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Anyonic Symmetries and Topological Defects in Abelian Topological Phases: an Application to the $ ADE $ Classification

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Anyonic Symmetries and Topological Defects in Abelian Topological Phases: an Application to the $ ADE $ Classification Anyonic Symmetries and Topological Defects in Abelian Topological Phases: an application to the ADE Classification Mayukh Nilay Khan, Jeffrey C.Y. Teo, and Taylor L. Hughes Department of Physics, Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, IL 61801, USA We study symmetries and defects of a wide class of two dimensional Abelian topological phases characterized by Lie algebras. We formulate the symmetry group of all Abelian topological field theories. The symmetries relabel quasiparticles (or anyons) but leave exchange and braiding statis- tics unchanged. Within the class of ADE phases in particular, these anyonic symmetries have a natural origin from the Lie algebra. We classify one dimensional gapped phases along the inter- face between identical topological states according to symmetries. This classification also applies to gapped edges of a wide range of fractional quantum spin Hall(QSH) states. We show that the edge states of the ADE QSH systems can be gapped even in the presence of time reversal and charge conservation symmetry. We distinguish topological point defects according to anyonic symmetries and bound quasiparticles. Although in an Abelian system, they surprisingly exhibit non-Abelian fractional Majorana-like characteristics from their fusion behavior. I. INTRODUCTION AND MOTIVATION edge, or interface phases. In addition we can apply our re- sult to predict when edge theories of certain time-reversal invariant fractional quantum spin Hall states (FQSH) Topologically ordered phases with Abelian anyons are made from time-reversed copies of the ADE FQH states usually considered to be the simplest examples of topo- can be gapped without breaking symmetries. One re- logical order (TO), however recent exciting work has markable result we find is an exact mapping between the shown that the theory is still far from complete. Two well-known triality symmetry of the Lie algebra so(8) notable developments related to our current work are: and the AS of the associated topological state. In fact, (i) the generation of semi-classical defects in Abelian we prove that the AS for these theories are exactly the topological phases that exhibit similar features to non- symmetries of the Dynkin diagrams that represent the Abelian quasiparticles1–14 and (ii) the bulk-boundary ADE Lie algebras. This is not only applicable to a 2D correspondence for topological phases with and without FQH state that carries an so(8) edge algebra, but also the symmetry protection, and the resulting stability of the spin liquid surface state of a three dimensional bosonic gapless edge theories15–21. symmetry protected phase32–35. Some aspects of these two lines of research can be To begin, we need to introduce the well-known K- unified by applying the concept of anyonic symmetry matrix formalism for Abelian TO states. An Abelian (AS). TO phases support an additional AS structure if FQH state is described by an effective Chern-Simons the quasiparticle (QP) fusion and braiding are invariant topological field theory = 1 K α dα in 2 + 1 di- under a set of anyon relabeling operations. This is a 4π IJ I J mensions, where α is anLr-component∧ set of U(1) gauge common feature in many topological states, such the Ki- I fields. The topological state is characterized by the sym- taev toric code22 which has an electromagnetic-duality, metric, integral-valued K-matrix25. QP excitations of and the Abelian (mmn)-fractional quantum Hall (FQH) the theory are labeled as r-component vectors (a, b ...) states23–30 which have a bi-layer symmetry. An element in an integer (anyon) lattice Γ∗ = Zr. Vector addition of the AS group might, for example, switch a particu- a b a b corresponds to QP fusion ψ ψ = ψ + . lar anyon-type between the two layers in bi-layer FQH a × 1 − a at 1a states. The AS is not necessarily a symmetry of the quan- The spin h of a QP ψ is given by 2 K . The topological spin (or exchange statistics) of a QP ψa tum Hamiltonian, but rather a symmetry of the anyon − πiaT K 1a a arXiv:1403.6478v2 [cond-mat.str-el] 25 Nov 2014 content. For example, an AS could permute QP excita- is given by θa = e , and encircling a QP ψ b tions with different energies. In general, a ground state once around another QP ψ gives the braiding phase aT −1b in a closed system will not be invariant under an AS Sab = e2πi K , for = det(K) 1. The topo- operation, and therefore the symmetry can be regarded logicalD spin of the quasiparticlesD | is often|≥ stated in terms 1,31 2πiha p as being weakly broken . However, unlike a classical of the T matrix, Tab = e = δa,bθa. We note that for symmetry-broken phase, the AS may not be associated completely chiral theories, (using bulk boundary corre- with a physical quantity, and cannot be measured by a spondence) the spin ha is the same as the scaling dimen- finite vacuum expectation value of any local observable. sion of the primary fields(corresponding to the quasipar- In this work we construct a class of Abelian bosonic ticles) of the (1+1-d) conformatl field theory (CFT) on FQH states associated with elements of the ADE Car- the edge. tan classification of Lie-algebras and show that they have The QPs that occupy the sublattice Γ = KZr Γ∗ are AS. For these systems the AS can be used to create non- called local and only contribute trivial braiding⊆ phases Abelian twist defects and topologically distinct gapped with all other QPs. Intuitively they are the fundamen- 2 tal building blocks that are “fractionalized” to form the relevant to several important lines of research. The im- topological state; we will enforce that all local parti- portance of the states in the A series for the study of cles be bosonic by requiring the diagonal entries of K FQH hierarchy states has already been pointed out in a be even. Topological information encoded in the non- series of papers as early as Ref. 42 and the structure local braiding and exchange statistics of fractionalized was further clarified by39,40,43. This is particularly rele- m QPs is left invariant upon the addition of local parti- vant for the hierarchy states at filling fractions mp±1 with cles Γ. We can remove this redundancy by labeling symmetry u(1) su(m) 41, where p is an even positive ∈ × 1 distinct QPs with elements of the anyon quotient lattice integer. The factor of su(m)1 is exactly the symmetry of =Γ∗/Γ= Zr/KZr. A the A series of Lie algebras. QPs are electromagnetically∗ charged in the presence of As an example, the second hierarchy state at filling e 2 2 the additional coupling term 2π tI A dαI where A is the fraction ν = 2×2+1 = 5 has external electromagnetic gauge field∧ and e∗ is the unit charge of the fundamental local boson. We will assume 3 1 K = − (1) a symmetric coupling t = (tI )=(1,..., 1) which, for ex- 1 2 ample, is the natural choice in multi-layer systems. The − a ∗ − charge of a QP ψ is qa = e tT K 1a. At zero temper- 1 with charge vector t = and symmetry u(1) su(2) . ature, the ground state is a Bose-Einstein condensate of 0 × 1 local bosons with broken U(1) symmetry/number conser- 3 The third hierarchy state at filling fraction ν = 3×2−1 = vation. Physically, the boson condensate could describe 3 has symmetry u(1) su(3)1, K matrix an anyonic superconductor36,37 where local particles are 5 × Cooper pairs of electrons or perhaps a strongly correlated 1 1 0 cold atomic system. QPs that differ by local bosons are 1 −2 1 , (2) indistinguishable and interchangeable up to the boson −0− 1 2 condensate vacuum. Thus, due to the boson condensate, − the QP charges qa are only defined modulo integral units ∗ 1 of e at zero temperature. This motivates an intuitive and a charge vector t = 0 . The general structure in Zr Zr way to think about the anyon quotient lattice as /K , 0 i.e., the anyons are only defined modulo the local bosons these cases follows this basic pattern. We also note that that make up the lattice KZr. Laughlin states are stably equivalent to su(n)1 states in We are interested in Abelian topological states which 1 the A series as was explicitly noted for the 3 state in Ref. carry chiral Kac-Moody (KM) current algebras at level 1 20. In our work we will not focus on the hierarchy states along their edges. These include a range of FQH states because these have an extra fermionic u(1) (charge) sec- under the Cartan ADE classification of simply-laced Lie tor which complicates the anyonic symmetry analysis to 38–40 algebras . The set of Ar and Dr form infinite se- follow. We instead present the bosonic case first and quences while there are only three exceptional Er=6,7,8. then discuss the strategy to solve the fermionic problem In this article, we consider Abelian topological states in future work. where the K-matrix is given by the Cartan matrix of a Beyond their relevance for hierarchy states, the ADE corresponding (simply-laced) Lie algebra (which has rank states have also been featured in the recent discussion of r). We will henceforth refer to these models as ADE topologically ordered and symmetry protected topologi- states since the Lie algebras with symmetric Cartan ma- cal states. For example, the E8 state has been in focus trices that are suitable to form K-matrices are the An,Dn because it is a bosonic short range entangled phase with and En series from the Cartan classification of Lie alge- no topological order44.
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