Color degeneracy of topological defects in quadratic band touching systems
CColor degeneracy of topological defects in quadratic band touching systems
Bitan Roy Department of Physics, Lehigh University, Bethlehem, Pennsylvania, 18015, USA (Dated: April 29, 2020)We study two-dimensional fermionic quadratic band touching (QBT) systems in the presence ofvortex and skyrmion of insulating and superconducting masses. A prototypical example of suchsystems is the Bernal bilayer graphene that supports eight zero-energy modes in the presence of amass vortex with the requisite U(1) symmetry. Inside the vortex core, additional ten masses thatclose an SO(5) algebra can develop local expectation values by splitting the zero modes in five andten different ways by lifting its SO(4) and SU(2) chiral symmetries, respectively. In particular, eachSU(2) chiral symmetry can be broken by three distinct copies of chiral-triplet mass orders, givingrise to the notion of the color or flavor degeneracy among the competing orders. By contrast, askyrmion of three anticommuting masses supports additional six masses in its core, and possessesan SU(2) isospin quantum number, besides the usual generalized U(1) charge. Consequently, charge4 e Kekule pair-density-waves can develop in the skyrmion core of N´eel layer antiferromagnet, whilea skyrmion of quantum spin Hall insulator in addition supports a mundane s -wave pairing. We alsoanalyze the internal algebra of competing orders in the core of these defects on checkerboard orKagome lattice that supports only a single copy of QBT. I. INTRODUCTION
The transition between two distinct broken symmetryphases, even though commonly believed to be first-order,can be continuous when two orderparameters are relatedvia a chiral rotation (see below). Such unconventionalcontinuous phase transition possibly takes place throughproliferation of real space singularities, known as topo-logical defects, when one order resides inside the defectcore of the other, giving rise to the notion of dual or-ders and deconfined criticality [1, 2]. One well studiedexample of such dual or competing orders is the N´eel an-tiferromagnet and valence bond solid in two-dimensionalfrustrated spin models of insulating systems [3–7].The notion of competing orders becomes more trans-parent, when they can be described as composite ob-jects of underlying fermionic degrees of freedom. In thisrespect, massless Dirac fermions, realized in monolayergrpahene (MLG), d -wave superconductor, honeycombKondo-Heisenberg model, constitute an ideal platformto capture the competing orders [8–20]. Namely, in amulticomponent spinor basis (arising from the sublatticeor orbital, valley, and spin degrees of freedom) orderedphases are represented by Dirac bilinears. Two compet-ing orders are then described by mutually anticommut-ing Dirac matrices, which when in addition anticommutewith the Dirac Hamiltonian, are named masses . Natu-rally, the generators of the chiral rotation between anytwo competing masses commute with the Dirac Hamilto-nian, manifesting its chiral symmetry [21, 22].However, representation of ordered phases in termsof Dirac matrices is not limited to the Dirac materials,rather quite natural for any multiband systems. And herewe address competing orders that can be found in thecore of topological defects, such as vortex and skyrmion,in planar fermionic systems, where the valence and con-duction bands in the normal state display biquadratictouching, also known as Luttinger materials. The Bernal stacked bilayer graphene (BLG) is an ideal place to real-ize such unusual gapless fermionic excitations [23]. The-oretical studies have shown that quadratic band touch-ing (QBT) in Bernal BLG can be unstable toward theformation of various broken symmetry phases, the exactnature of which depends on a number of microscopic de-tails, such as the relative strength of various finite rangecomponents of the Coulomb interactions [24–39], even ifthat may require a finite interaction couplings [40–42]. Anumber of ordered phases has also been observed in ex-periments in the presence or absence of external magneticand electric fields [43–50]. Therefore, understanding therole of topological defects and competing orders in QBTsystems is a timely topic of pressing importance.Various theoretical works in the recent past have dis-cussed the