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

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QSH) nlgpe hssaogteinter- the along phases gapped onal πi ysrrsnl xii non-Abelian exhibit surprisingly ey u falAeintplgclfield topological Abelian all of oup triality h n nycnrbt rva riigphases braiding trivial contribute only and a θ T a a nlAeintplgclphases topological Abelian onal K faQP a of = − T α 1 ab b I e for , san is ymtyo h i algebra Lie the of symmetry πi h = a a T e stesm stesaigdimen- scaling the as same the is L K 2 K D ψ πih r so ry, − = -matrix ψ r 32–35 a cmoetstof set -component ψ = cmoetvcos( vectors -component 1 a 8 deagba u lothe also but algebra, edge (8) oia hss an Phases: logical a a b sgvnby given is 4 n nicigaQP a encircling and , × 1 p π = ie h riigphase braiding the gives K . ∗ ψ | n charge and l δ det( IJ b a = 25 , b = α Pecttosof excitations QP . θ Z I K a ψ r ent htfor that note We . ∧ ADE . ) a ≥ | + dα etraddition Vector 2 1 b a K . J t .Tetopo- The 1. K Z Q states FQH n2+1di- 1 + 2 in U r − 1 gauge (1) ⊆ 1 a a , The . Γ b so ∗ . . . are (8) K ψ ψ a a ) - 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. The so(8) state, which lies in the bras. This construction ensures the presence of a Kac- D series, exists on the surface of a 3D bulk Symmetry Moody algebra corresponding to the same Lie algebra at protected topological(SPT) phase protected by time re- level 1 at the edge of the system, and that the modular S versal symmetry33–35. As we will soon show, the so(8) matrix of the bulk topological state is the same as that of state has very special properties in the context of anyonic 38 the corresponding affine Lie Algebra at level 1. Strictly symmetry that could be probed if such a bulk 3D topo- speaking, the presence of a sector whose K (sub-)matrix logical phase were discovered. Additionally, as mentioned is identical to the Cartan matrix of the Lie algebra (i.e. further below, the D series forms half of the sixteen- we can identify a sub-matrix of the full K-matrix which fold classification scheme first predicted by Kitaev for is the Cartan matrix of the Lie algebra) is enough to de- 2D topologically ordered states1, and further considered fine a set of currents which obey Kac-Moody algebra at in recent work on interacting 2D45,46 and 3D topological 38–41 level 1 at its edge. . Examples of such K-matrices phases47–49. are given in Eqs. (1),(2). However, we will not explicitly From a mindset of pure convenience, the ADE states deal with such states in this manuscript. also have the advantage that their anyonic symmetries In the remainder of this section we will motivate the can be fully classified using powerful results from the reasons for studying the ADE fractional quantum Hall mathematics of Lie algebras as we will see, whereas iden- states. In fact, these fractional quantum Hall states are tifying the exact anyonic symmetry group is still an open 3 problem for states with generic K-matrices/anyon lat- equivalent to K E , and K E is stably − su(3) ⊕ 8 − su(4) ⊕ 8 tices. With these motivations in mind we will now pro- equivalent to Kso(10) σx σx σx. ceed to the discussion of anyonic symmetry and its con- ⊕ ⊕ ⊕ sequences. 1

e2 e2

II. ANYONIC SYMMETRIES OF THE 120 ◦ ADE-SERIES 1 e 1 e

Throughout this article we will use two explicit examples of ADE states to illustrate our results: A = su(3) 2 (a) Inner Automorphism (b) Inner automorphism and D4 = so(8) which are described by where the lattice is corresponding to rotated through 120◦ reﬂection about the 2 1 1 1 horizontal dotted grey 2 1 12− − 0− 0 K = ,K = . line su(3) 1− 2 so(8)  −10 2 0  − −10 0 2  −  e2 e2  (3) 60 ◦ The su(3) state has 3 QPs: 1 = (0, 0), e = ( 1, 1) 1 e 1 e and e2 = (1, 1), which form the anyon quotient− lat- Z− 2 2 tice su(3) = 3 with fusion e e = e and e e = 1 up to localA bosons. The QPs have× neutral electric× charge, 2πi/3 (c) Outer Automorphism (d) Outer automorphism but have non-trivial spin θe = θe2 = e and braiding −2πi/3 2πi/3 where the lattice is corresponding to √ √ 2 2 √ 2 phases 3See = 3Se e = e and 3See = e . rotated through 60◦ reﬂection about the The anyon quotient lattice of su(3) is two dimensional vertical dotted grey line and can be easily illustrated in 2D. It is shown in Fig. 1 where the white, red and blue circles refer to the quasiparticles 1, e and e2. The inner product between vectors FIG. 1. Anyon lattice of su(3) with inner and outer automor- a, b of the lattice is given by aT K−1 b. Thus, the in- phisms. The white, red and blue circles refer to the distinct su(3) 2 ner product between vectors (1, 0)( e) and (0, 1)( e2) quasiparticles 1, e and e respectively in the anyon quotient lattice Z2/KZ2. (a) and (b) are examples of inner automor- is (1, 0), (0, 1) = 1 prompting the≡ representation≡ as a h i 2 phisms where anyon labels (colors of the circles) are preserved triangular lattice. whereas (c) and (d) the outer automorphisms exchange e and The so(8) state has 4 QPs: 1 = (0, 0, 0, 0), e = e2 (blue and red circles) (0, 1, 0, 1), m = (0, 1, 1, 0) and ψ = (0, 0, 1, 1) form- ing− the anyon quotient− lattice = Z− Z with Aso(8) 2 × 2 fusion rules e2 = m2 = 1 and ψ = e m up to lo- × cal bosons. They have neutral electric charge, carry Anyon fusion lattice r r Anyon labels fermionic spin θe = θm = θψ = 1, and braiding phases = Z /KZ − A r+1 r 2See =2Smm =2Sψψ =1, and 2Sem =2Smψ =2Sψe = Ar Zr+1 1= e ,e,...,e 1. Thus, the three non-trivial anyons are fermions. Z Z − D2n 2 2 1,e,m,ψ = e m The anyon labels and fusion rules for general ADE × 4 2 ×3 D2n+1 Z4 1= e ,e,e ,e states are listed in Table I and Table V. In the Lie alge- Z 3 2 Zr E6 3 1= e ,e,e bra language the lattice Γ = K of local bosons is the Z 2 root lattice, while the anyon lattice Γ∗ = Zr is called the E7 2 1= e ,e weight lattice20 and is dual to Γ under the bilinear prod- TABLE I. Quasiparticle labels of the A = su(r + 1), D = uct a, b = aT K−1b. We have omitted the E state with r r 8 so(2r) and E6,7 Abelian topological states at level 1. trivialh topologicali order ( = 1)44. We note in passing D that there is an eightfold periodicity in the Dr series with 20 rank r 3 such that the Dr state is stably equivalent to We will now construct the anyonic symmetry groups ≥ 50 the Dr+8 theory up to an additional E8 state , and both for the ADE states and discuss applications of this re- theories have identical anyon fusion and braiding con- sult in the context of symmetry enhanced topological tent. Taken together with the non-simply laced Br series phases and semi-classical twist defects. One requirement (which are non-Abelian at level-1 and will be discussed of an AS is that its operation commutes with the modular elsewhere50), they form a class of topological states with S and T transformations of topologically ordered (TO) sixteenfold periodicity which matches the structure found states on a torus. For Abelian theories in the K-matrix in Refs. 1, 51, and 52. Also, from the braiding phase and formalism, a unitary anyon relabeling symmetry can be spin of the quasiparticles we expect that KE7 σx is sta- represented by a unimodular (integral entries, unit de- bly equivalent to K E ,K σ σ⊕ is stably terminant) matrix M that leaves the K-matrix invariant − su(2) ⊕ 8 E6 ⊕ x ⊕ x 4

T Ar = su (r + 1) under MKM . This forms a group of automorphisms Dr = so (2 r) ...... Aut(K)= M GL(r; Z): MKM T = K . (4) ∈

Since exchange and braiding are completely determined E6 by the K-matrix, the modular transformations are un- D4 = so (8) changed under the anyon relabeling,

