-local way. The information is therefore immune to decoherence – the destructive effect of the noisy environment. Thus, non-Abelian systems may hold the key for overcoming one of greatest challenges in contemporary physics and engineering: the realization of a fault-tolerant quantum computer.### Selected Work

#### Universal topological quantum computation from a superconductor-Abelian quantum Hall heterostructure

Non-Abelian anyons promise to reveal spectacular features of quantum mechanics that could ultimately provide the foundation for a decoherence-free quantum computer. A key breakthrough in the pursuit of these exotic particles originated from Read and Green’s observation that the Moore-Read quantum Hall state and a (relatively simple) two-dimensional p+ip superconductor both support so-called Ising non-Abelian anyons. Here, we establish a similar correspondence between the Z3 Read-Rezayi quantum Hall state and a novel two-dimensional superconductor in which charge-2e Cooper pairs are built from fractionalized quasiparticles. In particular, both phases harbor Fibonacci anyons that—unlike Ising anyons—allow for universal topological quantum computation solely through braiding. Using a variant of Teo and Kane’s construction of non-Abelian phases from weakly coupled chains, we provide a blueprint for such a superconductor using Abelian quantum Hall states interlaced with an array of superconducting islands. Fibonacci anyons appear as neutral deconfined particles that lead to a twofold ground-state degeneracy on a torus. In contrast to a p+ip superconductor, vortices do not yield additional particle types, yet depending on nonuniversal energetics can serve as a trap for Fibonacci anyons. These results imply that one can, in principle, combine well-understood and widely available phases of matter to realize non-Abelian anyons with universal braid statistics. Numerous future directions are discussed, including speculations on alternative realizations with fewer experimental requirements.- Roger S. K. Mong, David J. Clarke, Jason Alicea, Netanel H. Lindner, Paul Fendley, Chetan Nayak, Yuval Oreg, Ady Stern, Erez Berg, Kirill Shtengel, Matthew P. A. Fisher, Universal topological quantum computation from a superconductor-Abelian quantum Hall heterostructure,Phys. Rev. X 4, 011036 (2014); arXiv:1307.4403

#### Topological quantum computation – From basic concepts to first experiments

Quantum computation requires controlled engineering of quantum states to perform tasks that go beyond those possible with classical computers. Topological quantum computation aims to achieve this goal by using non-Abelian quantum phases of matter. Such phases allow for quantum information to be stored and manipulated in a nonlocal manner, which protects it from imperfections in the implemented protocols and from interactions with the environment. Recently, substantial progress in this field has been made on both theoretical and experimental fronts. We review the basic concepts of non-Abelian phases and their topologically protected use in quantum information processing tasks. We discuss different possible realizations of these concepts in experimentally available solid-state systems, including systems hosting Majorana fermions, their recently proposed fractional counterparts, and non-Abelian quantum Hall states.- A. Stern and N. H. Lindner, Topological quantum computation – From basic concepts to first experiments, Science 339, 1179 (2013), Invited review article;

#### Fractionalizing Majorana Fermions: Non-Abelian Statistics on the Edges of Abelian Quantum Hall States

We study the non-Abelian statistics characterizing systems where counterpropagating gapless modes on the edges of fractional quantum Hall states are gapped by proximity coupling to superconductors and ferromagnets. The most transparent example is that of a fractional quantum spin Hall state, in which electrons of one spin direction occupy a fractional quantum Hall state of ν= -1/m , while electrons of the opposite spin occupy a similar state with ν= -1/m . However, we also propose other examples of such systems, which are easier to realize experimentally. We nd that each interface between a region on the edge coupled to a superconductor and a region coupled to a ferromagnet corresponds to a non-Abelian anyon of quantum dimension √¯2m. We calculate the unitary transformations that are associated with the braiding of these anyons, and we show that they are able to realize a richer set of non-Abelian representations of the braid group than the set realized by non-Abelian anyons based on Majorana fermions. We carry out this calculation both explicitly and by applying general considerations.- N. H. Lindner, E. Berg, G. Refael, and A. Stern, Fractionalizing Majorana Fermions: Non-Abelian Statistics on the Edges of Abelian Quantum Hall States, Physical Review X 2, 041002 (2012); arXiv:1204.5733