Toward Kitaev's Sixteenfold Way in a Honeycomb Lattice Model

Toward Kitaev’s sixteenfold way in a honeycomb lattice model

Shang-Shun Zhang,1 Cristian D. Batista,1, 2 and G´abor B. Hal´asz3 1Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996, USA 2Neutron Scattering Division and Shull-Wollan Center, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA 3Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

Kitaev’s sixteenfold way is a classification of exotic topological orders in which Z2 gauge theory is coupled to Majorana fermions of Chern number C. The 16 distinct topological orders within this class, depending on C mod 16, possess a rich variety of Abelian and non-Abelian anyons. We realize more than half of Kitaev’s sixteenfold way, corresponding to Chern numbers 0, ±1, ±2, ±3, ±4, and ±8, in an exactly solvable generalization of the Kitaev honeycomb model. For each topological order, we explicitly identify the anyonic excitations and confirm their topological properties. In doing so, we observe that the interplay between lattice symmetry and anyon permutation symmetry may lead to a “weak supersymmetry” in the anyon spectrum. The topological orders in our honeycomb lattice model could be directly relevant for honeycomb Kitaev materials, such as α-RuCl3, and would be distinguishable by their specific quantized values of the thermal Hall conductivity.

I. INTRODUCTION mathematically described in the language of topological quantum field theory [10]. Topological order is an important cornerstone of mod- The simplest and most widely studied topological order ern condensed matter physics which facilitates a classi- is Z2 gauge theory [6], which gives rise to an entire class fication of gapped phases of matter beyond the classical of topological orders when coupled to gapped Majorana paradigm of spontaneous symmetry breaking [1]. While fermions of Chern number C [8]. This class contains an topologically ordered phases may be fully symmetric and infinite number of topologically distinct edge theories as locally featureless, they are characterized by particular the number of chiral Majorana edge modes is given by the patterns of long-range quantum entanglement [2] which Majorana Chern number C itself. Interestingly, however, manifest in robust global features, such as a topological the bulk topological order is determined by C mod 16, ground-state degeneracy [3] and a universal correction to and the infinitely many edge theories thus correspond to the bipartite entanglement entropy [4,5]. only 16 bulk topological orders with distinct topological Arguably, the most exciting feature of topological or- properties of the bulk anyons. der is the fractionalization of fundamental particles into This classification, commonly known as Kitaev’s six- emergent nonlocal quasiparticles. Because of their non- teenfold way [8], contains both Abelian and non-Abelian local nature, these fractionalized quasiparticles possess topological orders, corresponding to even and odd Ma- unusual “anyonic” particle statistics in two dimensions jorana Chern numbers, respectively. The topological or- that is distinct from both bosons and fermions. In partic- ders of Kitaev’s sixteenfold way are relevant for a wide ular, moving one anyon around another one (“braiding”) range of topological materials, including fractional quan- may correspond to a nontrivial operation on the under- tum Hall systems, topological superconductors, as well as lying quantum state [6]. For Abelian topological orders, quantum spin liquids. In particular, recent thermal Hall these braiding operations act on a single quantum state, conductivity measurements in the quantum spin liquid while for non-Abelian topological orders, they act on a candidate α-RuCl3 [11] indicate a single Majorana edge set of degenerate quantum states within an internal space mode for a range of applied magnetic fields, correspond- spanned by the anyons themselves. In addition to their ing to the non-Abelian C = 1 topological order. fundamental scientific appeal, such non-Abelian anyons The search for topological orders in such magnetic ma- are highly promising from the perspective of topological terials was fueled by the discovery of the Kitaev honey- quantum computation [7]. comb model [8], which realizes the C = 0 and C = 1 Each topological order is uniquely characterized by the topological orders in an exactly solvable spin model± on arXiv:1910.00601v1 [cond-mat.str-el] 1 Oct 2019 topological properties of its anyonic quasiparticle excita- the honeycomb lattice. Indeed, the bond-dependent Ising tions: the distinct classes of anyons as well as the fusion interactions of this exactly solvable model were first pro- and braiding rules between them [8]. To a large extent, posed to emerge between transition-metal ions in the d5 anyons generalize the concept of topological defects in [12, 13] and d7 [14, 15] configurations as well as between classically ordered systems [9]. Indeed, the anyon classes rare-earth ions [16, 17], and then these proposals led to a are topologically distinct in the sense that they cannot wide range of honeycomb candidate materials, including be locally transformed into each other, while the fusion (Na,Li)2IrO3 [18–25], H3LiIr2O6 [26], α-RuCl3 [27–35], rules between these classes are analogous to the combi- Na3Co2SbO6 [36], and YbCl3 [37, 38]. However, it should nation rules between topological defects. Together with be emphasized that, while the original Kitaev model only the fundamentally quantum braiding rules, these topo- contains C 1 topological orders, there is no reason to logical properties fully define a given topological order, believe that| | ≤only these topological orders can emerge in 2 such honeycomb magnets. (a) K1 (b) K2 In this work, we study an exactly solvable generaliza- x y k tion [39] of the Kitaev model that respects all symmetries 6 l of the honeycomb lattice and realizes more than half of 5 1 B j z p the topological orders in Kitaev’s sixteenfold way, corre- 4 2 A sponding to Majorana Chern numbers 0, 1, 2, 3, 4, 3 and 8. These topological orders contain± both± Abelian± ± and non-Abelian± anyons with a rich variety of fusion and braiding rules, and are experimentally distinguishable by (c) K3 (d) K their different quantized values of the thermal Hall con- 3 ductivity. For each topological order, we use the exact solution of our model to explicitly identify the anyon classes and verify their fusion rules. In some cases, we find that lattice symmetry becomes intertwined with anyon permutation symmetry, corresponding to weak symmetry j j k k breaking [8], and gives rise to a “weak supersymmetry” in k the excitation spectrum. Since the additional four-spin l l l interactions of our generalized Kitaev model arise natu- m m rally from time-reversal-symmetric perturbations [39], in the same way as the three-spin interactions in the original Kitaev model arise from an external magnetic field, FIG. 1. Generalized Kitaev model. (a) Bond-dependent Ising we believe that the C > 1 topological orders described | | interactions of the K1 term corresponding to the pure Kitaev in this work are likely to be realized in spin-orbit-coupled model: the spin components σx,y,z at neighboring honeycomb honeycomb magnets, such as α-RuCl3. sites are coupled along x (red), y (green), and z (blue) bonds, respectively. The site-labeling convention around a plaquette p is also illustrated. (b) Representative (orange) path hjkliyx associated with the K2 term in Eq. (3). (c)-(d) Representa- tive (orange) paths hjklmiyzx (c) and hjklmiyzy (d) associ- II. LATTICE MODEL 0 ated with the K3 and K3 terms in Eq. (4), respectively. Spin interactions along these paths give rise to Majorana hopping We consider a generalization of the Kitaev spin model terms along the dashed arrows. Note that the K3 interactions on the honeycomb lattice, come in symmetry-related pairs (orange and blue) that correspond to the same Majorana hopping term and may interfere constructively or destructively. In general, sites in sublattice = + + , (1) A (B) are marked by black (white) dots. H H1 H2 H3 where the first term notation, the third term then takes the form