SMaMb = Sab,TMaMb = Tab (5) FIG. 2. Mirror symmetry of Dynkin Diagrams of Ar, Dr,E6 and the fusion rules remain unaltered as a direct conse- and S3 = Dih3 symmetry of D4. quence of the linearity of M or the Verlinde formula53 in general. Outer(K) anyonic symmetry action Out of the full group of automorphisms there are cer- 1 Ar, D2n+1,E6 Z2 σ : e e− tain trivial symmetry operations M0 that only rearrange ↔ D2n, n = 2 Z2 σ : e m local particles without changing the QP types. That is, 6 ↔ σψ : e m these operations do not change the anyon equivalence D4 S3 = Dih3 ↔ Zr ρ : e m ψ e classes [a] = a + K since they rotate the anyon → → → lattice vector up to a∈ local A particle in Γ = KZr. M 0 TABLE II. Symmetry action Outer(K) on anyon labels. forms a normal subgroup of Aut(K) called the inner automorphisms

Inner(K)= M0 Aut(K) : [M0a] = [a] . (6) The only ADE state with more than just a Z AS { ∈ } 2 group is the D = so(8) state which has a triality sym- To construct the relevant AS group we must remove this 4 metry. The AS group is S3 = Dih3 which is the permu- redundancy of trivial symmetry operations by quotient- tation group of three elements generated by “reﬂection” ing, to generate the group known as the outer automor- and threefold “rotation” in its Dynkin diagram. This phisms group is non-Abelian, contains a total of six elements, Aut(K) and the generators are represented by Outer(K)= . (7) Inner(K) 1000 1000 Thus, Outer(K) is the AS group of the Abelian topolog- 0100 0001 ical phase characterized by K. If the topological state is Mσ =   ,Mρ =   (8) ψ 0001 0100 strictly charge conserving, a charge compatible AS ele-      0010   0010  ment must keep the charge vector Mt = t ﬁxed which     will ensure the charge of a QP is unchanged, qMa = qa.    

For a U(1)-breaking bosonic state at zero temperature, which act on four dimensional anyon lattice vectors. Mσψ the charge compatibility condition can be relaxed mod- interchanges e m but fixes ψ up to local boson. Mρ ↔ ulo the image of K so that the fractional charge is pre- rotates e m ψ e which is an example of a three- → → → served by the symmetry only up to units of e∗ through fold symmetry operation. All ADE symmetry operations the addition of local particles. Imposing these charge can be chosen to strictly preserve U(1) symmetry and conservation conditions will further restrict the group of leave the charge vector unchanged so we have no fur- automorphisms in Aut(K). ther restrictions. In fact, to our knowledge this is the Remarkably, in an Abelian state in the ADE classi- only known example of a non-Abelian anyonic symmetry fication, Outer(K) is exactly the group of outer auto- group. morphisms of the simply-laced Lie algebra, and coincides Now that we have introduced the basic idea behind with the symmetry group of the Dynkin diagram54,55 AS we will move on to discuss the consequences. The (see Fig. 2). The explicit AS actions are listed in Ta- remainder of the article is organized as follows: (i) we ble II. We present a brief discussion of the outer auto- first show/review that the generic consequence of a non- morphisms of Lie algebras Appendix C where we moti- trivial AS group is the existence of topologically dis- vate this connection in more mathematical detail. We tinct gapped phases along quasi-one dimensional inter- omit E7 from this list because it has the anyon content faces/edges with identical, but oppositely propagating, 3 1,e , h1 = 0; he = 4 (see Table VIII ). Thus there are no edge modes which chiral central charge c− = 0 (see { } 19 anyonic symmetries for E7. For A2 = su(3) the Dynkin Fig. 3) ; (ii) since counter-propagating modes at an in- diagram has a Z2 “reflection” AS which is represented terface can be mapped to a single edge of a fractional by the Pauli matrix Mσ = σx that acts on the rank two quantum spin Hall phase we can apply our results to de- anyon lattice vectors and simply interchanges the QPs e termine the stability of the edges of ADE FQSH states; and e2 while leaving the vacuum fixed. Examples of ex- (iii) finally we will use the AS group to determine the plicit inner and outer automorphisms as represented on distinct set of semiclassical twist defects in ADE FQH the anyon lattice are shown in Fig. 1. states and some of their properties. 5

III. ANYONIC SYMMETRY AND DISTINCT The backscattering terms simultaneously pin the boson GAPPED INTERFACE PHASES vacuum expectation values

φL + M φR =2π(K−1) m , for m Zr (11) h I JI J i IJ J ∈ Ltop 1 R R = KIJ ∂xφI ∂t φJ Ma edge 4π as the operators for each value of I mutually commute. LM This pinning effectively condenses the local bosonic QP (M) L R − ′ ′ gI cos £KIJ (φJ + MJ J φJ )¤ pairs m − 1 Ma † a −i(a·φL+(Ma)·φR) −2πiaT K 1m Lbottom = − K ∂ φL∂ φL a ψ ψ = e = e edge 4π IJ x I t J R L h i D E (12) FIG. 3. Local boson tunneling in (10) gives a gapped interface phase M represented by a branch cut (dashed wavy blue line) along the interface. The gapped interface state is then and aL parallel quasiparticle string (solid blue line). Passing characterized by this QP pair condensation, and can be anyon changes type a Ma and accumulate a crossing phase diagrammatically represented by a branch cut associated → Sam in (12). with M that is decorated with a parallel QP string m D localized near the cut (see Fig. 3). The branch cut it- We begin our study with a quasi-one dimensional in- self will change the anyon type of a passing QP from a Ma, while the attached m string contributes the terface with identical but oppositely propagating edge → modes. The gapless edge modes on an interface have additional U(1) crossing phase required from Eq. (12). Such QPs localized at defects have also been studied by the (1 + 1)-d bosonic Lagrangian density edge = 1 L in Refs. 19, 56–59. Two anyonic symmetry matrices M 4π KIJ ∂xφI ∂tφJ , where the K-matrix is identical to that ′ in the bulk, and QPs are expressed as vertex operators and M correspond the same gapped interface phase if ψa = eia·φ. In limit of weak coupling between oppo- the backscattering terms pin and condense the same set site sides of the interface, the chiral gapless edge modes of bosonic QP pairs, i.e., the gapped edge phases are a ′a Zr along the opposite sides of the interface in question are identical if M = M modulo K . Gapped interface described by the boson Lagrangian density phases are therefore in one-to-one correspondence to the group Outer(K) of anyonic relabeling symmetries defined 1 ′ ′ 1 in (7). We further note that, although we focus on charge top + bottom = Kσσ ∂ φσ∂ φσ + tσǫµν ∂ φ σA Ledge Ledge 4π IJ x I t J 2π I µ I ν conserving edge tunneling terms in this work, our formal- (9) ism also applies to gapping terms which describe super- conducting pairing. This includes terms like where σ =0, 1= R,L labels right and left moving modes, R L (M) ′ ′ R − L φ (φ ) are the boson fields living along the top (bot- δ = g eiKIJ MJ J φJ e iKIJ φJ + h.c. I I ′ ′ M I σσ σ σσ L − tom) edge, and KIJ = ( 1) δ KIJ (see Fig. 3). We I σ L −R X assume that t = t = t = t. Corresponding to each (M) L R = g cos KIJ φ MJ′J φ ′ quasiparticle vector a in the bulk, the vertex operator on − I J − J I a ia·φR a −ia·φL X the right(left) edge is ψR = e (ψL = e ). This a a Ma a convention ensures that the charge of both ψR and ψL is which condense local bosonic pairs ψR ψL on the edge. e∗aT K−1t . Remarkably, given any element M of the AS group, the interface can be gapped by a corresponding set of IV. APPLICATION I: STABILITY OF ADE backscattering terms BOSONIC FRACTIONAL QUANTUM SPIN HALL STATES (M) L R δ = g cos K φ + M ′ φ ′ (10) LM − I IJ J J J J I In this section we argue that the Lagrangian for the in- X terface is identical to that of the edge of a bosonic quan- where repeated indices J and J ′ are summed over. This R tum spin hall state. This enables us to classify gapped iK M ′ φ ′ describes tunneling between the local boson e IJ J J J edges of bosonic FQSH using anyonic symmetries. We − L on the top edge and e iKIJ φJ on the bottom. We as- will outline the action of time reversal operators on the sume that the representative matrix M of the AS group edge degrees of the quantum spin hall state (with more element is charge conserving so that the tunneling term details in the appendix). To understand the stability of preserves boson number, however, this condition can be the FQSH edge states we must carefully analyze whether relaxed if the global U(1) charge symmetry is broken as the gapping terms we add to destabilize the gapless de- discussed earlier. grees of freedom on the edge explicitly or spontaneously In the strong-coupling limit the terms in Eq. (10) lead break TR symmetry. Such questions have been the focus to a gapped interface phase associated with the symmetry of recent works like60,61. Once we understand the sym- M which we denote by the symbol for convenience. metry properties of the possible gapped edge phases we LM 6

