Anyonic quantum spin chains: Spin-1 generalizations and topological stability

There are many interesting parallels between systems of interacting non-Abelian anyons and quantum magnetism occurring in ordinary SU(2) quantum magnets. Here we consider theories of so-called su(2)k anyons, well-known deformations of SU(2), in which only the first k+1 angular momenta of SU(2) occur. In this paper, we discuss in particular anyonic generalizations of ordinary SU(2) spin chains with an emphasis on anyonic spin S=1 chains. We find that the overall phase diagrams for these anyonic spin-1 chains closely mirror the phase diagram of the ordinary bilinear-biquadratic spin-1 chain including anyonic generalizations of the Haldane phase, the AKLT construction, and supersymmetric quantum critical points. A novel feature of the anyonic spin-1 chains is an additional topological symmetry that protects the gapless phases. Distinctions further arise in the form of an even/odd effect in the deformation parameter k when considering su(2)k anyonic theories with k≥5, as well as for the special case of the su(2)4 theory for which the spin-1 representation plays a special role. We also address anyonic generalizations of spin-12 chains with a focus on the topological protection provided for their gapless ground states. Finally, we put our results into the context of earlier generalizations of SU(2) quantum spin chains, in particular so-called (fused) Temperley-Lieb chains.

05.30.Pr
03.65.Vf
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spin: chain
stability: topological
symmetry: topological
angular momentum: 1
anyon: nonabelian
critical phenomena
deformation

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Virasoro minimal models 34

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N = 1 superconformal minimal models 35

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S3 minimal models 35

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The Zk parafermion CFT. 35

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The Z2 orbifold theories 35 a. The S-matrix 35 References 36 since the early days of condensed matter physics, quantum magnets have played an integral role in shaping our understanding of interacting quantum many-body systems. Following the experimental discovery of the high-temperature superconductors whose undoped parent compounds typically are antiferromagnets, the study of quantum magnets has further intensified yielding a plethora of deeper insights. Early on, quantum spin chains - typically one-dimensional arrangements of SU(2) spins - have become prototypical systems that proved to be fruitful ground for analytical descriptions and quasi-exact numerical analysis1. One seminal result was the exact solution of the antiferromagnetic spin-1/2 Heisenberg chain via the Bethe ansatz and its description in terms of conformal field theory. Another crucial contribution was Haldane’s realization2 that the antiferromagnetic spin-1 Heisenberg chain forms a gapped state with characteristic zeroenergy edge states for open boundary conditions - a principle observation that holds true for all half-integer and integer spin chains. More recently, it has been found that the

I. INTRODUCTION Ever