X X α α X X α α β β γ γ = K σ σ (2) 3 = K3 σj σk σ σ σ σm H1 − 1 j k H − k l l α jk α (αβγ) jklm αβγ h i h i X X α α β β α α K0 σ σ σ σ σ σ − 3 j k k l l m is the pure Kitaev model [8] with Ising interactions be- (αβγ) jklm αβα α h i tween the spin components σ along each α = x, y, z X X α γ α γ { } = K3 σ σ σ σ (4) bond jk α [see Fig.1(a)], while the remaining two terms j k l m h i (αβγ) jklm αβγ r with r = 2, 3 contain products of such Ising interac- h i tionsH along paths consisting of r bonds each. If we deﬁne X X α γ γ α K30 σj σk σl σm, jkl αβ to be the path consisting of the two bonds jk α − (αβγ) jklm αβα h i h i h i and kl β [see Fig.1(b)], the second term reads h i where jklm αβγ and jklm αβα are paths consisting of three bondsh eachi [see Figs.h 1(c)i and1(d)]. As it is clear X X α α β β 2 = iK2 (αβγ) σj σk σk σl from our construction, the term r for general r contains H − H (αβγ) jkl αβ h i (r+1)-spin interactions and thus breaks (preserves) time- X X α γ β reversal symmetry for even (odd) r. We remark that the = K2 σj σk σl , (3) term 2 was already introduced in Ref. [8] while the term (αβγ) jkl αβ H h i 3 was ﬁrst considered in Ref. [39]. It is also important toH note that these two terms are respectively generated by where (αβγ) is a general permutation of (xyz), and (αβγ) time-reversal-breaking and time-reversal-symmetric per- is +1 ( 1) for even (odd) permutations. Using analogous turbations on top of the pure Kitaev model. − 3

Remarkably, the generalized Kitaev model in Eq. (1) 0.6 is exactly solvable in the same way as the original Kitaev 1-flux (Fermi III) model [8]. By expressing each physical spin component 0.5 α α as a product of two Majorana fermions, σj = ibj cj, the 1-flux (Fermi II) Hamiltonians n in Eqs. (2)-(4) become H 0.4 2/3-flux (Dirac) X X α 1/2-flux (Gapped) 1 = iK1 ujkc c , H j k 0.3 0-flux (Dirac) α jk α 3/4-flux (Dirac) h i X X β 1/4-flux (Dirac) = iK uα u c c , H2 − 2 (αβγ) jk lk j l 0.2 (αβγ) jkl αβ 1/3-flux (Gapped) h i 2/3-flux 1-flux X X = iK uα uβ uγ c c (5) 0.1 (Gapped) (Fermi I) H3 3 jk lk lm j m (αβγ) jklm αβγ 1-flux (Dirac) h i X X α β α 0 +iK30 ujkulkulmcjcm, 0 0.05 0.1 0.15 0.2 0.25 (αβγ) jklm αβα h i α α α α where the Z2 gauge fields ujk = ukj ibj bk along the − ≡ FIG. 2. Phase diagram of the time-reversal-symmetric Hamil- bonds jk α are conserved quantities that commute with tonian H +H as a function of K /K and K0 /K [39]. Black each other.h i Therefore, the Hamiltonian in Eq. (1) de- 1 3 3 1 3 1 H solid lines are first-order transitions between different flux sec- scribes free fermions coupled to a static Z2 gauge theory, tors, denoted by distinct colors, while white dashed lines are and cj can be identified as deconfined Majorana fermion second-order Lifshitz transitions between different Majorana (“spinon”) degrees of freedom. In terms of these Majo- nodal structures, specified in parentheses. rana fermions, each term r in Eq. (5) corresponds to r-th-neighbor hopping [40].H Also, unlike the gauge fields themselves, the product of the gauge fields around any mally small time-reversal-breaking perturbation can have plaquette p [see Fig.1(a)] is a gauge-invariant quantity a dramatic effect on their low-energy physics [8]. that can be expressed in terms of the physical spins:

z x y z x y x y z x y z Wp = u12u32u34u54u56u16 = σ1 σ2 σ3 σ4 σ5 σ6 . (6) III. MAJORANA PROBLEMS Thus, Wp = 1 can be identified as static Z2 gauge flux (“vison”) degrees± of freedom. III.1. Quadratic Hamiltonians While Eq. (5) reduces to a quadratic fermion problem in each flux sector, Wp = 1 , represented with an ap- { ± } α propriate gauge-field configuration, ujk = 1 , it is not For each ground-state flux sector in Fig.2, the gauge- immediately clear which flux sector{ contains± the} ground field configuration in Fig.3 gives rise to a quadratic Ma- state of the physical spin model . For the pure Kitaev jorana problem [see Eq. (5)]. The unit cell of this Majo- H model 1, it is guaranteed by Lieb’s theorem [41] that the rana problem may consist of n > 1 honeycomb unit cells groundH state belongs to the 0-flux sector characterized by for two distinct reasons. First, the physical unit cell is Wp = +1 for all p. However, Lieb’s theorem no longer ap- enlarged in the fractional-flux sectors because translation plies if the additional terms 2 and/or 3 are included in symmetry is spontaneously broken. This enlargement is the spin model. Indeed, it wasH demonstratedH in Ref. [39] twofold for the 1/2-flux sector, threefold for the 1/3-flux that the frustration between and can stabilize and 2/3-flux sectors, and fourfold for the 1/4-flux and H1 H3 a wide range of flux sectors as a function of K3/K1 and 3/4-flux sectors. Second, if the physical unit cell has an K /K (see Fig.2), including the 1-flux sector character- odd number of Wp = 1 plaquettes, translation sym- 30 1 − ized by Wp = 1 for all p, as well as fractional-flux sec- metry acts projectively on the Majorana fermions. In tors in which a− nontrivial fraction of the plaquettes have this case, the Majorana unit cell, as characterized by the Wp = 1 rather than Wp = +1. In these fractional-flux gauge-field configuration, must consist of two physical − sectors, the plaquettes with Wp = 1 form crystalline unit cells. Consequently, the Majorana unit cell has an structures (“vison crystals”) that spontaneously− break additional twofold enlargement in all flux sectors except translation symmetry (see Fig.3). for the 0-flux and 2/3-flux sectors. Assuming an infinitesimally small coupling constant For each flux sector, we label the honeycomb sites as K2, we start from the time-reversal-symmetric Hamilto- j = (r, λ) and the corresponding Majorana fermions as nian 1 + 3 [39] and treat the Hamiltonian term 2 as cj = cr,λ, where r is the lattice vector of the Majorana a time-reversal-breakingH H perturbation. Due to theH finite unit cell, and λ = (µ, ν) in terms of the sublattice in- flux gap, the ground-state flux sectors in Fig.2 are ro- dex µ = A, B and the index ν = 1, . . . , n specifying the bust against small perturbations. In contrast, if the Ma- particular honeycomb unit cell within the Majorana unit jorana fermions are originally gapless, even an infinitesi- cell. Using this labeling convention, the quadratic Majo- 4