† can determine the conditions under which the edge can for ψx andx ¯ ψx . The A = su(3) R/L R/L ≡ R/L 2 be destabilized without breaking any protective symme- Z tries. state has AS Outer(Ksu(3) ) = 2 generated by 1 and To approach the analysis, we first note that the two σ. These generators correspond to the gapped inter- edge theories coupled at an interface in the previous sec- face/edge phases 1 and σ with QP pair conden- L2 2 L 2 2 tion (c.f. Eq. (9)) can be regarded as a single-edge of sates 1L1R,eLeR,eLe R and 1L1R,eLe R,eLeR re- ′ { } { } a doubled system with K-matrix Kσσ = K ( K) if spectively. To test if these phases break TR explic- ⊕ − itly we calculate −1 = 1 1 ,e e ,e2 e2 and the topological states on the two sides of the interface T L1T { R L R L R L} 18,19 −1 = 1 1 ,e e2 ,e2 e . To compare with are folded on top of each other . The K matrix of a T LσT { R L R L R L} L1 and σ we take the Hermitian conjugate, and we see K 0 L † † such a theory looks like with bosonic degrees that −1 = ( ) and −1 = ( ) . While this 0 K T L1T L1 T LσT Lσ − ! might initially seem problematic for TR preservation, we † φR note that and actually represent the same set of of freedom on the edge represented by , where φR condensedL bosons.L We see by using equations (11) and φL! (12) that and φL are counterpropagating edge modes on the same interface. We see that this Lagrangian is the same as (9). a † a a a † ψM ψ =0 = ψM (ψ ) =0. This represents a fractional quantum spin Hall state of R L 6 ⇒ R L 6 60,62 bosons . This identification enables us to characterize Explicitly,D we can seeE this by takingD the HermitianE con- the gapped edges of these states using anyonic symmetry. ′ ′ jugate of equation (12) σσ σσ The time reversal (TR) matrix is TIJ = (σx) δIJ act- a † a † a a aT −1m ing on the spin-momentum locked σ = , = R,L degree ψM ψ = ψM (ψ )† = e2πi K . of freedom. The TR operator is anti-unitary↑ ↓ and acts R L R L according to T h i D E Thus, we see that we can safely add both δ M and δ T M ′ ′ ′ ′ L L −1 σ σσ σ −1 σσ σ (i.e., the terms corresponding to the time-reversed part- φI = TIJ φJ + π(K )IJ χJ (13) top bottom T T ner of the δ m) to the edge Lagrangian edge + edge L Z L L R2r 60 in this case(and in all cases with 2 anyonic symmetry for some TR vector χ = (χ↑, χ↓) . For our classes of ADE FQSH systems (and∈ many others) we as we show in the next subsection) while staying in the same phase (that is condensing the same set of bosons) . show in Appendix F 2 that χ can be set to 0 and hence, top a a bottom −1 L/R R/L −1 Hence, edge+ edge +δ M +δ T M is TRI. Thus, we see φI = φI . Thus, ψR = ψL. Indeed L L L L T a T a T T that when the time reversed (i.e., Hermitian conjugate) since ψR and ψL are time-reversed partners, their T − a a ∗ iπaT K 1b phase condenses the same bosons as the original one, i.e. and S matrices obey θR = (θL) = e and −1 it does not break time reversal explicitly. These state- ab ab ∗ iπaT K b SR = SL = e . Also, under time reversal ments pertain to the explicit breaking of time-reversal. the gapping term transforms as We will deal with spontaneous symmetry breaking below where we check that the expectation value of the con- δ = −1δ T M M densate and its TR partner are the same. As we will L T L(MT) R ′ L = g cos KIJ φJ + MJ J φJ′ . show the condition for the absence of spontaneous time- − I I reversal breaking will be Eq. (16). Before we discuss this, X Generically, since the edge is non-chiral, one can desta- let us explicitly consider the phases of the so(8) state. bilize the edge and open a gap via, for example, con- The D4 = so(8) state has six symmetry opera- densing bosons on the edge. However, we are not only tors in Outer(Kso(8)) = S3, and these correspond interested if a gap can form, but what symmetries the to six gapped phases. For example the trivial one resulting gapped state breaks or preserves, e.g., some 1 condenses 1L1R,eLeR,mLmR, ψLψR , the twofold oneL condenses{ 1 1 ,e m ,m e , ψ} ψ , and the gapped phases may break time-reversal and some may Lσψ { L R L R L R L R} threefold one condenses 1 1 ,e m ,m ψ , ψ e . preserve time-reversal. For our problem, the edge con- Lρ { L R L R L R L R} † The Lagrangian subgroup for preserves TR (anal- Ma a Lσψ densate is formed from the QP pairs ψ↑ ψ↓ , and is ogous to for su(3)) while that for breaks Lσ Lρ a maximal collection of mutually local bosons, known as it upon adding the gapping terms. We con- a Lagrangian subgroup17 in the TR symmetric doubled sider the example for explicitly, −1 = Lρ T LρT anyon lattice system ↑ ↓. 1 1 ,e m ,m ψ , ψ e . Taking hermitian conju- A ⊗ A { R L R L R L R L} gate we see that 1 1 ,m e , ψ m ,e ψ = −1 = { L R L R L R L R} Lρ 6 ρ. Hence the edge phase determined by the threefold A. Explicit T-Breaking on the edge of bosonic Lsymmetry explicitly breaks time reversal symmetry, be- FQSH systems cause the time-reversed phase condenses a different set of bosons. Let us look at a few examples before making a gen- After the intuition gained from examining these cases eral statement. To simplify notation let x stand we will now prove that the gapped edge phase does R/L LM 7 not break TR explicitly if the Lagrangian subgroup is TR automorphisms. In this case we require M = M −1, thus invariant, i.e., the TR of the gapped edge describes the M 2 = 1 up to inner automorphisms. same phase, which is the case if and only if M 2 = 1 (up to Thus if the gapped edge of the bosonic quantum spin inner automorphisms). Microscopically this arises from hall effect is in the phase M and preserves TRI it must the fact that if M 2 = 1 then the sine-Gordon gapping satisfy M 2 = 1. We, haveL already seen the example of terms in (10) can be made TR symmetric by adding time- the three fold symmetry ρ of so(8), which acts on the reversed counterparts that pin the same set of QP pairs. anyon labels by sending (e,m,ψ) (m,ψ,e) . It satisfies 3 → This includes all symmetries in the ADE states except ρ = 1. δ T ρ however represents the inverse threefold the threefold symmetry ρ of so(8). One cannot write symmetry Lρ−1, by equation (15). Since ρ−1 = ρ this down a time-reversal invariant Lagrangian that is gapped breaks TRI explicitly as mentioned above. 6 by the three-fold symmetry group element. We can continue our microscopic treatment by expand- ing the edge Lagrangian density in (9), B. Spontaneous T-breaking on the edge of Bosonic FQSH systems 1 1 = K ∂ φR∂ φR K ∂ φL∂ φL Ledge,bare 4π IJ x I t J − 4π IJ x I t J ∗ ∗ This concludes our discussion of the explicit breakdown e µν R e µν L of TRI. There is a further complication in that, even + ǫ tI ∂µφI Aν + ǫ tI ∂µφI Aν . 2π 2π though the Lagrangian is TR invariant, the ground state condensate itself can break TR spontaneously. As such, edge is obviously TRI. But, now we need to understand L even when a gapped interface may pin the same QPs as what happens when we add the gapping term δ M in (10) ; we reproduce it here for convenience L its TR partner, the expectation value of the condensate in the time reversed phase may be different than that in (M) L R the original phase. In order to prevent the spontaneous δ M = g cos KIJ φ + MJ′J φ ′ . L − I J J breaking of TR symmetry we must constrain the conden- I X sate phase as follows. First, let us specialize to gapped 2 δ M breaks TRI, to restore it we add its time reversed edges where M = 1 so that TR is not broken explic- partnerL itly. Time reversal (13) operates on the QP condensate † ψMa ψa along a fractional quantum spin Hall edge by δ = −1 (δ ) R L LT M T LM T (M) R L a † a a R a L = g cos KIJ φ + MJ′J φ ′ . −1 M −1 −i((M )·φ + ·φ ) − I J J ψR ψL = e I T T T T X h a ·φL a·φiR h a ·φR a·φL i = ei((M ) + ) = ei((MM ) +M ). The full Lagrangian + δ + δ is time Ledge,bare LM LT M reversal invariant, but we also need to make sure that Using Eq. (11) we find δ and δ describe the same gapped phase, i.e. both LM LT M the terms condense the same set of bosonic quasiparticles. a † a aT −1 m −1[ ψM ψ ] = e2πi K (M ), This is equivalent to the statement that T R L T D E R ′ L Z a † a KIJ φJ + MJ J φJ′ =2π (p1) ; (p1) M I I ∈ which should be compared to ψR ψL = L R − = K φ + M ′ φ ′ =2π (p ) ; (p ) Z. −2πiaT K 1m ⇒ IJ J J J J 2 I 2 I ∈ e from Eq. (12). Thus,D to preserve E time- m Next we find the conditions when the above is true (we reversal we must have satisfy use vector notation from here on and K and M are ma- m Mm mod Km (16) trices) ≡− spontaneously R T L otherwise the ground state will break TR . K φ + M φ =2πp1 (14) For all of the ADE cases, we have examples of M which obey M 2 = 1, and conserve charge Mt = t (Eqs. (8) and using MKM T = K we see and (B4)). We can also satisfy Eq. (16) with the case when m = 0. Thus in all the ADE FQSH cases, we L T −1 R K φ + M φ =2πMp1. (15) can gap out the edge even if we demand TRI and charge conservation. Now, drawing upon the correspondence between gapped In fact, using the criterion in Ref. 60, one can show interface phases and AS outlined before, this implies that in general that whenever χ = 0, the edge can be gapped if δ M leads to the gapped interface which acts on the without breaking TR or charge conservation. In particu- L −1 2 anyons labels a Ma, δ T M leads to a M a. lar these gapped edges represent twofold defects M = 1. Further, as remarked→ before,L two anyon symmetries→ M However, there exist gapped edges which do break time and M ′ lead to the same gapped interface if Ma = reversal symmetry explicitly while conserving charge as M ′a + KZr a, which is to say M = M ′ up to inner well, e.g., the threefold defect ρ for so(8). ∀ 8

