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arXiv:1303.4290 · Physical Review B

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

Charlotte Gils, Eddy Ardonne, Simon Trebst, David A. Huse, A. W. W. Ludwig, Matthias Troyer, Zhenghan Wang

Published Jun 17, 2013• 85 citations
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Abstract

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 $k\ensuremath{\ge}5$, 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-$\frac{1}{2}$ 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.

DOI
10.1103/physrevb.87.235120

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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{}_{k}k​ anyons, well-known deformations of SU(2), in which only the first k+1k+1k+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=1S=1S=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 kkk when considering su(2)k{}_{k}k​ anyonic theories with k\ensuremath≥5k\ensuremath{\ge}5k\ensuremath≥5, as well as for the special case of the su(2)4{}_{4}4​ theory for which the spin-1 representation plays a special role. We also address anyonic generalizations of spin-12\frac{1}{2}21​ 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.
Quantum ladder models, consisting of coupled chains, form intriguing systems bridging one and two dimensions and have been well studied in the context of quantum magnets and fermionic systems. Here we consider ladder systems made of more exotic quantum mechanical degrees of freedom, so-called non-Abelian anyons, which can be thought of as certain quantum deformations of ordinary SU(2) spins. Such non-Abelian anyons occur as quasiparticle excitations in topological quantum fluids, including px+ipy{p}_{x}+i{p}_{y}px​+ipy​ superconductors, certain fractional quantum Hall states, and rotating Bose-Einstein condensates. Here we use a combination of exact diagonalization and conformal field theory to determine the phase diagrams of ladders with up to four chains. We discuss how phenomena familiar from ordinary SU(2) spin ladders are generalized in their anyonic counterparts, such as gapless and gapped phases, odd and even effects with the ladder width, and elementary ``magnon'' excitations. Other features are entirely due to the topological nature of the anyonic degrees of freedom. In general, two-dimensional systems of interacting localized non-Abelian anyons are anyonic generalizations of two-dimensional quantum magnets.
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.
Boundary conformal field theory is brought to bear on the study of topological insulating phases of non-Abelian anyonic chains. These phases display protected anyonic end modes. We consider spin-1/2 su(2)k{}_{k}k​ chains at any level kkk, focusing on the most prominent examples: the case k=2k=2k=2 describes Ising anyons (equivalent to Majorana fermions) and k=3k=3k=3 corresponds to Fibonacci anyons. The method we develop is quite general and rests on a deep connection between boundary conformal field theory and topological symmetry. This method tightly constrains the nature of the topological insulating phases of these chains for general kkk. Emergent anyons which arise at domain walls are shown to have the same braiding properties as the physical quasiparticles. This suggests a ``solid-state'' topological quantum computation scheme in which emergent anyons are braided by tuning the couplings of non-Abelian quasiparticles in a fixed network.
We discuss how to construct models of interacting anyons by generalizing quantum spin Hamiltonians to anyonic degrees of freedom. The simplest interactions energetically favor pairs of anyons to fuse into the trivial (“identity") channel, similar to the quantum Heisenberg model favoring pairs of spins to form spin singlets. We present an introduction to the theory of anyons and discuss in detail how basis sets and matrix representations of the interaction terms can be obtained, using non-Abelian Fibonacci anyons as example. Besides discussing the “golden chain”, a one-dimensional system of anyons with nearest neighbor interactions, we also present the derivation of more complicated interaction terms, such as three-anyon interactions in the spirit of the Majumdar-Ghosh spin chain, longer range interactions and two-leg ladders. We also discuss generalizations to anyons with general non-Abelian SU(2)k statistics. The k→∞ limit of the latter yields ordinary SU(2) spin chains.
Gapped quantum spin chains with symmetry PSU(N)=SU(N)/ZN\text{PSU}(N)=\text{SU}(N)/{\mathbb{Z}}_{N}PSU(N)=SU(N)/ZN​ are known to possess NNN distinct symmetry protected topological phases. Besides the trivial phase, there are $N\ensuremath{-}1$ Haldane phases which are distinguished by the occurrence of massless boundary spins. We give an expression for a universal AKLT Hamiltonian for an arbitrary symmetry group, including the case where inversion symmetry is broken. Using a diagrammatic method, we construct Hamiltonians for two of the topological classes in SU(N)\text{SU}(N)SU(N) that arise when the physical spin is in the adjoint representation. Since our results are valid for all NNN, we also discuss the large-NNN limit.
We discuss generalizations of quantum spin Hamiltonians using anyonic degrees of freedom. The simplest model for interacting anyons energetically favors neighboring anyons to fuse into the trivial ("identity") channel, similar to the quantum Heisenberg model favoring neighboring spins to form spin singlets. Numerical simulations of a chain of Fibonacci anyons show that the model is critical with a dynamical critical exponent z=1, and described by a two-dimensional (2D) conformal field theory with central charge c=7/10. An exact mapping of the anyonic chain onto the 2D tricritical Ising model is given using the restricted-solid-on-solid representation of the Temperley-Lieb algebra. The gaplessness of the chain is shown to have topological origin.