(a) 0-flux (b) 1-flux (c) 1/2-flux

R R2 2 2 a1 1 2 3 4 1 R1 1 R1 R1

(d) (e) 1/3-flux 2/3-flux

R2 4 5 6 R2

1 2 3 1 2 3 R1 R1

(f) (g) 1/4-flux 3/4-flux

R R2 3 4 7 8 2 3 4 7 8

1 2 5 6 1 2 5 6 R1 R1

FIG. 3. Flux configurations and representative gauge-field configurations in the ground-state flux sectors of Fig.2. In each case, α α plaquettes with Wp = +1 (Wp = −1) are marked by white (gray) filling, while bonds with ujk = +1 (ujk = −1) are marked by thin (thick) lines. The physical unit cell, spanned by the lattice vectors R1,2, is marked by a yellow shaded parallelogram, while the Majorana unit cell is marked by a blue dashed parallelogram. Note that the Majorana unit cell may contain several honeycomb unit cells, indexed by ν = 1, . . . , n, each containing one A site (black dot) and one B site (white dot).

rana Hamiltonian takes the general form By diagonalizing the matrix Hq at each momentum q, one obtains Majorana bands at both positive and nega- i X X ˜ = Hr0 r,λ,λ0 cr,λcr0,λ0 , (7) tive energies. However, since ψ q,λ = ψq†,λ by deﬁnition, H 2 − r,r0 λ,λ0 there is a redundancy in our description,− and only the bands with positive energies are physical. where each H˜r0 r,λ,λ0 is proportional to the product of the static gauge− ﬁelds uα = 1 along a path connecting jk ± the sites (r, λ) and (r , λ ). Introducing the momentum- 0 0 III.2. Projective symmetries space complex fermions

1 X iq r ψ = c e− · , (8) We now discuss the general symmetries of the Majo- q,λ √ r,λ N r rana problem in Eq. (10). Since the Majorana fermions are fractionalized degrees of freedom, symmetries may where N is the number of honeycomb sites, and arranging act on them projectively [42], i.e., the classical relations them into the 2n-component vector between symmetry operations may only be satisfied up h iT to an overall complex phase factor eiϕ. However, as the ψ ψ , . . . , ψ , ψ , . . . , ψ , q ≡ q,(A,1) q,(A,n) q,(B,1) q,(B,n) Majorana fermions are coupled to Z2 gauge fields, this (9) phase factor must actually be a sign 1 [43]. ± the Hamiltonian in Eq. (7) can then be written as We first consider the translation symmetries and T1 T2 X along the lattice vectors R1 and R2 of the physical unit = ψq† Hq ψq, (10) cell (see Fig.3). Note that the physical unit cell depends H · · q on the particular flux sector and may be larger than the original honeycomb unit cell due to spontaneous breaking where Hq is a 2n 2n matrix with elements × of translation symmetry in the fractional-flux sectors. If X iq r ˜ 0 the physical unit cell has no overall Z2 flux, correspond- (Hq)λλ0 = iHr,λ,λ e · . (11) ing to an even number of Wp = 1 plaquettes, trans- r − 5

sal, inversion is a unitary operation, [ , i] = 0, and is a general symmetry of our model. WhileP the action of inversion on the Majorana fermions depends on the given flux sector, it always exchanges the two sublattices A and B, and thus necessarily anticommutes with time reversal: , = 0. Also, inversion satisfies 2 = 1 [43] in all flux{P T} sectors except for the 3/4-flux sector.P For− simplicity, we ignore the 3/4-flux sector in the rest of this work and only return to it briefly in Sec.VI. Finally, the redundancy in our description, corresponding to H q = Hq∗ [see Eq. (11)], gives rise to an emergent antiunitary− − particle-hole symmetry , which satisfies [ , ] = 0, [ , ] = 0, and 2 = +1.C We emphasize thatC particle-holeT C symmetryP is actuallyC an antisymmetry as it anticommutes with the Hamiltonian. While , , and each reverse the fermion momentum q, theirT twoP independentC products = and = transform FIG. 4. Schematic illustrations of the conventional (a) and the fermions at momentumS qTCamongR eachPC other: the compact (b) Brillouin zones when translation symmetry acts projectively on the Majorana fermions. The conventional : ψq S ψq, Brillouin zone is equivalent to two identical copies (“×2”) of S → · : ψq R ψq. (13) the compact Brillouin zone. R → · The 2n 2n transformation matrices are given by × lation symmetry acts linearly (i.e., not projectively) on I 0 0 P the Majorana fermions, and the two elementary transla- S = ,R = T , (14) 0 I P 0 K tions commute: [ , ] = 0. The Brillouin zone is then − − T1 T2 spanned by the reciprocal lattice vectors G1 and G2 cor- where denotes complex conjugation, I is the n n unit responding to the physical unit cell, and different points matrix,K and P is an n n permutation matrix× satisfy- in the Brillouin zone are labeled by different eigenvalues ing P P T = I. The unitary× antisymmetry can thus of the translations and . · S T1 T2 be identified as sublattice symmetry, while the antiuni- Conversely, if the physical unit cell has an overall Z2 tary antisymmetry can be interpreted as an effective flux, corresponding to an odd number of Wp = 1 pla- R − momentum-conserving particle-hole symmetry. quettes, translation symmetry acts projectively on the Since is a general antisymmetry of our model, the Majorana fermions, and the two elementary translations R Hamiltonian matrix Hq in Eq. (11) satisfies R,Hq = 0, anticommute: , = 0. The eigenvalues of the trans- { } {T1 T2} which implies that the Majorana spectrum is symmetric lations 1 and 2 are then no longer compatible quan- around zero energy at each momentum q. Furthermore, tum numbersT forT the Majorana fermions. Nevertheless, in the time-reversal-symmetric limit of K2 = 0, the an- since [ 2, ] = 0, one may consider a larger Majorana T1 T2 tisymmetry requires S, Hq = 0 and therefore con- unit cell spanned by 2R1 and R2, which translates into S { } 1 strains the Hamiltonian matrix to the form a smaller Brillouin zone spanned by G1 and G2. In 2 fact, the Majorana spectrum is periodic with respect to 0 Mq 1 1 Hq = . (15) an even smaller Brillouin zone spanned by 2 G1 and 2 G2 Mq† 0 [see Fig.4(a)] because the residual symmetry 1 anticom- T 1 mutes with 2 and hence corresponds to a shift G2 in In this limit, the eigendecomposition of the 2n 2n ma- T 2 × the Majorana momentum. Thus, one may use a compact trix Hq is equivalent to the singular value decomposition 1 Brillouin zone spanned by 2 G1,2 and indicate that each of the n n matrix Mq. Majorana band has a twofold “translation degeneracy” × [see Fig.4(b)]. Different points in this compact Brillouin 2 2 zone are labeled by 1 and 2 , while the two degenerate III.3. Generic Majorana nodes Majorana fermions atT a givenT point are labeled by and T1 mapped onto each other by 2 (or vice versa). In terms of the low-energy physics, the gapless nodes of We next consider time-reversalT symmetry and inver- T the momentum-space Majorana spectrum are of partic- sion symmetry . Time reversal is an antiunitary oper- ular interest. Because of the antisymmetry , a generic ation, , i = 0,P and is only a symmetry for K = 0. In R {T } 2 nodal momentum Q has two zero-energy fermions that each flux sector, it acts on the Majorana fermions as correspond to distinct Majorana bands of opposite energies. By projecting onto these two low-energy Majorana : cr,(A,ν) cr,(A,ν), cr,(B,ν) cr,(B,ν), (12) T → → − bands around q = Q, one then obtains an effective low- and hence satisfies 2 = +1. In contrast to time rever- energy theory of the given Majorana node. T 6