(a) heuristic way to understand this constraint is that if the L emitted quasi-particle n is changed to Mn by the twist LM M defect then the defect itself must be able to absorb the µ + difference between n and Mn to give a physically consis- L1 m tent result. As we will see, this means that the defects themselves can be thought of heuristically to “contain” (b) (c) m + (M− 1) n internal structure that increases their quantum dimension and allows them to compensate for this emission- µ µ + reabsorption process. µ + + = Mn a n Take the the Ar = su(r + 1) state for example. By Ma m m solving the redundancy constraint we can see that the non-redundant species label only represents the parity p FIG. 4. (a) Twist defect µ (red cross) as a disclination at of an e QP bound to a twofold defect σ. However, the the end of a gapped interface M (dashed curvy line) and parity (i.e., even or oddness of p) is only well defined L dressed with a quasiparticle string m (solid line). (b) Defect only when the particular Ar has well defined even and action on a circling quasiparticle a Ma. (c) Redundancy odd QP sectors, i.e. when there are even number of QPs, → in defect-quasiparticle fusion. i.e., when r is odd. Thus Ar, when r is even, only has one type of defect species (equivalent to the bare defect), and when r is odd there are two types, an even and odd V. APPLICATION II: ADE TWIST DEFECTS defect σ0 and σ1. It is interesting to note that when r FROM ANYONIC SYMMETRY is even the defect internally harbors all of the distinct QPs leading to a quantum dimension of √r + 1 while In addition to gapping symmetry protected edge for r odd the burden is shared between the two different states, we can apply our results to classify twist defects defect species which each carry (r + 1)/2. For the D4 = associated with an AS element M, i.e., a topological so(8) state, while each twofold defect σ , σ , σ comes p e m ψ point defect that changes the anyon type of QPs that with two distinct species labels (e.g., (σe)0,1), there is no travel around it according to a Ma (see Fig. 4b). As non-trivial species label for threefold defects ρ, ρ as their → discussed above, twist defects can essentially be consid- symmetries mix even and odd QP sectors. We notice ered as a domain wall sandwiched between two distinct that in the cases when charge is fractionalized (i.e. the gapped interface phases19. They can also appear at- A states for r odd, and D states for r 2, 3 mod 4), the r r ≡ tached to dislocations or disclinations (see Fig. 4a) where Z2 species label also counts the fractional electric charge non-trivial boundary conditions (i.e. local boson tunnel- in units of e∗/2 carried by the defect (c.f. Ref. 14). ing) are applied at the extra inserted half-layer or wedge respectively. The classical defect interpretations are par- defects and species d ticularly relevant when the TO intertwines with (liquid) A2n σ = σ e √2n + 1 crystalline order where a broken discrete spatial symme- × 2 13,22,63,64 A2n+1 σ0 = σ0 e , σ1 = σ0 e √n + 1 try matches an AS . × × According to Eq. (12) a twist defect µ can bind a QP m D2n σ0 = σ0 ψ, σ1 = σ0 e,m √2 × 2 × { } (see Fig. 4a,c). The defect-QP composite is summarized D2n+1 σ0 = σ0 e , σ1 = σ0 e √2 × × by a defect species label λ where µ µ m, where µ D4 ρ, ρ 2 λ ≡ 0 × 0 is a bare defect with no attached QP, and means fu- (σψ)0 =(σψ)0 ψ, (σψ)1 =(σψ)0 e √2 × × × sion. Because of the intrinsic nature of the twist defect, (σe)0 =(σe)0 e, (σe)1 =(σe)0 m √2 there is an important consistency constraint that must × × (σm)0 =(σm)0 m, (σm)1 =(σm)0 ψ √2 be satisfied when determining the possible attached QP × × E6 σ = σ e √3 types, i.e., there are redundancies in defect-quasiparticle × fusion. For instance, as shown in Fig. 4c, when the QP m TABLE III. Defects and species for the A D E states, and is fusing with the defect µ it can emit a QP n that travels their quantum dimensions d. − − around the twist defect and is re-absorbed after (possibly) changing its anyon type. If the twist defect does not Defect species can also undergo a mutation by absorb- transform n then it is simply re-absorbed without issue, ing or releasing a quasiparticle. This can be microscopi- however if n is transformed then a physical consistency cally controlled, for example, by adding phase parameters condition must be satisfied that leads to a redundancy in the sine-Gordon terms in (10), and letting them wind in the possible defect species types. The constraint that adiabatically by multiples of 2π. The mutation process must be satisfied is is summarized by the defect-quasiparticle fusion µ m = µ (m + (M 1)n) (17) p × × − σλ+p = σλ e (18) and the defect species labels are thus classified by equiv- × alence classes λ = JmK [m] modulo (M 1) 14. A where λ Z = 0, 1 for the A and D states. ≡ − A ∈ 2 { } 2n+1 2n+1 9

For the D2n state, point out the possibility for (conﬁned) excitations with non-symmetric fusion rules. Though they do not give any σ = σ e = σ m, σ = σ ψ. (19) λ+1 λ × λ × λ λ × explicit examples, it would be interesting to compare the Or for the D4 state, σψ obeys (19) and the other two underlying mechanisms in future work. twofold defects σe and σm follow (19) up to cyclic per- Another interesting property of the D4 = so(8) is that mutation of quasiparticles. the three-fold defects can be used to form a kind of twist The anti-partner of a defect µ with symmetry M is vortex defect. This defect can be thought of as a point a defect µ with symmetry M −1 and reciprocal species where (at least) three distinct gap edge phases meet at label λ = λ. As MM −1 = 1, a QP will not change a point18,19, whereas, until now we have only considered type when− dragged around the µ µ pair. This gives a point defects at the junction between two lines with dif- Wilson measurement of the overall× QP type associated ferent mass terms. We show an example of this “vortex- with the defect pair, known as an Abelian fusion chan- like” defect in Fig. 5a. This defect is equivalent to the nel. These channels are restricted only by defect species since QP parity (or half e∗ charge) is a conserved property. Fusions of twofold defect pairs in the ADE states are summarized in Table IV. They ﬁx the quantum dimensions of defects (shown in Table III) by identifying (a) (b) the total dimensions on both sides of the fusion equations. We see that fusion of two defects can give rise to a large number of types of QPs, and these possible internal states of two defects are precisely the same structure that allows them to compensate for the QP attachment emission-re-absorption constraint discussed above.