possibilities of topological defects and dualorders in BLG in the absence [51–53] as well as in thepresence of magnetic fields [54, 55], and also predicted acharge 4 e s -wave superconductor induced by skyrmion oftopological quantum spin Hall insulator (QSHI) [51, 52].Despite these commendable efforts, the internal algebraof competing orders in QBT systems still remains unex-plored, and constitutes the central theme of the presentwork. Here we use the real Clifford algebra of anticom-muting matrices to address this question [56].The most tantalizing outcomes are possibly the fol-lowing. We find that skyrmions of both N´eel antiferro-magnet and QSHI in BLG accommodate charge 4 e spin-singlet pair-density-waves, assuming two distinct Kekulepatterns on honeycomb lattice, whereas the later one inaddition sustains a mundane s -wave pairing. On theother hand, all three singlet pairings support topologicalQSHI in the mixed phase, while the N´eel order can onlybe realized in the vortex phase of spin-singlet Kekule su-perconductors. Thus, in BLG the competing orders arenot unique , giving rise to the notion of a flavor or colordegeneracy (defined precisely below) among them. Nowwe present an extended summary of our main findings. a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r A. Extended summary of results
The differences in the internal algebra of competingorders in MLG and BLG root into the dispersion of non-interacting fermions, which respectively scales linearlyand quadratically with the momentum in these two sys-tems. Consequently, the number of mass matrices thatcan develop a uniform spectral gap at the band touchingpoints via spontaneous lifting of discrete and/or continu-ous symmetries are 36 [12] and 28 [57] in MLG and BLG,respectively, despite both of them possessing the samesymmetry [23], see Table II. Also in stark contradis-tinction to Dirac systems, we show that QBT does notnecessarily encounter the fermion doubling, and one canrealize a two-component QBT for spinless femrions ontwo-dimensional, such as checkerboard and Kagome [58],lattices. A simple algebraic proof of this statement is of-fered in Appendix A. Here, such a realization is named‘single-flavored QBT’, while the QBTs in Bernal BLG iscoined ‘valley-degenerate QBT’.(1) In a single-flavored QBT system, a vortex ofany two mutually anticommuting masses [see Table I]hosts two states at precise zero-energy and each of themare two-fold degenerate, yielding total four zero-energystates. But, three competing mass matrices can splitthe zero-energy manifold by developing finite expectationvalues, and they close an SU(2) algebra, see Fig. 1. Forexample, the zero-energy modes bound to the vortex ofan s -wave superconductor supports all three componentsof the QSHI. On the other hand, a skyrmion of QSHIaccommodates the s -wave pairing.(2) A real space vortex of two anticommuting masseswith the requisite U(1) symmetry accommodates doubly-degenerate four, thus total eight states at zero energy ina valley-degenerate QBT system. The sub-space of zero-energy states altogether supports ten masses. For exam-ple, a vortex of translational symmetry breaking Kekulecurrent orders sustains the layer polarized state (1), N´eellayer antiferromagnet (3), the real (3) and imaginary (3)components of the spin-triplet f -wave pairing. Quan-tities in the parentheses indicate the number of matri-ces required to describe a particular order, see Table II.Other examples are discussed in Sec. IV B 1. Irrespectiveof these details, the ten masses close an SO(5) algebra. (3) Any set of ten masses can be organized into fivesets of four mutually anticommuting masses, closing anSO(4) algebra [see Fig. 2 and Appendix B]. Therefore,if the system chooses to split the zero-energy manifoldby lifting its SO(4) chiral symmetry, there are five suchchoices. On the other hand, an SO(5) group has ten The maximal number of mutually anti-commuting masses is five in MLG [12, 16], while that is six in BLG [57]. Such two-fold degeneracy of each zero mode is protected by apseudo time-reversal symmetry [51], discussed in Sec. III A. In a Dirac material, such as MLG, the six masses bound to thevortex zero-modes close an SU(2) ⊗ SU(2) ∼ =SO(4) algebra [16]. Mass order Matrix I uv (cid:126)S I T QAHI τ ⊗ σ ⊗ α − (cid:51) − QSHI τ ⊗ (cid:126)σ ⊗ α − (cid:55) + s -wave pairing ( τ , τ ) ⊗ σ ⊗ α + (cid:51) (+ , − )TABLE I: All masses in a single-flavored QBT system incheckerboard lattice [65], and their transformation under theexchanges of two sublattices ( I uv ), rotation of spin quanti-zation axes ( (cid:126)S ), and reversal of time ( I T ). Here, +( − ) cor-responds to even(odd), and (cid:51) and (cid:55) reflect weather a massoperator preserves a particular symmetry or not, respectively.Three sets of Pauli matrices { τ µ } , { σ µ } and { α µ } operateon the Nambu, spin and sublattice indices, respectively, with µ = 0 , · · · ,
