There are many interesting parallels between systems of interacting non-Abelian anyons and quantum magnetism, occuring 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 manuscript, 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-1/2 chains with a focus on the topological protection provided for their gapless ground states. Finally, we put our results into context of earlier generalizations of SU(2) quantum spin chains, in particular so-called (fused) Temperley-Lieb chains.

The degrees of freedom in our microscopic models are so-called Fibonacci anyons, one of the simplest types of non-abelian anyons [1,2]. The Fibonacci theory has two distinct particles, the trivial state 1 and the Fibonacci anyon τ , which can be thought of as a generalization (or more precisely a 'truncated version') of an 'angular momentum' when viewing the Fibonacci theory as a certain deformation [3] of SU(2). We will now make this notion more precise and illustrate it in detail. In analogy to the ordinary angular momentum coupling rules, we can write down a set of 'fusion rules' for the anyonic degrees of freedom which are analogs of the Clebsch-Gordon rules for coupling of ordinary angular momenta,where the last fusion rule reveals what is known as the non-abelian character of the Fibonacci anyon: Two Fibonacci anyons τ can fuse to either the trivial particle or to another Fibonacci anyon. In more mathematical terms, these fusion rules can also be expressed by means of so-called fusion matrices N j whose entries (N j ) j1 j2 equal to one if and only if the fusion of anyons of types j 1 and j 2 into j is possible. The fusion rules are related to the so-called 'quantum dimensions' d j of the anyonic particles bywhere |d j � is the ('Perron Frobenius') eigenvector corresponding to the largest positive eigenvalue of the 2 × 2-matrix N j . [The sense in which these numbers are 'dimensions' will become apparent in section II A 1 below.] For the particles in the Fibonacci theory the quantum dimensions are d 1 = 1 and d τ = ϕ ≡ (1 + √ 5)/2 and the total quantum dimension of the theory is then given by D = (To define our Hamiltonian, some additional indegredients of the theory of anyons are required. In analogy to the 6j-symbols for ordinary SU(2) spins, there exists a basis transformation F that relates the two differents ways three anyons can fuse to a fourth anyon, depicted asThe left hand side (l.h.s.) represents the quantum state that arises when anyon a first fuses with anyon b into an anyon of type e, which, subsequently, fuses with anyon c into an anyon of type d. Similarly, the right hand side (r.h.s.) denotes the quantum states that arises when anyon b first fuses with anyon c into anyon type f which, in turn, fuses with anyon a into anyon type d. Whilst keeping all external labels, the types of the three anyons (a, b, c) as well as the resulting anyon type d fixed, the states on the l.h.s and r.h.s. are fully specified by the labels e and f , respectively. Eq. (3) says the so-specified states are linearly related to each other by the so-called F -matrix [4] with matrix elements (Ff e . In general, the F -matrix is uniquely defined (up to 'gauge transformations') by the fusion rules through a consistency relation called the pentagon equation [6]. Similarly, the braiding properties of anyons are given by the so-called R-matrix (which however is not needed here) that is uniquely determined by the hexagon equation [6].For the Fibonacci theory, it is straightforward to verify that in most cases there is only o...

Quantum mechanical systems, whose degrees of freedom are so-called su(2)k anyons, form a bridge between ordinary SU(2) quantum magnets (of arbitrary spin-S) and systems of interacting non-Abelian anyons. Anyonic spin-1/2 chains exhibit a topological protection mechanism that stabilizes their gapless ground states and which vanishes only in the limit (k-->infinity) of the ordinary spin-1/2 Heisenberg chain. For anyonic spin-1 chains the phase diagram closely mirrors the one of the biquadratic SU(2) spin-1 chain. Our results describe, at the same time, nucleation of different 2D topological quantum fluids within a "parent" non-Abelian quantum Hall state, arising from a macroscopic occupation with localized, interacting anyons. The edge states between the "nucleated" and the parent liquids are neutral, and correspond precisely to the gapless modes of the anyonic chains.