Fusion rules FIG. 5. (a)Three ρ branch cuts emerging from the center to 2n A2n σ σ =1+ e + . . . + e create a vortex-like defect (b) A quasiparticle string e being × 2 2n A2n+1 σ0 σ0 = σ1 σ1 =1+ e + . . . + e absorbed by the defect configuration × × 3 2n+1 σ0 σ1 = e + e + . . . + e × fusion of three ρ defects. The fusion of three ρ defects is D2n σ0 σ0 = σ1 σ1 =1+ ψ × × also the same as considering the fusion of ρ and 2ρ from σ0 σ1 = e + m × 2 (21). Combining Eqs. (21) and (20) we find D2n+1 σ0 σ0 = σ1 σ1 =1+ e × × 3 σ0 σ1 = e + e ρ ρ ρ = ρ 2ρ = 2(1+ e + m + ψ). (24) × 2 × × × E6 σ σ =1+ e + e × We can check this by matching the quantum dimensions: TABLE IV. Defect pair fusion in the A D E states. the quantum dimension of ρ is 2 and the quantum di- − − mensions of the Abelian particles on the right add up to the correct value. Furthermore, we see that the defect There is more structure in the D4 = so(8) state due to configuration makes it possible to have a quasiparticle the S3 triality symmetry. (σψ)λ (σψ)λ′ obeys the same × strings of all four types, i.e. 1, e, m, ψ, to be absorbed by fusion rules as the D2n states in Table IV, while the two the defect. In Fig. 5 we show an example of an e particle other twofold defects σe, σm satisfy similar rules up to a cyclic permutation of quasiparticles. The threefold defect being absorbed at the defect site, similar configurations ρ annihilates its anti-partner ρ and gives can also be drawn for m, ψ by permuting the particle labels. This explains the existence of the different Abelian ρ ρ = ρ ρ =1+ e + m + ψ. (20) fusion channels. × × We can also explain the factor of 2 in Eq. 24 by draw- This implies the quantum dimension d = d = 2 and ρ ρ ing extra Wilson loops which form an anti-commuting al- the degenerate fusion of the pair gebra, thus explaining the doubling of the Hilbert space. ρ ρ =2ρ, ρ ρ =2ρ. (21) This doubling already happens at the fusion of two ρ × × defects as seen in Eq. 21. The extra structure already The non-Abelian symmetry group S3 results in non- present for two ρ defects allows us to draw additional Wil- commutative fusion rules son loops which anticommute. This has been illustrated σ σ = ρ, σ σ = ρ (22) in Fig. 6. The key point to note there is that the Wilson m × e e × m σ ρ = ρ σ = (σ ) + (σ ) . (23) loops involve only two ρ defects. Each of the blue and e × × m ψ 0 ψ 1 green loops exist independently and are separate Wilson All other fusion rules can be written down by cyclic per- lines. There are 4 intersections between the blue and mutation of the quasiparticle labels e,m,ψ. It is interest- green loops, and since 2S = 2S = 1 we pick up a eψ em − ing to note here that anyon condensation induced transi- phase of -1 three times (the grey dots) and Smm = 1(the tions in topological field theories studied in Ref. 65 also black dot) once. Thus we pick up a net phase of 1 when − 10 the two loops are exchanged. This explains the doubling additional basis transformation W ). L describes a the- of the Hilbert space. ory with equal numbers of right and left movers so that the chiral central charge is unchanged. In these cases the edge theory is no longer described by K but by K′ . The defects and gapped edges that can exist on the edge of K′ can again be analyzed using anyonic symmetry as defined in this article and will be comprehensive for that edge theory. We would like to emphasize that simply stating the anyonic symmetry in terms of the braiding matrix was not the goal of this article, but instead the focus has been on realizing distinct gapped edges and defects associated with that anyonic symmetry. Hence, once the edge theory has been fixed by the K matrix, we do not need to worry about extra anyonic symmetries that might arise from stably equivalent matrices. Any FIG. 6. The quasiparticle labels for each loop are marked by gapped edge that can be written down for the theory their respective colors. The intersections which give a phase must involve an expression of the form in equation (10), of 1 upon exchange are marked by grey circles. The inter- and all such terms can be fully classified by looking at section− with a phase of 1 is marked by a solid black circle. the symmetries of the K matrix alone. While we mentioned the connection between the fractional quantum Hall hierarchy states and the ADE se- VI. DISCUSSION, OPEN QUESTIONS, AND ries, the cases studied in this paper do not refer to states CONCLUSIONS with a local fermionic sector which is more relevant for experimentally observed filling fractions in two dimen- While we have not considered it here, we note that sional electron gas systems. The fermionic problem is consideration of anyonic symmetries up to stable equiv- more challenging. For example, if b is a local boson and alence leads to an even richer structure of symmetries. a an arbitrary anyon, then we identify a and a + b and 2πiha 2πiha+b So far we have considered anyonic symmetries satisfy- θa = θa+b = e = e . However, once we include ing Eq. (4). These symmetries preserve the K matrix fermions as local particles this can lead to complications 1 − and the spins ha of the quasiparticles, ha = aT K 1a. such as θa+f = θa because fermions have half integral 2 − Z In other words, the conformal structure on the edge spin. This leads to anyonic symmetries being 2 graded is respected. However, we might relax this constraint according to fermion parity (which should be physically and only require keeping the T matrix invariant where conserved), which adds more complexity but should be a 2πiha Ta,a = e , then ha will only be preserved mod 1. straightforward, though perhaps technically challenging This will lead to more anyonic symmetries as considered extension. in Ref. 66. For example, in the case of su(12) the sym- To conclude, we associated the anyon relabeling symmetries which preserve spin lead to QP transformations metry of a general Abelian topological phase with the generated by e e−1 = e11 (c.f. Table II) and we have group of outer automorphisms of the K-matrix. We pre- 11→ −1 he = he = 24 (c.f. Table VIII). However, if we only sented the AS of the bosonic ADE Abelian topological require keeping the T -matrix invariant this leads to an- states, and discussed thoroughly the Z2 symmetry for 7 other anyonic symmetry generated by e e . In this su(3) and S3 triality symmetry for so(8). One dimen- 11 → 7 case he = 1+ 24 . This is an allowed symmetry be- sional gapped interface phases for chiral ADE states were 2πih 7 2πih cause, e e = e e . Thus, if we require the full set of shown to be naturally classified by AS. A similar method anyonic symmetries which preserve the T matrix, it will was applied to gapped edge phases for bosonic ADE frac- be generated by both the transformations e e7 and tional quantum spin Hall states, where extra constraints e e11. However demanding that the symmetries→ pre- (in addition to the even and odd criterion in Refs. 60 and serve→ conformal structure on the edge restricts us to only 62) on the AS and the QP pair condensate were required consider e e11. We expect that these types of anyonic for TR to be unbroken explicitly or spontaneously. It symmetries→ up to stable equivalence can be realized by would be interesting to explore the interplay between AS considering the spin preserving symmetries of the the set and TR as well as the compatibility between QP pair con- of K matrices K′ : K′ is stably equivalent to K . These densates and TR in general fermionic phases. We studied will be the subject{ of future work. } topological point defects, each associated with an AS op- The above comments should not be taken to mean eration, and exhaustively described the fusion behavior that our classification of defects and domain walls is of all possible twist defects in the ADE states. Although incomplete or that our method is insufficient. As dis- not shown explicitly in this article, the F -symbols for cussed in Appendix D, considering a matrix K′ which defect state transformations should take a similar form is stably equivalent to K involves enlarging K by some to certain previously studied exactly solvable models in L GL(n,z) such that K′ = K L (possibly with some Refs. 13 and 14. These twist defects therefore form a ∈ ⊕ 11 consistent fusion category1, and are powerful enough to 2 1000000 − construct a measurement-only topological quantum com- 1 2 10 0 0 0 0 67  − −  puter . 0 1 2 10 0 0 0 Note: During the preparation of the manuscript a  − −  66  0 0 1 2 10 0 0  recent work of Lu and Fidkowski appeared . Their nice KE8 =  − −  (A5)  0 0 0 1 2 1 0 1  work has similar themes and results to part of our work    − − −  and both works compliment each other.  0 0 0 0 1 2 1 0   − −  Acknowledgements We acknowledge useful discussions  00000 12 0   −  with M. Barkeshli, L. Fidkowski, L. Kong, Y.-M. Lu, M.  0 0 0 0 10 0 2   −  Mulligan, C. Nayak and X.-L. Qi. We thank J. Alicea for   pointing us to Ref.66. TLH was supported by ONR grant no. N0014-12-1-0935 (TLH). J.CY.T. acknowledges sup- Since all diagonal entries are 2, any such state is port from the Simons Foundation Fellowship. We also bosonic as all local particles are bosons. The E8 state thank the support of the UIUC ICMT. does not have topological order as det(KE8 ) = 1 so that Z8 Z8 the anyon content E8 = /KE8 = 1 is trivial and all QPs are mutuallyA local. The K-matrices (or Cartan matrices) can also be read off from the Dynkin diagrams Appendix A: Anyon theory of the bosonic Abelian ADE states of the Lie algebras (see Fig. 2 for Ar,Dr, E6 and Fig. 7 for E7, E8). By assigning an enumeration I =1,...,r of the dots in the Dynkin diagram, the non-zero entries of Here we summarize two dimensional bosonic Abelian the K-matrices are given by KII = 2 and KIJ = 1 if quantum Hall states with ADE chiral Kac-Moody (KM) dot I and J are connected. − current algebras at level one along boundary edges. We provide their K-matrices, quasiparticle (QP) labels, E braiding, spin, and electric charge. 7 E 8 The simplest K-matrices of the Chern-Simons actions that describe Abelian ADE topological states are given by the Cartan matrices of the corresponding simply-laced FIG. 7. Dynkin diagrams for the exceptional simply-laced Lie 50 algebras . Ar = su(r + 1), for r 2, and Dr = so(2r), algebras E7,E8. for r 4, form infinite series of Abelian≥ states, each with a K-matrix≥ of rank r. Even though we are considering purely bosonic states, it is important to note that each Abelian ADE state can (KA ) =2δIJ (δI,J+1 + δI,J−1) (A1) r IJ − be realized in an electronic system exhibiting a fractional (K ) =2δ (δ + δ − )+ Dr IJ IJ − I,J+1 I,J 1 quantum Hall (FQH) state. The K-matrix of such an (δI,rδJ,r−1 + δI,r−1δJ,r δI,rδJ,r−2 δI,r−2δJ,r) . electronic ADE state needs to be modified to include − − (A2) local fermionic electrons. For instance a K-matrix could take the form of a direct sum = K σ 39,40, where K ADE ⊕ z There are also three exceptional simply-laced (i.e. with σz only introduces local fermions. The electric charge symmetric Cartan matrix) Lie algebras KEr=6,7,8 with vector would take the form t = (2,..., 2, 1, 1) so that K-matrices primitive local bosons are treated as charge e∗ =2e pairs of electrons. As the additional σz does not contribute to 2 10 0 0 0 the topological order ( det(σ ) = 1), all results in the − z 1 2 10 0 0 main text extends to such| electronic| FQH states.  − −  0 1 2 1 0 1 Next we describe the anyon lattice vectors correspond- KE6 =  − − −  , (A3)  0 0 1 2 1 0  ing to the QPs in an Abelian ADE topological state.  − −   0 0 0 12 0  The bulk boundary correspondence identifies a bulk QP  −  to an edge vertex operator, ψa eia·φ, where a is an  0 0 10 0 2  ∼ ∗ r  −  r-dimensional anyon lattice vector in Γ = Z . It has   ∗ − charge qae for qa = tT K 1a, where the charge vector t = (1, 1, , 1) describes the external electromagnetic 2 100 0 0 0 ··· ∗ − coupling with fundamental charge e for local bosons. As 1 2 10 0 0 0 explained in the main text, due to the local boson con-  − −  0 1 2 10 0 1 densate at zero temperature, the QPs are defined only up  − − −  to local particles, i.e. a a + KZr. And the fractional KE7 =  0 0 1 2 10 0  (A4)   ≡ ∗  − −  electric charge is defined modulo integer multiples of e .  0 0 0 1 2 1 0   − −  Table V lists a particular representation of each set of  0 0 0 0 12 0  equivalent anyons [a]= a + KZr.  −   0 0 10 0 0 2   −  QP braiding is summarized by the S matrix, where the   12