3. The real and imaginary components of the s -wave pairing appear with τ and τ , respectively. SO(3) or SU(2) subgroups. Thus, zero-energy subspacecan also be split by breaking its SU(2) chiral symmetryin ten different ways. But, each set of SU(2) generatorsrotate between three distinct set of three mutually anti-commuting masses, see Fig. 3. Hence, each SU(2) chiralsymmetry of zero modes can be lifted in three differentpatterns, leading the notion of the flavor or color degen-eracy among the competing orders inside the vortex core.For example, either the N´eel layer antiferromagnet or thereal and imaginary components of triplet f -wave pairingcan split the zero modes bound to the vortex of singletKekule current orders by spontaneously lifting the SU(2)spin rotational symmetry.(4) In the presence of an underlying skyrmion of threemutually anticommuting masses, there is no bound stateat zero energy. But, the bound states at finite energiespossess an SU(2) ⊗ U(1) chiral symmetry. While the gen-erator of U(1) rotation captures the generalized chargeof the skyrmion, the SU(2) generators correspond to itsisospin, see Fig. 4. Altogether a skyrmion core supportssix induced masses. The U(1) charge causes rotationamong three distinct copies of induces masses, while eachSU(2) generator rotates between two distinct flavors ofmasses. Thus by developing finite expectation value of itscharge or isospin quantum number, a skyrmion core cansupport degenerate flavors of competing induced masses,giving rise the color degeneracy among competing or-ders in its core. Consequently, one can construct mul-tiple copies of five mutually anticommuting masses [seeSec. IV B 2], the right number to sustain a Wess-Zumino-Witten (WZW) term in d = 2 [59, 60], after integratingout the fermions [8, 61]. However, due to the color degen-eracy one can construct charge-WZW and isopin-WZWterms (defined more precisely in Sec. IV B 2), which canbe responsible for continuous and possibly deconfinedphase transitions between competing phases that can alsobe tested in quantum Monte Carlo simulations [62–64]. Mass order Matrix I uv I K (cid:126)S I tr I T SymbolLayer polarized τ ⊗ σ ⊗ η ⊗ α − + (cid:51) (cid:51) + LPQAHI τ ⊗ σ ⊗ η ⊗ α − − (cid:51) (cid:51) − QAHIOdd-Kekule charge current τ ⊗ σ ⊗ η ⊗ α − − (cid:51) (cid:55) − K O Even-Kekule charge current τ ⊗ σ ⊗ η ⊗ α − + (cid:51) (cid:55) − K E Layer antiferromagnet τ ⊗ (cid:126)σ ⊗ η ⊗ α − + (cid:55) (cid:51) − (cid:126) NQSHI τ ⊗ (cid:126)σ ⊗ η ⊗ α − − (cid:55) (cid:51) + (cid:126) SHOdd-Kekule spin current τ ⊗ (cid:126)σ ⊗ η ⊗ α − − (cid:55) (cid:55) + (cid:126) K O Even-Kekule spin current τ ⊗ (cid:126)σ ⊗ η ⊗ α − + (cid:55) (cid:55) + (cid:126) K E s -wave pairing ( τ , τ ) ⊗ σ ⊗ η ⊗ α + + (cid:51) (cid:51) (+ , − ) (S , S ) s -Kekule pairing ( τ , τ ) ⊗ σ ⊗ η ⊗ α + + (cid:51) (cid:55) (+ , − ) (sK , sK ) p -Kekule pairing ( τ , τ ) ⊗ σ ⊗ η ⊗ α + − (cid:51) (cid:55) (+ , − ) (pK , pK ) f -wave pairing ( τ , τ ) ⊗ (cid:126)σ ⊗ η ⊗ α + − (cid:55) (cid:51) (+ , − ) ( (cid:126) F , (cid:126) F )TABLE II: All masses in Bernal BLG (supporting valley-degenerate QBTs) that anticommute with ˆ H BLG0 , see Eq. (4) [57]. Firsteight candidates represent insulating, and last four to fully gapped superconducting states. Among the insulating masses, firstfour are spin-singlet, while the remaining ones are spin-triplet. From the third to seventh column we display the transformationsof these masses under the exchanges of the layers ( I uv ), valleys ( I K ), rotation of the spin quantization axis ( (cid:126)S ), U(1) translationalsymmetry ( I tr ), and reversal of time ( I T ). The Pauli matrices { τ µ } , { σ µ } , { η µ } and { α µ } operate on Nambu or particle-hole,spin, valley and layer indices, respectively, where µ = 0 , · · · ,
