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

TL;DR

This work analyzes anyonic spin chains built from $su(2)_k$ theories, focusing on spin-1 generalizations and their topological stability. By constructing Hamiltonians from fusion-channel projectors with $F$-symbol basis changes, the authors map rich phase diagrams that parallel the ordinary SU(2) spin-1 chain, including Haldane-like gapped phases, AKLT-like ground states, and multiple critical regimes protected by a topological symmetry. A central finding is a robust even-odd distinction in the level $k$ and a special role for $k=4$, yielding distinct critical content (parafermion, coset, and orbifold CFTs) and a topological mechanism that shields gapless states from generic perturbations. The results connect to broader classes of topological quantum matter and generalized TL/kink-type chains, providing a unifying view of how non-Abelian anyon fusion physics shapes one-dimensional quantum magnetism with potential insights for topological phases and quantum information.

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>4, 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.

Figures (22)

• Figure 1: The anyonic spin-1 chain.
• Figure 2: The basis transformation for the anyonic spin-1 chain.
• Figure 3: (color online) Phase diagrams of the ordinary SU(2) spin-1 chain in a projector representation \ref{['eq:spin1-hamiltonian']} with $J_1=-\sin(\theta_{2,1})$ and $J_2=\cos(\theta_{2,1})$.
• Figure 4: (color online) Phase diagrams of the anyonic su(2)$_k$ spin-$1$ chain with odd $k$ in a projector representation \ref{['eq:spin1-hamiltonian']} where $J_1=-\sin(\theta_{2,1})$ and $J_2=\cos(\theta_{2,1})$. With increasing (odd) index $k \geq 5$ the phase boundaries move as indicated by the arrows.
• Figure 5: (color online) The su(2)$_5$ spin-1 chain: The energy spectra for the various phases of the phase diagram are shown in the upper left panel. For the critical phases/point the energy spectra have been rescaled to match the conformal field theory prediction given in Eq. \ref{['CFT_energy_levels']}. Green squares indicate the location of the primary fields, red circles the descendant fields. The energies predicted by conformal field theory are given in green (red) for primary (descendant) fields. The topological symmetry sector is indicated by the violet index. Data shown are for system sizes $L=18$ and $L=15$, respectively.