Algebra Anyon Anyon vector The spin of the quasiparticle is

Ar su(r + 1) 1 (0, , 0) ≡ i · · · r 2 e (1 i r) (0, , 1 , , 0) 1 t −1 ≥ ≤ ≤ · · · · · · ha = a K a. (A8) th |{z}i 2 Dr(r odd) 1 (0, , 0) Algebra S matrix · · · 1 µν r > 4 e (0, , 0, 1, 0) A S µ ν = exp 2πi · · · r e ,e h r+1 i e2 (1, , 0) √r + 1 − 3 · · · 0 µ, ν r e (0, , 0, 1) ≤ ≤ · · · Dr(r mod 4= 0) See = Smm = Sψψ = 1 Dr(r even) 1 (0, , 0) · · · Sem = Seψ = Smψ = 1 r > 4 e (0, , 0, 1, 0) − · · · 1 µν m (0, , 0, 1) Dr(r mod 4= 1) Seµ,eν = exp πi 2 0 µ, ν 3 · · · 2 ≤ ≤ ψ (1, , 0) Dr(r mod 4= 2) See = Smm = Seψ = Smψ = 1 · · · − E6 1 (0, 0, 0, 0, 0, 0) Sem = Sψ,ψ = 1 e (1, 0, 0, 0, 0, 0) 1 µν Dr(r mod 4= 3) Seµ,eν = exp πi 2 0 µ.ν 3 2 2 − ≤ ≤ e (0, 0, 0, 0, 1, 0) 1 2πi 2 2 E6 Se,e = Se ,e = Se,e∗ 2 = √3 exp[ 3 ] E7 1 (0, 0, 0, 0, 0, 0, 0) 1 E7 Se,e = e (0, 0, 0, 0, 0, 1, 0) − √2

TABLE V. r-dimensional quasiparticle vectors of the ADE TABLE VII. Braiding S-Matrix of anyons Abelian topological states at level 1. Algebra Spin Algebra Fractional electric charge µ µ Ar hµ = 2 (1 r+1 ) A (r even) All anyons are neutral − r 0 µ r (or have integral charge) ≤ ≤ r 1 3 2 i Dr(r odd) h1 = 0; he = he = 8 ; he = 2 Ar(r odd) e i even integral charge r 1 i → Dr(r even) h1 = 0; he = hm = 8 ; hψ = 2 e i odd half-integral charge 2 → E6 h1 = 0; he = he2 = 3 Dr(r mod 4= 0, 1) All anyons are neutral 3 E7 h1 = 0; he = (or have integral charge) 4

Dr(r mod 4= 2) 1, ψ neutral (or integral charge) TABLE VIII. Quasiparticle Spin. → e,m half-integral charge 2 → Dr(r mod 4= 3) 1,e neutral (or integral charge) → e,e3 half-integral charge → The braiding phase and exchange phase (topological E6 All anyons are neutral spin) are invariant under the addition of local bosons. (or have integral charge) The QP’s spin and braiding phases are listed in Ta-

E7 1 neutral ble VII, and VIII respectively, and are labeled according → e half-integral charge to the anyon labels in Table V. For instance they ver- → ify the triality S3 = Dih3 symmetry for so(8) and the TABLE VI. Fractional electric charges of the quasiparticles eightfold periodicity Dr Dr+8. → in units of fundamental boson charge e∗.

phase of Appendix B: Anyonic relabeling symmetry Outer(K)

− of su(3) 1 2πiaT K 1b Sab = e (A6) D In the main text, we deﬁned the notion of anyon re- corresponds the braiding phase if the QP ψa is dragged b labeling symmetry by the group of outer automorphisms once around ψ . The normalization = det(K), in Eq. 7. Here we demonstrate this explicitly for the known as the total quantum dimension,D is to ensure uni- A2 = su(3) state. The group of automorphisms Aut(K) tarity of the S matrix. The topological spin ofp a QP is can be identiﬁed with the dihedral group Dih6, the sym- given by the exchange phase metry group of a hexagon. It is is generated by a sixfold

T −1 “rotation” P and a twofold “reﬂection” R, and has the 2πiha πia K a θa = e = e . (A7) two-dimensional representation 13