For simplicity, we start our discussion from the time- IV. MAJORANA CHERN NUMBERS reversal-symmetric limit of K2 = 0. Since the Hamilto- nian matrix HQ takes the form of Eq. (15), we can choose IV.1. Definition and numerical results the low-energy subspace to be spanned by two fermions located on the two respective sublattices A and B, Since the Majorana spectrum is symmetric around zero (1) X energy at each momentum q and generically gapped for ψq = uQ∗ ν ψq,(A,ν), ν K2 = 0 in each ground-state flux sector, diagonalizing the quadratic6 Hamiltonian in Eq. (10) gives equal numbers of (2) X ψq = vQ ν ψq,(B,ν), (16) Majorana bands at strictly positive and strictly negative ν energies. Therefore, one can define a ground-state Chern where uQ (vQ) is the left (right) eigenvector of the matrix number of the Majorana fermions by summing the Chern MQ corresponding to zero eigenvalue [44]. If we project numbers of all the negative-energy Majorana bands. As onto these two low-energy fermions, the antisymmetries it was argued in Ref. [8], the low-energy physics of the and are represented with the 2 2 matrices corresponding topological order is completely determined S R × by this Majorana Chern number C. ˆ ˆ S = τ3, R = iτ2 , (17) Mathematically, the eigendecomposition of the 2n 2n K × matrix Hq in Eq. (11) gives 2n eigenvalues εq,κ and 2n and the most general Hamiltonian matrix anticommuting ˆ ˆ corresponding eigenvectors wq,κ with κ = 1,..., 2n at with both S and R takes the form each Majorana momentum q. The Chern number of each Majorana band, labeled with κ, is then obtained as Hˆq = β1(q)τ1 + β2(q)τ2, (18) 1 Z where τ1,2,3 are the Pauli matrices. Since there are two Cκ = dq Fq,κ, (23) independent real coefficients, β1(q) and β2(q), that must 2π BZ vanish at the nodal momentum Q itself, the generic nodal where the Berry curvature at momentum q is given by structures in two dimensions are point nodes. Expanding β1,2(q) up to linear order in δq q Q = (δqx, δqy), Fq,κ = q Aq,κ (24) ≡ − ∇ × β1(q) = γ1 δq = γ1,xδqx + γ1,yδqy, · in terms of the corresponding Berry connection β (q) = γ δq = γ ,xδqx + γ ,yδqy, (19) 2 2 · 2 2 A = iw∗ w . (25) these point nodes are generically Dirac nodes with linear q,κ q,κ · ∇q q,κ dispersions [45]. Also, by considering the complex phase We note that the cross product of two vectors is a scalar of β (q)+iβ (q) at q = Q+(cos ϑ, sin ϑ)δq as a function 1 2 in two dimensions. Also, since the integral in Eq. (23) is of ϑ, one can assign a winding number W to each Dirac Q defined over the conventional Brillouin zone in Fig.4(a), node, which is generically given by it must be multiplied by 2 when calculated over the compact Brillouin zone in Fig.4(b). Using Eqs. (23)-(25), the γ1,x γ1,y WQ = sgn det = 1 (20) Majorana Chern number can then be numerically com- γ2,x γ2,y ± puted in each flux sector via with sgn x x/ x . Therefore, each Dirac node is a sta- n ble U(1) vortex≡ | protected| by time-reversal symmetry or, X C = C , (26) equivalently, by sublattice symmetry. κ κ=1 If we then break time-reversal symmetry with an in- ˆ finitesimally small K2 = 0, the Hamiltonian matrix Hq where the bands κ = 1,..., 2n are arranged by increas- 6 still anticommutes with Rˆ but no longer with Sˆ. Thus, ing energy eigenvalues so that the summation is over all its most general form reads negative-energy Majorana bands. The numerical results for the Majorana Chern numbers ˆ Hq = β1(q)τ1 + β2(q)τ2 + β3(q)τ3, (21) are summarized in Fig.5. For most of the flux sectors in Fig.2, the Chern number only depends on K and is oth- where the third coefficient may be expanded up to lin- 2 erwise the same throughout the entire flux sector. In con- ear order in K such that β (q) = m K . Since there 2 3 q 2 trast, for the 2/3-flux sector, there are two disconnected are three independent real coefficients, nodes can only (“upper” and “lower”) phases with distinct Chern num- emerge as a result of fine tuning, and the spectrum is bers, while for the 1-flux sector, there are several phases generically gapped. In particular, at each Dirac node of with distinct Chern numbers that are separated by topo- the K2 = 0 limit, the Hamiltonian matrix becomes logical transitions as a function of K30 /K1. We note that K /K is an irrelevant parameter in the 1-flux sector as HˆQ = mQK2τ3, (22) 3 1 symmetry-related pairs of K3 interactions [see Fig.1(c)] and the Dirac node at momentum Q is thus gapped out give rise to equivalent Majorana hopping terms with a by a fermion mass term mQK . perfect destructive interference between them [39]. ∝ 2 7

FIG. 5. (a)-(f) Topological phase diagrams for the various flux sectors in Fig.2 as a function of an infinitesimal K2 at fixed 0 representative values of K3 and K3. Gapped phases are labeled by their Majorana Chern numbers C, while gapless phases are labeled by their Majorana nodal structures: Dirac nodes or Fermi surfaces (i.e., line nodes). (g) Topological phase diagram for 0 the 1-flux sector as a function of K3/K1 and an infinitesimal K2. The critical points of the time-reversal-symmetric model at 0 1 K2 = 0 are marked by black circles, while the dashed arrows next to the multicritical point at K2 = 0 and K3 = 2 K1 illustrate the two-step scheme for obtaining the Majorana Chern numbers of the gapped phases surrounding the multicritical point.

IV.2. Analytical understanding

By studying the phase transitions between the various phases in Fig.5, we can also understand their Majorana Chern numbers analytically. If the Majorana spectrum is gapped for K2 = 0, the Chern number C vanishes due to time-reversal symmetry and is robust against an infinitesimally small K2 = 0. Thus, the 1/3-flux phase, the 1/2-flux phase, and the6 “lower” 2/3-flux phase of Fig.2 are all characterized by C = 0. If the Majorana spectrum has gapless nodes for K2 = 0, these nodes are all gapped out by an infinitesimally small K2 = 0, and the resulting gapped phases have opposite Chern6 numbers C for opposite signs of K . Since a change in the Chern± number 2 FIG. 6. Half-skyrmion configuration of the vector field d(q) is always connected to a closing gap, the Chern number around a Dirac node of winding number WQ = +1 that is C at K2 > 0 can be understood as a sum of contributions gapped out by a fermion mass mQK2 > 0. from the various nodes at K2 = 0.

