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

Abstract

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.

Anyonic Chains, Topological Defects, and Conformal Field Theory

“…Note that (1.3) is the CFT version of the topological protection mechanism discussed in [8,9]. In particular, the vanishing of the microscopic commutators 4) ensures that if a putative deformation of the effective theory, δ H eff = λ m, − → A , does not satisfy (1.3), then it cannot be generated upon coarse graining.…”

“…We will also use the fact that C in contains a set of isomorphisms, called "F-symbols", that allow us to move between different fusion channels of the objects in an associative way. The case C in = Rep [su (2) k ] was studied in [8,9,15,[17][18][19], but we will be more general.…”

“…The phase diagram of this model is quite complicated (see [9] for an overview). One important handle on the behavior of the chain is a general result (due to Haldane [23]) that states that in the AF Heisenberg case, if the spin s is half-integer, then the spin chain is gapless (and conformal).…”

“…After briefly mentioning its generalizations, we move on to discuss anyonic chains [8,9,15,[17][18][19], and we review the construction of the resulting Hilbert spaces, Hamiltonians, and topological symmetries.…”

“…While similar behavior is known to occur in spin chains like the Heisenberg model, the mechanism that gives rise to this critical effective behavior in anyonic chains is somewhat different. Indeed, we will see that certain types of non-local operators (also known as "topological symmetries,"), Y , play a starring role [8,9]. Roughly speaking, the Y are in one-to-one correspondence with the simple elements of C in , and they can be used to rule out relevant terms in the effective action if and only if these deformations are in "topologically non-trivial" sectors of the theory.…”