0 1 0 1 6 2 −1 Aut Ksu(3) = P = − , R = P = R =1,RPR = P . (B1) * 1 1 ! 1 0 ! +

We notice that these matrices are isometries with respect Appendix C: Outer Automorphisms of Lie Algebras to the K-matrix for su(3), P KP T = RKRT = K. Also, and anyonic symmetry P and R act on anyon labels by taking e e2. P and ↔ R can be visualized geometrically in Figs. 1(c) and 1(d) We provide some further details and references to elu- respectively. On the other hand the matrices PR,P 2 cidate the structure of the outer automorphisms of a Lie preserve anyon labels up to local particles, and there- Algebra and its relation to anyonic symmetry. This sec- fore generate the group of inner automorphisms. P 2 is tion is rather mathematical and is not necessary to un- geometrically represented in Fig. 1(a) and Fig. 1(b) rep- derstand the rest of the paper. resents the effect of RP 3. Given a Lie algebra we have a set of generators g. An automorphism of a Lie algebra is a map ω :g g which Inner K = PR,P 2 = Z ⋉ Z = S . (B2) su(3) 2 3 3 is linear and respects the structure of the Lie→ algebra, The quotient i.e. ω([x, y]) = [ω(x),ω(y)],x,y g. The set of automorphisms Aut(g) form a group called∈ the automorphism Aut(K) Dih6 Z group of the Lie algebra. It has a normal subgroup, de- Outer Ksu(3) = = = 2 = 1, σ Inner(K) S3 { } noted by Inn(g) which is generated by (B3) exp (ad ): x g; x ∈ describes a mirror symmetry of the A2 = su(3) state, where adx :g g, adx(y) = [x, y], x,y g e e2. We further notice that the equivalence → ∈ class↔ 1 can be represented by any of the elements and is called the set of inner automorphisms. 1, P 2, P 4,PR,P 3R, P 5R, while the equivalence class σ Since Inn(g) is a normal subgroup of Aut(g) we can can be represented by R,P,P 3, P 5, P 2R, P 4R. However, define the coset if we further impose the constraint of charge conserva- Aut(g) Out(g) = , tion, we must leave t = (1, 1) invariant, and thus we are Inn(g) restricted to the identity matrix for conjugacy class 1 and R for mirror σ. which is the group of outer automorphisms. It is a well For a general ADE state, Outer(K) = Z2 except for known theorem in mathematics that Out(g) is isomorphic D4 = so(8), where Outer(Kso(8)) = S3 as explained in to the automorphisms of the Dynkin diagram. The inter- the main text. Given any ADE state, it is always possible ested reader can look up the proofs in proposition D.40 to find a charge conserving symmetry matrix which real- of Ref. 68, Ch 11 of Ref. 55 or section 12.2, in particular izes the symmetry. It has been already explicitly written Table 1 of Ref. 54.This connection has been exploited down in the main text for the A2 = su(3) and D4 = so(8) in the main text to deduce the outer automorphisms of states. The charge conserving matrices that represent the various Lie algebras and, as we discuss below, has the mirror anyonic symmetries σ in Table II for other motivated our definition of anyonic symmetries. simply-laced Lie algebras are listed in Eq. (B4). There The discussion henceforth will be limited to simply is no mirror anyonic symmetry for E7 as its symmetry laced Lie algebras with symmetric Cartan matrices. All group Outer(K) is trivial. the information about a Lie algebra is encoded in the Cartan matrix and the associated Dynkin diagram (c.f. Figs. 2,7) of the Lie algebra. The number of nodes of the 0 ... 0 1 Dynkin diagram is equal to the rank of the Lie algebra. 0 ... 1 0 11 0 Let us consider a simply laced Lie algebra with rank r. σ =   σ = , Ar . . . . Dr In this case the weight lattice (the anyon lattice) Γ∗ = Zr . . . . 0 σx  . . . .  !r×r r   and the root lattice Γ = KZ where K is the Cartan  1 ... 0 0   r×r matrix. The Cartan matrix induces an inner product in   000010 the weight space 000100 T −1 Zr   a, b = a K b a, b . 001000 h i ∈ σE6 =   . (B4) A basis for the root lattice of the Lie algebra (i.e., the  010000    simple roots) is given by the vectors  100000       000001  αi = (Ki1,Ki2, ,Kir), i [1, , r].   ··· ∈ ··· 14

In mathematics the components of αi are known as the As shown in appendix B, inner automorphisms are gen- Dynkin labels. In the normalization α2 = 2 the Cartan i 2 0 1 0 1 matrix is K = α , α . Thus, if ω Aut(g), α , α = erated by PR,P where P = − , R = . ij i j i j 1 1 1 0 ωα , ωα = K h i ∈ h i ! ! h i ji ij Now we are in a position to motivate the definition Inner Automorphisms of Aut(K) in equation (4). The automorphisms in- • The action of inner automorphisms are duce a linear transformation in the space of roots such that the Cartan matrix is preserved. The transforma- P Rα1 = α1 − tion M can be interpreted simply as a basis change P Rα2 = α1 + α2 for the Cartan matrix. But since the Cartan matrix Using equation C1 α , α = P Rα ,PRα , [i, j] [1, 2] stays invariant, we have defined the set of automor- h i j i h i ji ∈ phisms Aut(K)= M GL(r; Z): MKM T = K . The PRe1 = e1 α1 ∈ − fact that M GL(r; Z) just implies that this is a vol- PRe2 = e2 (C2) ume preserving∈ basis transformation which preserves the number of ground states det(K). The roots transforming 2 among themselves indicate that local particles stay local. P α1 = α2 2 Before we carry on, we note that the the automor- P α2 = α1 α2 (C3) phisms preserve the Dynkin diagrams and the Cartan − − α , α = P 2α , P 2α , [i, j] [1, 2] matrix. However, they induce rotations in the space of h i j i h i j i ∈ 2 weights Zr. The anyons live in the weight space. Each P e1 = e1 α1 e1 − ≡ node of the Dynkin diagram can be associated with a 2 P e2 = e2 α1 α2 e2. (C4) corresponding fundamental weight. The i-th node will − − ≡ have Dynkin label (0, , 1, , 0) with 1 in the i-th Thus we see that the inner automorphisms induce ··· ··· entry. Thus inner automorphisms act trivially in the linear maps on the set of roots which preserve the weight space, thus preserving the weight labels(up to Cartan matrix. They act trivially on the weight roots) whereas outer automorphisms act non trivially on space too, thus the weights (anyon labels) are pre- the weight space rotating distinct weights (anyons) into served up to local particles. Hence the inner auto- each other. This can help one understand the definition morphisms act trivially on the weight space(up to of inner automorphism in the context of anyons. roots). Outer Automorphisms Inner(K)= M0 Aut(K) : [M0a] = [a] . { ∈ } • The outer automorphisms act by Since we are only interested in transformations which P α = α + α interchange weights we quotient out Inn(K) to get Aut(K) 1 1 2 Inn(K) P α = α as the group of anyon symmetries. 2 − 1 α , α = P α , P α , [i, j] [1, 2] h i j i h i j i ∈ Pe1 = e2 1. An example su(3)1 Pe = e α (C5) 2 1 − 1 Now let us consider an example in particular: su(3) . 1 Rα = α We have already seen explicit realization of the automor- 1 2 phism group of su(3) in Appendix B. We now see the Rα2 = α1 action of inner an outer automorphisms on the roots and α , α = Rα , Rα , [i, j] [1, 2] h i j i h i j i ∈ weights of su(3). Re1 = e2 2 1 Re = e . (C6) The Cartan matrix of su(3) is K = − . 2 1 1 2 − ! A basis set for the root system is given by α = (2, 1) Hence we see that while the linear transformations in- 1 − and α2 = ( 1, 2). Hence, the roots obey duced by the outer automorphisms preserve the Cartan − matrix they act nontrivially on the weight space (anyon α , α = α , α = 2; α , α = 1. (C1) labels) interchanging them. h 1 1i h 2 2i h 1 2i − These roots correspond to local particles. On the other Appendix D: Stable equivalence hand the weights correspond to the anyon labels e1 = (1, 0) and e2 = (0, 1). Now we explicitly examine the action of the inner and aouter automorphisms on the We introduce the basic definitions of stable equivalence roots and the weights to reinforce the statements made in this appendix. More detailed expositions and stronger above. results can be found in Refs. 19 and 20. 15