negative-energy band in the low-energy theory. For the IV.2.1. Dirac nodes Hamiltonian matrix in Eq. (27), this quantity can be calculated by means of a standard formula [8]: The low-energy theory around a Dirac node at K = 0 2 Z and momentum Q takes the general form [see Eq. (21)] 1 CˆQ = dq d(q) ∂q d(q) ∂q d(q) , (29) 4π · x × y Hˆq = β(q) τ , (27) · where d(q) β(q)/ β(q) . Geometrically, CˆQ is simply ≡ | | where τ (τ1, τ2, τ3) and, up to linear order in both K2 the number of “skyrmions” in the vector field d(q). Since ≡ and δq q Q [see Eqs. (19) and (22)], the vector-field configuration in Eq. (28) corresponds to ≡ − a half skyrmion or meron (see Fig.6), the contribution β(q) = (γ δq, γ δq, mQK ) . (28) 1 · 2 · 2 of the given Dirac node to the Chern number becomes For K2 = 0, the contribution to the Chern number from 1 1 6 ˆ CˆQ = WQ sgn (mQK ) = (30) the given Dirac node, CQ, is the Chern number of the 2 2 ±2 8

(a) 0-flux (b) 1/4-flux (a) 1-flux (Fermi I) (b) 1-flux (Fermi II & III)

K5 < 0 K5 < 0 2 2 + − × × + + + + − − + + + 2 + + − − − + + − − × − − − − + − C = 1 C = 2 C = 2 C = 8 ± ± ± ± (c) 2/3-flux (upper) (d) 1-flux (Dirac) (c) 1-flux (Fermi I) (d) 1-flux (Fermi II & III)

K5 > 0 K5 > 0 + 2 2 + × × + + − − − + + − + + + 2 + + − − − − − + + − − × + − − + − − C = 3 C = 2 C = 2 C = 4 ± ± ± ∓ FIG. 7. Dirac nodes within the compact Brillouin zone in the FIG. 8. Dirac nodes within the compact Brillouin zone in Dirac phases of Fig.2. For each Dirac node at momentum Q, the first Fermi phase (left) and in the second and the third a winding number WQ of +1 (−1) is marked by an anticlock- Fermi phases (right) of the 1-flux sector for the two opposite wise (clockwise) arrow, while a positive (negative) mass coef- signs (top and bottom) of a generic fifth-neighbor Majorana ficient mQ is marked by a “+” (“−”) label. For K2 > 0, the hopping amplitude K5. The notation is identical to Fig.7. corresponding contribution to the Majorana Chern number is either +1/2 (red) or −1/2 (blue). If translation symmetry acts projectively on the Majorana fermions, each contribution ping amplitude K . While each line node is gapped out must be doubled (“×2”) due to translation degeneracy. Sum- 5 ming these contributions, the total Majorana Chern number into six Dirac nodes at the same momenta for K5 > 0 and K < 0, the winding number of each Dirac node changes C is given by the upper (lower) sign for K2 > 0 (K2 < 0). 5 sign at K5 = 0 (see Fig.8). Thus, Dirac nodes with opposite winding numbers must be connected by line nodes at the phase transition so that they can exchange their in terms of its winding number WQ [see Eq. (20)]. These contributions of the individual Dirac nodes are illustrated winding numbers with each other. in Fig.7 for the 0-flux phase, the 1 /4-flux phase, the Instead of constructing a low-energy theory for each ac- “upper” 2/3-flux phase, and the Dirac 1-flux phase of cidental line node, it is then more natural to include an infinitesimally small K = 0 and consider the low-energy Fig.2. For K2 > 0, the resulting total Chern numbers, 5 P ˆ theories of the resulting Dirac6 nodes. Their contributions C = Q CQ, are 1, 2, 3, and 2, respectively. to the Chern number at K = 0, as given by Eq. (30), are In general, Dirac nodes emerge in pairs related by in- 2 6 version symmetry. Since the two nodes in any pair have illustrated in Fig.8 for all three Fermi 1-flux phases and for both signs of K . For the first Fermi phase, the Dirac opposite mass coefficients mQ as well as opposite wind- 5 nodes corresponding to each line node have a vanishing ing numbers WQ, each pair contributes 1 to the Chern number. If translation symmetry acts projectively± on the net contribution to the Chern number for both K5 > 0 and K < 0. Thus, we can deduce that the gapped phase Majorana fermions, corresponding to an overall Z2 flux 5 in the physical unit cell, the total Chern number is then at K5 = 0 and K2 > 0 is adiabatically connected to that necessarily even as these contributions 1 come in iden- obtained from the Dirac phase at K2 > 0 and that its tical pairs due to translation degeneracy.± total Chern number is C = 2. Conversely, for the second and the third Fermi phases, the Dirac nodes corresponding to each line node have opposite net contributions to IV.2.2. Line nodes the Chern number for K5 > 0 and K5 < 0. Thus, for K5 = 0, there are two possible gapped phases at K2 > 0 with total Chern numbers 8 and 4, respectively. To dis- While the generic Majorana nodal structures at K2 = 0 criminate between these two scenarios,− we consider the are point nodes, these point nodes seem to coexist with 1 multicritical point at K = 0 and K0 = K . line nodes in the three Fermi 1-flux phases of Fig.2. As 2 3 2 1 expected, these accidental line nodes are unstable against generic further-neighbor Majorana hopping terms that respect the projective symmetries of the system. In fact, IV.2.3. Multicritical point the accidental “phases” with line nodes can be understood as phase transitions between two generic phases The second and the third Fermi 1-flux phases of Fig.2 with point nodes as a function of a fifth-neighbor hop- are separated by a multicritical point [see Fig.5(g)] at 9

+ (a) δK<0 (b) δK>0 1-flux − (Multicritical point) Trivial configuration Double Skyrmion Q 2 × + − Q2 Q1

C = 4 ± FIG. 9. Majorana nodes within the compact Brillouin zone at the multicritical point of the 1-flux sector: a single Dirac FIG. 10. Vector-field configuration d(q) around the quadratic node at each momentum Q1,2 and a pair of Dirac nodes at point node on the two respective sides of the phase transition, momentum Q. The notation is identical to Fig.7. corresponding to (a) δK < 0 and (b) δK > 0.