3. Rest of the notations are the same as in Table I.
B. Organization
The rest of the paper is organized in the followingway. In the next section, we discuss the microscopic mod-els leading to both single-flavored and valley-degenerateQBTs, and all possible mass orders therein, see Tables Iand II. Topological defects, such as vortex and skyrmions,and the bound states in their cores are discussed inSec. III. Sec. IV is devoted to the derivation of the inter-nal algebra among competing orders in the defect coresusing the real representation of the Clifford algebra. Wesupport these findings through some concrete examplesin Sec. V, and summarize the results in Sec. VI. Addi-tional discussions are relegated to the appendices.
II. MASSES IN QBT SYSTEMS
We begin the discussion by considering microscopicmodels for QBTs. Unlike the situation in two-dimensionalDirac materials, displaying linear touching of the va-lence and conduction bands, for which the minimal rep-resentation must be four component [67, 68] (for spin-less fermions), a two-component QBT can be realized intwo-dimensional lattices with finite-range hopping. Suchrealizations are compatible with the requirement of thetime-reversal symmetry, see Appendix A. Nevertheless,it is also conceivable to realize four-component QBTs intwo-dimensional lattices, such as in Bernal stacked BLGin the presence of intralayer nearest-neighbor and inter-layer dimer hopping elements. In this system two copiesof two-component QBT are realized near two inequiva-lent corners, also known as the valleys, of the hexagonal Brillouin zone [23]. Below we write down the low-energymodels of these systems and tabulate all possible massorders therein, see Table I and II.
A. Single-flavored QBT
The simplest microscopic model, supporting a singlecopy of QBT can be realized on a checkerboard lattice.To accommodate all possible masses in such a system,we introduce an eight-component Nambu spinor Ψ =(Ψ p , Ψ h ) (cid:62) , where Ψ p and Ψ h are two four-componentspinors, with Ψ (cid:62) p = (Ψ p, ↑ , Ψ p, ↓ ) and Ψ (cid:62) h = (Ψ h, ↓ , − Ψ h, ↑ ).The two-component spinors areΨ (cid:62) p,σ = [ u σ , v σ ] ( k ) , Ψ (cid:62) h,σ = (cid:2) u † σ , v † σ (cid:3) ( − k ) . (1)Here u σ ( k ) and v σ ( k ) correspond to fermion annihilationoperators on two sublattices of the checkerboard latticewith momentum k , measured from the band touchingΓ = (0 ,
0) point, and spin projection σ = ↑ , ↓ . In thisbasis, the low-energy Hamiltonian near the Γ point isˆ H SF0 = τ ⊗ σ ⊗ [ α d ( k ) + α d ( k )] , (2)where d ( k ) = k x − k y m ∗ , d ( k ) = 2 k x k y m ∗ , (3)and m ∗ has the dimension of mass. Three sets of Paulimatrices { α µ } , { σ µ } and { τ µ } operate on the sublattice,spin and Nambu indices, respectively, where µ = 0 , , , ⊗ ’ represents a direct or tensor product. Through-out we neglect the particle-hole anisotropy.The above Hamiltonian ( ˆ H SF0 ) is invariant under the(1) exchange to two sublattices ( u ↔ v ), generated by I uv = τ ⊗ σ ⊗ α , under which ( k x , k y ) → ( k y , k x ), (2)reversal of time, generated by the antiunitary operator I T = ( τ ⊗ σ ⊗ α ) K , where K is the complex conju-gation, and (3) rotation of the spin quantization axis,generated by (cid:126)S = τ ⊗ (cid:126)σ ⊗ α .Various mass orders in this system that uniformly gapthe QBT point and their transformation under variousdiscrete ( I uv , I T ) and continuous ( (cid:126)S ) symmetries of ˆ H SF0 are shown in Table I. Altogether, a single-flavored QBTsupports three physical masses, namely the quantumanomalous Hall insulator (QAHI), QSHI, and spin-singlet s -wave pairing. But, it requires six matrices to describethem [65]. Notice that QAHI only anticommutes withˆ H SF0 , but commutes with remaining two masses. Hence,for the following discussion it does not play any role.