Two K matrices K1 and K2 of the same dimension(N) particular form of the K matrix or charge vector t. We and signature are stably equivalent if there exist signa- follow the discussions in Refs. 44 and 60. ture (n,n) unimodular matrices L1 and L2 such that Let us consider the bulk CS Lagrangian Z there exists W GL(N +2n, ) so that ∗ ∈ KIJ µνλ e λµν T bulk = ǫ αIµ∂ν αJλ tI ǫ Aλ∂µαIν , K1 L1 = W (K2 L2) W. (D1) L 4π − 2π ⊕ ⊕ I,J =1, 2, ,N. (F1) The signature of a matrix is determined by n n− where ··· +− n+ and n− are the number of positive and negative eigen- We impose TRI on the system and study the implications values. Physically this is the chiral central charge of the on the bulk and the edge. Here we have assumed that edge theory described by the K matrix. In general if there are NU(1) gauge fields αI , thus the K matrix is the S and T matrices characterizing the anyon content N N and the gauge group is U(1)N . The system being of two K matrices are the same, then they are stably bosonic,× the diagonal entries of the K matrix must be 20 equivalent . Physically, stable equivalence was intro- even duced in the context of edge reconstruction of a quantum Hall state by adding trivial degrees of freedom to the edge KII =0 mod2. and can lead to the interesting situation where multiple edge state theories can support the same bulk topological Under the action of the anti-unitary time reversal oper- phase. If we restrict ourselves to bosonic quantum Hall ator , the external electromagnetic gauge field A trans- T states (as we do in this article), L1 and L2 must be even forms as (they must have even entries on the diagonal). A0 A0 → Ai Ai; i =1, 2. Appendix E: Stability of gapped interface phase →− acts on the internal CS gauge fields α as T I The set of sine-Gordon coupling terms in Eq. 8 in- α T α (F2) troduces a finite energy gap for all of the edge de- Iµ →∓ IJ Jµ grees of freedom, and correspond to a particular gapped where the sign corresponds to the time index µ = 0 phase along the interface. By rearranging the bosons and the + sign− corresponds to the spatial indices µ =1, 2. φR = ϕ + (K−1) θ and φL = ϕ (K−1) θ , Eq. 8 I IJ J I I IJ J Here, T is an integer valued N N matrix which has becomes an ordinary gapped sine-Gordon− model to obey some constraints as outlined× below. When we 1 impose TRI on bulk, and using equation (F2), we find = ∂ ϕ ∂ θ g cos(2θ˜) (E1) L LM π x I t I − I the requirement

R L T where 2θ˜ = K (φ + M ′ φ ′ ) is pinned at 0 mod- T KT = K (F3) h I i h IJ J J J J i − ulo 2π when gI is large for all I. T t = t. (F4) In the strong coupling limit, the collection of backscattering terms g(M) with respect to a symmetry M in Eq. For a bosonic system, 2 = 1. Hence, I T 8 describes a{ fully} gapped interface phase . Their as- M 2 (M) L T = 1. (F5) sociated scaling dimensions ∆(gI ) determine the low energy relevance at the ﬁxed point in the renormalization Next, let us consider the edge of the system in (F1). The group sense. Similar to conventional Luttinger liquid the- Lagrangian density is ory, they depend on the forward scattering Hamiltonian ∗ ′ ′ KIJ e µν = V σσ ∂ φσ ∂ φσ . (E2) edge = ∂xφI ∂tφJ + ǫ tI ∂µφI Aν + H IJ x I x J L 4π 2π forward scattering terms. (F6) The backscattering terms g(M) can be simultaneously I ′ σσ 26,71 tuned to be relevant by an appropriate choice of VIJ Remembering that ∂µφ = αµ and using equation 20,69,70. (F2), we get

−1 φI = TIJ φJ + CI , CI R. Appendix F: Time Reversal Invariance(TRI) for T T ∈ bosonic topological theories (bulk and edge) Here CI is a constant which will be ﬁxed later from physical considerations. For notational convenience and to 1. General formulation align our expressions with previous work, let us replace −1 CI by π K IJ χJ , i.e., In this subsection we begin with a brief discussion of −1φ =T φ + π K−1 χ , χ RN . (F7) TRI for bosonic systems, without any reference to the T I T IJ J IJ J ∈ 16

T χ is often referred to as the time reversal vector. operators are of the form, ψ = eiλ Kφ; λ ZN . local ∈ Physically χ determines the action of time reversal Since the system under consideration is bosonic on vertex operators ψa = eia·φ on the edge. How- T −2ψ 2 = ψ . ever, different χ’s are not necessarily physically distinct. T localT local In fact, they might be gauge equivalent to each other. However, we also know that In order to understand this we start off by noting that edge is shift invariant in φ. Thus, φ φ + ξ leaves −2 2 −2 iλT Kφ 2 iλT Kφ iπλT χ −iπ(T λ)T ·χ L → ψlocal = e = e e e . edge unchanged. This should come as no surprise as T T T T L the gauge field αIµ is also left unchanged by this re- Since this must be true for all λ, we find the constraint definition. Indeed, αIµ = ∂µ(φI + ξI ) = ∂µφI . But, ia·φ ia·φ ia·ξ φ φ + ξ = e e e . I T T χ =0 mod2. (F13) Now→ we can consider⇒ the→ global U(1) gauge transforma- − tion on the edge associated with the shift invariance of Equations (F12),(F13) are very important in the defini- φ, tion of χ and we will use them in the next section. 2. Time reversal at the edge of bosonic fractional a a a a a ψ˜ = ψ ei ·ξ where ψ = ei ·φ. (F8) quantum spin Hall states

In the main text, we studied the edge of a bosonic ã a With this we can see how ψ and ψ transform under fractional quantum spin Hall system with K matrix TR: ′ K 0 Kσσ = r×r , where σ = R,L = , . 2r×2r 0 K ↑ ↓ † − r×r! −1 a (T T a) −iπ(K 1a)·χ − ψ = ψ e (F9) t T T The charge vector for this system is tσ = r×1 . −1 a h −1 a ia·ξ 2r×1 = ψ˜ = ψ e tr×1! ⇒ T T T T T a † −1a a A suitable value of T is = ψ(T ) e−iπ(K )·χe−i ·ξ. (F10) ′ ′ 0 1 h i T σσ = (σ )σσ δ = r×r (F14) IJ x IJ 1 We define the time reversal vector χ˜ in the new gauge r×r 0 ! a in terms of the action of on ψ˜ , analogous to equation ′ ′ T It obeys T 2 = 1; T T Kσσ T = Kσσ and T tσ = tσ. (F9). − χ Now, we need a time reversal vector χσ = ↑ which a T a † −1a χ↓! −1ψ˜ = ψ˜(T ) e−iπ(K )·χ˜ . (F11) T T obeys (F13). h i Combining equations (F10),(F11), we get 1 1 χ I T T χσ =0mod2 = − ↑ =0 mod 2 − ⇒ 1 1 χ 1 ! ↓! − χ˜ = χ + K(1 T )ξ (mod 2) (F12) = χ = χ mod 2 (F15) π − ⇒ ↑ ↓ χ χ 0 Thus we would say and ˜ are gauge equivalent to each The vector χ = trivially satisfies this condition. other. 0! The other constraint on χ is determined by the action Next, we claim that any valid time reversal vector χσ of on local operators ψ on the edge. Local vertex (which satisfies equation (F15)) is gauge equivalent to T local 0 . 0! Using equation (F12) an equivalent statement is ξ↑ 0 χ↑ 1 K 0 1 1 ξ↑ χ↑ : = + − mod 2 : χ↑ = χ↓ mod 2 ∃ ξ↓ ! 0! χ↓ ! π 0 K! 1 1 ! ξ↓ ! ∀ χ↓ ! − −

This reduces to ξ A solution to this equation is some ↑ such that ξ↓ χ 1 K ξ ξ ! ↑ = ↑ ↓ mod 2 1 − χ↑ = π K(ξ↑ ξ↓). Since det(K) = 0 such a so- χ↓ −π K ξ ξ − − 6 ! ↑ − ↓! 17

σ lution exists. Furthermore, since χ↑ = χ↓ mod 2, all other choices of χ are gauge equivalent to it. Thus 1 from now on we can and henceforth we will use the gauge χ↓ = π K(ξ↑ ξ↓) mod 2 is automatically satisﬁed. In conclusion− we− have shown that for the purposes of in which bosonic fractional quantum spin Hall states we can ﬁx −1 L/R R/L σ φ = φ . (F16) our gauge so that the time reversal vector χ = 0, since T I T I

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