1 K2 = 0 and K30 = 2 K1 [46] where the accidental line node shrinks to a single momentum Q (see Fig.9). Re- where the additional factor of 2 comes from translation markably, at this momentum Q, the 4 4 Hamiltonian degeneracy in the 1-flux sector. matrix in Eq. (11) takes the exact general× form Next, we fix a particular value of K2 > 0 and consider 1 HQ = (2K0 K )(τ η τ η τ η ) the phase transition at K30 = K1 + K2. At this phase 3 − 1 1 ⊗ 0 − 2 ⊗ 1 − 2 ⊗ 2 2 +2K (τ η τ η + τ η ) , (31) transition, the low-energy subspace at the critical mo- 2 0 ⊗ 3 − 3 ⊗ 1 3 ⊗ 2 mentum Q is spanned by theτ ˜ = 1 basis vectors in 3 − where “ ” denotes the Kronecker product, while τ , , , Eq. (32). By expanding Hq in Eq. (11) up to first order 0 1 2 3 1 ⊗ in δK K0 K K and second order in δq q Q, and η0,1,2,3 are the Pauli matrices acting on the µ = A, B ≡ 3 − 2 1 − 2 ≡ − and ν = 1, 2 degrees of freedom, respectively. Since the and projecting onto these basis vectors, we obtain two terms in HQ commute, we can use an appropriate canonical transformation to recast it in the simpler form 2 2 Hq = 2√3 δK θ δq + δq η˜ (35) − x y 3 HQ = √3 (2K30 K1)˜τ0 + 2K2τ˜3 η˜3. (32) 2 2 − ⊗ χ δqx δqy η˜1 + 2δqxδqyη˜2 , − − This matrix has zero eigenvalues, corresponding to phase 1 transitions, along the lines K2 = (K30 2 K1) in parameter space, while it vanishes identically± at− the intersection where θ and χ are positive numbers. Therefore, the low- of these lines, i.e., at the multicritical point. In the fol- energy theory of the phase transition is a quadratic point lowing, we determine the Chern numbers of the gapped node at momentum Q. The corresponding change in the ˆ phases around the multicritical point by following the Chern number, δCQ, across the phase transition is the two-step scheme shown in Fig.5(g). difference between the Chern numbers of the negative- 1 energy bands at δK > 0 and δK < 0. Since Eq. (35) as- First, we fix K30 = 2 K1 and construct the low-energy theory of the multicritical point node at momentum Q sumes the general form of Eq. (27), these Chern numbers are given by Eq. (29) in terms of the respective vector for K K . By expanding Hq in Eq. (11) up to first 2 1 fields d(q) plotted in Fig. 10. For δK < 0, the vector- order in both K2 and δq q Q, and projecting onto the basis vectors in Eq. (32≡), we− obtain field configuration is topologically trivial, and the Chern number is thus CˆQ, = 0. For δK > 0, the vector-field configuration corresponds− to a double skyrmion, and the Hq = 2√3 K2 (˜τ3 η˜3) (33) ⊗ Chern number is thus CˆQ,+ = 2. Remembering transla- +γ [δqx (˜τ η˜ ) + δqy (˜τ η˜ )] . 1 ⊗ 0 2 ⊗ 3 tion degeneracy, the change in the total Chern number Therefore, the low-energy theory contains a pair of Dirac across the phase transition is then nodes corresponding toη ˜ = 1. Since the two respec- 3 ± tive Dirac nodes have mass coefficients mQ = 2√3 and ˆ ˆ ˆ ± δC = 2 δCQ = 2 CQ,+ CQ, = 4, (36) winding numbers WQ = 1, they contribute CˆQ = 1 to − − ± the Chern number at K2 > 0. Together with the con- ˆ ˆ tributions CQ1 = CQ2 = 1/2 from the two Dirac nodes and the total Chern number of the gapped phase next to at momenta Q1,2 (see Fig.9), the total Chern number of the third Fermi phase is C = 4 + 4 = 8. Since the phase 1 1 the phase at K0 = K1 and K2 > 0 is then 3 2 transition at K30 = 2 K1 K2 is governed by an analogous low-energy theory, the gapped− phase next to the second ˆ ˆ ˆ C = 2 CQ + CQ1 + CQ2 = 4, (34) Fermi phase also has a total Chern number C = 8. 10

V. TOPOLOGICAL ORDERS

V.1. Anyon classes and fusion rules

Since the gapped phases described in the previous sec- tion are all topologically ordered, they have anyonic excitations characterized by particular fusion and braiding properties. Given that gapped Majorana fermions with a total Chern number C are coupled to a Z2 gauge theory, the topological classification is understood in terms of Kitaev’s sixteenfold way [8]. While all phases with different Chern numbers C are topologically distinct theories with different numbers of chiral Majorana edge modes, and are experimentally distinguishable by their thermal Hall conductivities, κxy = πCT/12, at temperature T , there are only 16 distinct classes in terms of their bulk anyon properties, determined by C mod 16. For each phase, we can use the exact solution of our lattice model to explicitly identify the topologically distinct anyon classes of the corresponding theory and to verify the expected fusion rules between them [8]. In general, the Majorana fermions are identified as the fermion excitations , while the gauge fluxes are related to the various classes of vortex excitations. If the Chern number C is odd, there is only one vortex class σ, and each flux excitation with respect to the ground-state flux sector corresponds to such a vortex excitation σ. If the Chern number C is even, there are two topologically distinct vortex classes denoted by e and m for C mod 4 = 0 and by a anda ¯ for C mod 4 = 2. Therefore, a flux excitation at any given plaquette may correspond to either of the FIG. 11. Maps of vortex excitations in our topological orders with even Majorana Chern number C. In each case, plaque- two vortex classes. Since the two vortex classes differ by ttes with ground-state eigenvalues Wp = +1 (Wp = −1) are a fermion in each case, as indicated by the fusion rules marked by white (gray) filling. At each plaquette, a flux excitation flips the eigenvalue of Wp and may correspond to e = m, m = e, × × vortex classes e or m for C mod 4 = 0 (a) and vortex classes a =a, ¯ a¯ = a, (37) a ora ¯ for C mod 4 = 2 (b). At each plaquette with no label, × × the two classes of vortex excitations are degenerate, signaling the vortex classes corresponding to the various plaquettes the existence of a “weak supersymmetry”. can be mapped out by considering the fermion parity of the ground state within the flux sector (p, p0) that contains two flux excitations at a general plaquette p and degenerate internal space. The two states in this internal at a far-away reference plaquette p0. If the ground-state space correspond to two fusion channels into a trivial ex- fermion parities of the flux sectors (p1, p0) and (p2, p0) are citation and a fermion excitation, respectively. In the lat- identical (opposite), the flux excitations at the plaquettes tice model, the internal space manifests as a zero-energy p1 and p2 correspond to identical (distinct) vortex exci- fermion in any flux sector containing two flux excitations tations. For each phase with even C, the resulting map far away from each other. Conversely, if the Chern num- of the vortex classes is depicted in Fig. 11. We note that ber C is even, the vortices are Abelian anyons with only the two vortex classes are related by an anyon permuta- one fusion channel. If C mod 4 = 0, the fusion rules tion symmetry and that the same maps are thus equally valid with e m and a a¯. e e = m m = 1 (38) ↔ ↔ × × In terms of the anyon fusion rules, the fermions have indicate that two identical vortices fuse into a trivial ex- similar properties in all phases. Indeed, the general fu- citation, whereas if C mod 4 = 2, the fusion rules sion rule = 1 indicates that two fermion excitations × fuse into a topologically trivial excitation. In contrast, a a =a ¯ a¯ = (39) there are three distinct scenarios for the fusion rules be- × × tween two identical vortices [8]. If the Chern number C indicate that two identical vortices fuse into a fermion is odd, the fusion rule σ σ = 1 + indicates that the excitation. In the lattice model, these fusion rules are re- vortices are non-Abelian× anyons as a pair of them has a flected in the ground-state fermion parity of a flux sector 11 containing two far-away flux excitations that correspond supersymmetry in the conventional sense because it re- to the same vortex class. For C mod 4 = 0 (C mod 4 = 2), lates the vortex excitations e and m that are both bosons this fermion parity is even (odd) with respect to the over- in terms of their self statistics [8]. Nevertheless, it may all ground state of the model. be interpreted as a generalized “weak supersymmetry” We finally note that the anyon braiding rules are even because it relates topologically distinct anyonic excita- more specific to the topological order than the anyon fu- tions that are different in their mutual statistics with re- sion rules. To explicitly check these braiding rules in our spect to each other. We also note that such a fermionic lattice model, we would need to calculate the complex symmetry does not manifest in the A phase of the origi- hopping matrix elements as one flux excitation is moved nal Kitaev model because translation symmetry does not around another one in such a way that the distance be- have a fixed point and does not map any plaquette onto tween the two flux excitations is much larger than the itself. However, we expect it to be a generic feature of correlation length at each step. Unfortunately, for most symmetry-enriched topological order whenever an anyon of our phases, the correlation length exceeds the maximal permutation symmetry is intertwined with a point-group system size for which such a calculation would be feasible symmetry, such as a rotation or a reflection. at all. Nevertheless, for all phases with sufficiently small correlation lengths, the results of such a calculation are in agreement with the braiding rules in Ref. [8]. VI. DISCUSSION