B. Valley-degenerate QBTs: Bernal BLG
Next we focus on the QBTs in Bernal BLG. Unlikethe previous example, BLG accommodates two copies ofQBT, yielding the valley degeneracy. The correspondingsixteen dimensional low-energy Hamiltonian readsˆ H BLG0 = τ ⊗ σ ⊗ [( η ⊗ α ) d ( k ) + ( η ⊗ α ) d ( k )] , (4)where the newly introduced set of Pauli matrices { η µ } operate on the valley index. The sixteen-componentNambu spinor basis is Ψ = (Ψ p , Ψ h ) (cid:62) , where Ψ (cid:62) p =(Ψ p, ↑ , Ψ p, ↓ ) and Ψ (cid:62) h = (Ψ h, ↓ , − Ψ h, ↑ ) are two eight-component spinors. The four-component spinors areΨ (cid:62) p,σ = [ u + ,σ , v + ,σ , u − ,σ , v − ,σ ] ( k ) , Ψ (cid:62) h,σ = (cid:104) v † + ,σ , u † + ,σ , v †− ,σ , u †− ,σ (cid:105) ( − k ) , (5)where u ± ,σ ( k ) and v ± ,σ ( k ) are the fermionic annihila-tion operators on two complimentary layers, with Fouriercomponent localized around the nonequivalent valleys at ± K , spin projection σ = ↑ , ↓ , and momentum k , mea-sured from the corresponding valley.The non-interacting Hamiltonian ( ˆ H BLG0 ) remains in-variant under the following symmetries.1. Exchange of two layers: I uv = τ ⊗ σ ⊗ η ⊗ α ,2. exchange of two valleys: I K = τ ⊗ σ ⊗ η ⊗ α ,3. reversal of time: I T = ( τ ⊗ σ ⊗ η ⊗ α ) K ,4. rotation of the spin quantization axis: (cid:126)S = τ ⊗ (cid:126)σ ⊗ η ⊗ α ,5. U(1) translational symmetry: I tr = τ ⊗ σ ⊗ η ⊗ α .It should be notated that the exchange of two layers andvalleys are accompanied by the inversions of momentum axes k y → − k y and k x → − k x , respectively. Therefore,all mass orders can be classified according to their trans-formation under these symmetries, see Table II.Altogether Bernal BLG supports twelve different sym-metry breaking mass orders, among which eight (four)are insulators (superconductors). But, one requires 28matrices to describe them [57]. Note that in BLG Kekulevalence bond solids [69] (both spin-singlet and spin-triplet) no longer represent masses. They are replacedby Kekule current orders (K E , K O , (cid:126) K E and (cid:126) K O ). Fur-thermore, two Kekule spin-triplet mass superconductorsin the pairing channels in MLG [70] are replaced by spin-singlet Kekule pairings in BLG (sK , sK , pK and pK ),reducing the number of mass matrices in BLG to 28 from36 in MLG [12]. These differences will play importantroles in the internal algebra of competing orders insidethe core of topological defects, which we discuss next. III. TOPOLOGICAL DEFECTS
In this section, we introduce topological defects insidevarious mass ordered phases. Specifically, we considervortex and skyrmion, and highlight the structure of thebound states in their cores. This will allow us to con-struct the internal algebra of competing orders in thecore of these defects, discussed in Secs. IV and V.
