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arXiv:0812.3158 · Physical Review B

Permutation-symmetric critical phases in disordered non-Abelian anyonic chains

Lukasz Fidkowski, H.-H. Lin, Paraj Titum, Gil Refael

Published Apr 27, 2009• 25 citations
arXivHTMLPDF

Abstract

Topological phases supporting non-Abelian anyonic excitations have been proposed as candidates for topological quantum computation. In this paper, we study disordered non-Abelian anyonic chains based on the quantum groups $\text{SU}{(2)}_{k}$, a hierarchy that includes the $\ensuremath{\nu}=5/2$ fractional quantum Hall state and the proposed $\ensuremath{\nu}=12/5$ Fibonacci state, among others. We find that for odd $k$ these anyonic chains realize infinite-randomness critical phases in the same universality class as the ${S}_{k}$ permutation symmetric multicritical points of Damle and Huse [Phys. Rev. Lett. 89, 277203 (2002)]. Indeed, we show that the pertinent subspace of these anyonic chains actually sits inside the ${\mathbb{Z}}_{k}\ensuremath{\subset}{S}_{k}$ symmetric sector of the Damle-Huse model, and this ${\mathbb{Z}}_{k}$ symmetry stabilizes the phase.

DOI
10.1103/physrevb.79.155120

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Topological phases supporting non-Abelian anyonic excitations have been proposed as candidates for topological quantum computation. In this paper, we study disordered non-Abelian anyonic chains based on the quantum groups SU(2)k\text{SU}{(2)}_{k}SU(2)k​, a hierarchy that includes the \ensuremathν=5/2\ensuremath{\nu}=5/2\ensuremathν=5/2 fractional quantum Hall state and the proposed \ensuremathν=12/5\ensuremath{\nu}=12/5\ensuremathν=12/5 Fibonacci state, among others. We find that for odd kkk these anyonic chains realize infinite-randomness critical phases in the same universality class as the Sk{S}_{k}Sk​ permutation symmetric multicritical points of Damle and Huse [Phys. Rev. Lett. 89, 277203 (2002)]. Indeed, we show that the pertinent subspace of these anyonic chains actually sits inside the Zk\ensuremath⊂Sk{\mathbb{Z}}_{k}\ensuremath{\subset}{S}_{k}Zk​\ensuremath⊂Sk​ symmetric sector of the Damle-Huse model, and this Zk{\mathbb{Z}}_{k}Zk​ symmetry stabilizes the phase.
Topological insulators supporting non-Abelian anyonic excitations are in the center of attention as candidates for topological quantum computation. In this paper, we analyze the ground-state properties of disordered non-Abelian anyonic chains. The resemblance of fusion rules of non-Abelian anyons and real-space decimation strongly suggests that disordered chains of such anyons generically exhibit infinite-randomness phases. Concentrating on the disordered golden chain model with nearest-neighbor coupling, we show that Fibonacci anyons with the fusion rule $\ensuremath{\tau}\ensuremath{\bigotimes}\ensuremath{\tau}=1\ensuremath{\bigoplus}\ensuremath{\tau}$ exhibit two infinite-randomness phases: a random-singlet phase when all bonds prefer the trivial fusion channel and a mixed phase which occurs whenever a finite density of bonds prefers the $\ensuremath{\tau}$ fusion channel. Real-space renormalization-group (RG) analysis shows that the random-singlet fixed point is unstable to the mixed fixed point. By analyzing the entanglement entropy of the mixed phase, we find its effective central charge and find that it increases along the RG flow from the random-singlet point, thus ruling out a ccc theorem for the effective central charge.
We consider the effect of quenched spatial disorder on systems of interacting, pinned non-Abelian anyons as might arise in disordered Hall samples at filling fractions $\ensuremath{\nu}=5/2$ or $\ensuremath{\nu}=12/5$. In one spatial dimension, such disordered anyon models have previously been shown to exhibit a hierarchy of infinite randomness phases. Here, we address systems in two spatial dimensions and report on the behavior of Ising and Fibonacci anyons under the numerical strong-disorder renormalization group (SDRG). In order to manage the topology-dependent interactions generated during the flow, we introduce a planar approximation to the SDRG treatment. We characterize this planar approximation by studying the flow of disordered hard-core bosons and the transverse field Ising model, where it successfully reproduces the known infinite randomness critical point with exponent $\ensuremath{\psi}\ensuremath{\approx}0.49$. Our main conclusion for disordered anyon models in two spatial dimensions is that systems of Ising anyons as well as systems of Fibonacci anyons do not realize infinite randomness phases, but flow back to weaker disorder under the numerical SDRG treatment.
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.
An important challenge in the field of many-body quantum dynamics is to identify nonergodic states of matter beyond many-body localization (MBL). Strongly disordered spin chains with non-Abelian symmetry and chains of non-Abelian anyons are natural candidates, as they are incompatible with standard MBL. In such chains, real space renormalization group methods predict a partially localized, nonergodic regime known as a quantum critical glass (a critical variant of MBL). This regime features a treelike hierarchy of integrals of motion and symmetric eigenstates with entanglement entropy that scales as a logarithmically enhanced area law. We argue that such tentative nonergodic states are perturbatively unstable using an analytic computation of the scaling of off-diagonal matrix elements and accessible level spacing of local perturbations. Our results indicate that strongly disordered chains with non-Abelian symmetry display either spontaneous symmetry breaking or ergodic thermal behavior at long times. We identify the relevant length and timescales for thermalization: Even if such chains eventually thermalize, they can exhibit nonergodic dynamics up to parametrically long timescales with a nonanalytic dependence on disorder strength.
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 $k\ensuremath{\ge}5$, 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.
We construct models of interacting itinerant non-Abelian anyons moving along one-dimensional chains. We focus on itinerant Ising (Majorana) and Fibonacci anyons, which are, respectively, related to SU(2)2{(2)}_{2}(2)2​ and SU(2)3{(2)}_{3}(2)3​ anyons and also, respectively, describe quasiparticles of the Moore-Read and Z3{\mathbb{Z}}_{3}Z3​-Read-Rezayi fractional quantum Hall states. Following the derivation of the electronic large-U effective Hubbard model, we derive effective anyonic ttt-JJJ models for the low-energy sectors. Solving these models by exact diagonalization, we find a fractionalization of the anyons into charge and (neutral) anyonic degrees of freedom---a generalization of spin-charge separation of electrons which occurs in Luttinger liquids. A detailed description of the excitation spectrum can be performed by combining spectra for charge and anyonic sectors. The anyonic sector is that of a squeezed chain of localized interacting anyons and, hence, is described by the same conformal field theory (CFT), with central charge c=1/2c=1/2c=1/2 for Ising anyons and c=7/10c=7/10c=7/10 or c=4/5c=4/5c=4/5 for Fibonacci anyons with antiferromagnetic or ferromagnetic coupling, respectively. The charge sector is the spectrum of a chain of hard-core bosons subject to phase shifts which coincide with the momenta of the combined anyonic eigenstates, revealing a subtle coupling between charge and anyonic excitations at the microscopic level (which we also find to be present in Luttinger liquids), despite the macroscopic fractionalization. The combined central charge extracted from the entanglement entropy between segments of the chain is shown to be 1+c1+c1+c, where ccc is the central charge of the underlying CFT of the localized anyon (squeezed) chain.