In this work, we have realized a wide range of distinct V.2. Weak symmetry breaking and supersymmetry topological orders in an exactly solvable spin model on the honeycomb lattice. Each of these topological orders For each map of vortex classes in Fig. 11, it is instruc- is a Z2 gauge theory coupled to Majorana fermions with a tive to consider the interplay between the unbroken lat- total Chern number C. Given their respective Majorana tice symmetries in the given flux sector and the relevant Chern numbers 0, 1, 2, 3, 4, and 8, these topo- anyon permutation symmetry (e m or a a¯). Inter- logical orders correspond± ± to± more± than half± of Kitaev’s estingly, for the 1/2-flux phase and↔ the “lower”↔ 2/3-flux sixteenfold way. In particular, the C = 3 phases realize phase of Fig.2, certain lattice symmetries become inter- non-Abelian topological orders that are± distinct from the twined with the anyon permutation symmetry e m in C = 1 phases of the Kitaev honeycomb model both in the sense that they map plaquettes corresponding↔ to the the number± of Majorana edge modes and in the braiding two vortex classes e and m onto each other. In each case, properties of the non-Abelian anyons. Also, the C = 8 one such lattice symmetry is a twofold rotation around a and C = 4 phases realize Abelian topological orders± in ± z bond separating two Wp = 1 plaquettes. which the gauge fluxes have fermionic and semionic par- This kind of interplay, commonly− known as weak sym- ticle statistics, respectively [8]. Since our model emerges metry breaking [8], was first discussed for the spatially naturally from the Kitaev honeycomb model in the pres- anisotropic gapped phase (“A phase”) of the original ence of perturbations [39], we expect that its topological Kitaev model, where the anyon permutation symmetry orders are likely to be realized in spin-orbit-coupled hon- e m is intertwined with translation symmetry. While eycomb magnets, such as α-RuCl3. there↔ is no symmetry breaking in the conventional sense From an experimental perspective, the key signature as all ground-state correlations are fully symmetric, there of each C = 0 topological order is a specific quantized 6 is a symmetry breaking in the topological properties of value of the thermal Hall conductivity, κxy = πCT/12, the anyonic excitations as symmetries map topologically at any temperature T below the bulk energy gap. This distinct anyons onto each other. quantized value is directly proportional to the chiral cen- Even more remarkably, for both the 1/2-flux phase and tral charge, c = C/2, of the edge theory, whose integer the “lower” 2/3-flux phase of Fig.2, there are certain (fractional) values correspond to Abelian (non-Abelian) plaquettes that do not correspond to any particular vor- bulk topological orders. In the fractional-flux sectors, the tex class e or m (see Fig. 11). Since these plaquettes are topological order is also accompanied by a spontaneous mapped onto themselves by a lattice symmetry that is in- breaking of translation symmetry which, in the presence tertwined with the anyon permutation symmetry e m, of any spin-lattice coupling, gives rise to a periodic lat- a vortex excitation at such a plaquette has a degenerate↔ tice distortion and can thus be picked up with nuclear internal space consisting of two states that correspond magnetic resonance or elastic x-ray scattering. Moreover, to the two vortex classes e and m. In the lattice model, the spontaneous breaking of discrete translation leads to this degenerate internal space manifests as a zero-energy a finite-temperature phase transition [39] that is readily fermion in any flux sector that contains a flux excita- observable in the specific heat. tion at such a plaquette. The creation and annihilation Conceptually, the generalization of the Kitaev honey- operators of the zero-energy fermion can then be iden- comb model in this work facilitates a convenient band- tified as the generators of a fermionic symmetry that is structure engineering for Majorana fermions coupled to a reminiscent of supersymmetry [47]. Z2 gauge theory. In many ways, the resulting topological We emphasize that this fermionic symmetry is not a phases are analogous to those studied in the context of 12 noninteracting electrons. For example, the C = 0 topo- generate, and the Dirac nodes thus come in pairs at each logical orders in this work correspond to Chern6 insulators nodal momentum. If time-reversal symmetry is broken, as the Majorana fermions form topologically nontrivial these Dirac nodes then expand into line nodes instead of bands with finite Chern numbers. In the future, it would gapping out. While the line nodes are stable in the nonin- be interesting to engineer other topological band struc- teracting Majorana theory, they are expected to have in- tures for the Majorana fermions and thereby realize Ma- stabilities against Majorana interactions [48]. The study jorana analogs of other noninteracting topological phases, of these instabilities and the resulting topological phases such as topological insulators. Alternatively, it could be is the subject of ongoing further work. worth completing the realization of Kitaev’s sixteenfold way by engineering Majorana band structures with total Chern numbers 5, 6, and 7. ACKNOWLEDGMENTS ± ± ± We finally note that, compared to the other flux sectors We thank Yong Baek Kim for useful discussions. S.- in this work, the Majorana fermions have a different be- S. Z. and C. D. B. are supported by funding from the havior in the 3/4-flux sector because inversion symmetry Lincoln Chair of Excellence in Physics. The work of acts differently on them. In the presence of time-reversal G. B. H. at ORNL was supported by Laboratory Di- symmetry, the entire Majorana spectrum is twofold de- rector’s Research and Development funds.

[1] X.-G. Wen, Quantum Field Theory of Many-Body Sys- C. Stock, and J. Taylor, Phys. Rev. Lett. 108, 127204 tems (Oxford University Press, Oxford, 2004). (2012). [2] X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 82, [22] F. Ye, S. Chi, H. Cao, B. C. Chakoumakos, J. A. 155138 (2010). Fernandez-Baca, R. Custelcean, T. F. Qi, O. B. Korneta, [3] X. G. Wen and Q. Niu, Phys. Rev. B 41, 9377 (1990). and G. Cao, Phys. Rev. B 85, 180403(R) (2012). [4] A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 [23] R. Comin, G. Levy, B. Ludbrook, Z.-H. Zhu, C. N. Veen- (2006). stra, J. A. Rosen, Y. Singh, P. Gegenwart, D. Stricker, [5] M. Levin and X.-G. Wen, Phys. Rev. Lett. 96, 110405 J. N. Hancock, D. van der Marel, I. S. Elﬁmov, and A. (2006). Damascelli, Phys. Rev. Lett. 109, 266406 (2012). [6] A. Y. Kitaev, Ann. Phys. 303, 2 (2003). [24] S. Hwan Chun, J.-W. Kim, J. Kim, H. Zheng, C. C. [7] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Stoumpos, C. D. Malliakas, J. F. Mitchell, K. Mehlawat, Das Sarma, Rev. Mod. Phys. 80, 1083 (2008). Y. Singh, Y. Choi, T. Gog, A. Al-Zein, M. M. Sala, M. [8] A. Y. Kitaev, Ann. Phys. 321, 2 (2006). Krisch, J. Chaloupka, G. Jackeli, G. Khaliullin, and B. [9] N. D. Mermin, Rev. Mod. Phys. 51, 591 (1979). J. Kim, Nat. Phys. 11, 462 (2015). [10] A. Bernevig and T. Neupert, arXiv:1506.05805. [25] S. C. Williams, R. D. Johnson, F. Freund, S. Choi, A. [11] Y. Kasahara, T. Ohnishi, Y. Mizukami, O. Tanaka, S. Jesche, I. Kimchi, S. Manni, A. Bombardi, P. Manuel, Ma, K. Sugii, N. Kurita, H. Tanaka, J. Nasu, Y. Motome, P. Gegenwart, and R. Coldea, Phys. Rev. B 93, 195158 T. Shibauchi, and Y. Matsuda, Nature 559, 227 (2018). (2016). [12] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, [26] K. Kitagawa, T. Takayama, Y. Matsumoto, A. Kato, R. 017205 (2009). Takano, Y. Kishimoto, R. Dinnebier, G. Jackeli, and H. [13] J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Takagi, Nature 554, 341 (2018). Lett. 105, 027204 (2010). [27] K. W. Plumb, J. P. Clancy, L. J. Sandilands, V. V. [14] H. Liu and G. Khaliullin, Phys. Rev. B 97, 014407 Shankar, Y. F. Hu, K. S. Burch, H.-Y. Kee, and Y.-J. (2018). Kim, Phys. Rev. B 90, 041112(R) (2014). [15] R. Sano, Y. Kato, and Y. Motome, Phys. Rev. B 97, [28] L. J. Sandilands, Y. Tian, K. W. Plumb, Y.-J. Kim, and 014408 (2018). K. S. Burch, Phys. Rev. Lett. 114, 147201 (2015). [16] F.-Y. Li, Y.-D. Li, Y. Yu, A. Paramekanti, and G. Chen, [29] J. A. Sears, M. Songvilay, K. W. Plumb, J. P. Clancy, Y. Phys. Rev. B 95, 085132 (2017). Qiu, Y. Zhao, D. Parshall, and Y.-J. Kim, Phys. Rev. B [17] S.-H. Jang, R. Sano, Y. Kato, and Y. Motome, Phys. 91, 144420 (2015). Rev. B 99, 241106(R) (2019). [30] M. Majumder, M. Schmidt, H. Rosner, A. A. Tsirlin, H. [18] Y. Singh and P. Gegenwart, Phys. Rev. B 82, 064412 Yasuoka, and M. Baenitz, Phys. Rev. B 91, 180401(R) (2010). (2015). [19] X. Liu, T. Berlijn, W.-G. Yin, W. Ku, A. Tsvelik, Y.-J. [31] R. D. Johnson, S. C. Williams, A. A. Haghighirad, J. Kim, H. Gretarsson, Y. Singh, P. Gegenwart, and J. P. Singleton, V. Zapf, P. Manuel, I. I. Mazin, Y. Li, H. O. Hill, Phys. Rev. B 83, 220403(R) (2011). Jeschke, R. Valent´ı, and R. Coldea, Phys. Rev. B 92, [20] Y. Singh, S. Manni, J. Reuther, T. Berlijn, R. Thomale, 235119 (2015). W. Ku, S. Trebst, and P. Gegenwart, Phys. Rev. Lett. [32] L. J. Sandilands, Y. Tian, A. A. Reijnders, H.-S. Kim, 108, 127203 (2012). K. W. Plumb, Y.-J. Kim, H.-Y. Kee, and K. S. Burch, [21] S. K. Choi, R. Coldea, A. N. Kolmogorov, T. Lancaster, Phys. Rev. B 93, 075144 (2016). I. I. Mazin, S. J. Blundell, P. G. Radaelli, Y. Singh, [33] A. Banerjee, C. A. Bridges, J.-Q. Yan, A. A. Aczel, L. Li, P. Gegenwart, K. R. Choi, S.-W. Cheong, P. J. Baker, M. B. Stone, G. E. Granroth, M. D. Lumsden, Y. Yiu, J. 13

Knolle, S. Bhattacharjee, D. L. Kovrizhin, R. Moessner, Phys. Rev. Lett. 123, 057201 (2019). D. A. Tennant, D. G. Mandrus, and S. E. Nagler, Nat. [40] In this work, two sites are “r-th neighbors” if the shortest Mater. 15, 733 (2016). path connecting them consists of r bonds. [34] A. Banerjee, J. Yan, J. Knolle, C. A. Bridges, M. B. [41] E. H. Lieb, Phys. Rev. Lett. 73, 2158 (1994). Stone, M. D. Lumsden, D. G. Mandrus, D. A. Tennant, [42] X.-G. Wen, Phys. Rev. B 65, 165113 (2002). R. Moessner, and S. E. Nagler, Science 356, 1055 (2017). [43] Y.-Z. You, I. Kimchi, and A. Vishwanath, Phys. Rev. B [35] S.-H. Do, S.-Y. Park, J. Yoshitake, J. Nasu, Y. Motome, 86, 085145 (2012). 2 Y. S. Kwon, D. T. Adroja, D. J. Voneshen, K. Kim, T.- [44] Since det HQ = ±| det MQ| = 0 at the nodal momentum H. Jang, J.-H. Park, K.-Y. Choi, and S. Ji, Nat. Phys. Q, the matrix MQ must have a zero eigenvalue. 13, 1079 (2017). [45] In principle, non-generic quadratic point nodes could also [36] J.-Q. Yan, S. Okamoto, Y. Wu, Q. Zheng, H. D. Zhou, appear in the Majorana spectrum. However, since they H. B. Cao, and M. A. McGuire, Phys. Rev. Materials 3, are absent in all flux sectors, we restrict our analysis to 074405 (2019). generic Dirac nodes. [37] J. Xing, H. Cao, E. Emmanouilidou, C. Hu, J. [46] We note that this multicritical point only appears to be Liu, D. Graf, A. P. Ramirez, G. Chen, and N. Ni, a “multicritical line” in Fig.2 because one of its axes is arXiv:1903.03615. the irrelevant parameter K3/K1. [38] G. Sala, M. B. Stone, B. K. Rai, A. F. May, D. S. [47] T. H. Hsieh, G. B. Halász,and T. Grover, Phys. Rev. Parker, G. B. Halász,Y. Q. Cheng, G. Ehlers, V. O. Lett. 117, 166802 (2016). Garlea, Q. Zhang, M. D. Lumsden, A. D. Christianson, [48] M. Hermanns, S. Trebst, and A. Rosch, Phys. Rev. Lett. arXiv:1907.10627. 115, 177205 (2015). [39] S.-S. Zhang, Z. Wang, G. B. Halász,and C. D. Batista,