A. Vortex
The effective single-particle Hamiltonian for a vortex-like point defect involving two anticommuting masses inQBT systems assume the following universal form H vor = γ ∂ y − ∂ x m ∗ + γ ∂ x ∂ y m ∗ + | m ( r ) | ( γ C nφ + γ S nφ ) , (6)where C nφ = cos( nφ ), S nφ = sin( nφ ), with φ as thepolar angle and r as the radial coordinate in the xy plane. The radial profile of m ( r ) is m ( r →
0) = 0 and m ( r → ∞ ) = m , otherwise arbitrary, where m is aconstant. For concreteness, we consider vortex of unitvorticity ( n = 1), as it is the most stable and energeti-cally favored topological defect. Here, γ j s are Hermitianmatrices satisfying the anticommuting Clifford algebra { γ j , γ k } = 2 δ jk . Therefore, γ and γ are the mass ma-trices. Since H vor involves four mutually anticommuting γ matrices, their minimal dimensionality is four.It was shown by Herbut and Lu [51] that due to theQBT in the normal phase, the above Hamiltonian de-scribing a unit vortex supports two modes at precisezero energy. Such two-fold degeneracy of the zero-energymanifold and rest of the spectrum is assured by an anitiu-nitary operator J K = U K , where U is a unitary operator,such that [ H vor , J K ] = 0 and J K = −
1. Therefore, J K M M M E E E FIG. 1: Triangle of three mutually anticommuting masses, M , M and M , that also anticommute with the eight-dimensional vortex Hamiltonian H Namvor [see Eq. (14)] in asingle-flavored QBT system. Three arms represent SU(2) ro-tations, generated by E jk = iM j M k . plays the role of a pseudo time-reversal operator. Exis-tence of such antiunitary operator does not depend onthe choice of representation of the γ matrices. With-out any loss of generality, we choose γ and γ to bepurely imaginary, and γ and γ to be purely real. Then, U = γ γ . While J K endows each energy eigenvaluea two-fold degeneracy, existence of the midgap states isguaranteed by the spectral symmetry, generated by anunitary operator γ , such that { H vor , γ } = 0. In par-ticular, γ = γ γ γ γ is the fifth anticommuting four-dimensional Hermitian γ matrix [71].From the above discussion, we can also infer thecompeting order in core of the mass vortex. Since { H vor , γ } = 0, the two zero-energy modes are the eigen-states of γ with eigenvalue +1 or −
1. Therefore, filledor empty zero modes yields a finite expectation value ofthe mass operator γ , i.e. (cid:104) γ (cid:105) (cid:54) = 0. Then in the core ofthe vortex, constituted by the γ and γ masses, the sys-tem supports their competing mass γ , since { γ , γ j } = 0for j = 1 and 2 (thus qualifying as a mass), as well as j = 3 and 5 (hence, a competing order). For the minimalmodel in Eq. (6), a finite expectation value of γ massplaces the zero modes at a finite energy. However, forsingle-flavored and valley-degenerate QBT systems, soonwe will find out that competing orders can split themsymmetrically about the zero enegry.A single-flavored QBT system is described by eight-dimensional Hermitian matrices. Since the effectivesingle-particle Hamiltonian describing a vortex configura-tion involves only four mutually anticommuting matrices,and their irreducible representation is four-dimensional,the vortex Hamiltonian can always be cast as a orthog-onal sum of two copies of H vor . As a result, the core of a mass vortex hosts four zero energy modes. Followingthe same line of arguments, one can convince herself thatthere exists eight zero energy modes in the core of a massvortex in valley-degenerate QBT systems. In the follow-ing sections, we will discuss the possible competing or-ders and their internal algebra in such higher-dimensionalzero-energy manifolds. B. Skyrmion
Next we consider a skyrmion of mass orders. It involvesthree mutually anticommuting mass matrices. The cor-responding effective single-particle Hamiltonian is H skyr = γ ∂ y − ∂ x m ∗ + γ ∂ x ∂ y m ∗ + m ( r ) γ + m ( r ) γ + m ( r ) γ . (7)For an underlying skyrmion of unit skyrmion number m ( r ) = m (cid:18) rλr + λ C φ , rλr + λ S φ , r − λ r + λ (cid:19) , (8)where the parameter λ determines its core size. Notethat H skyr exhausts all five mutually anticommuting four-dimensional γ matrices. Therefore, we cannot find anyunitary (or antiunitary) matrix that fully anticommuteswith H skyr and all states (including the bound ones) re-side at finite energies. Nonetheless, they continue to en-joy the two-fold degeneracy, as [ H skyr , J K ] = 0.One can render the loss of the spectral symmetry inthe following way. Say, we begin with two zero-energymodes (the eigenstates of γ with eigenvalues +1 or − γ such that itchanges sign as we approach the boundary of the system( r → ∞ ) from its origin ( r = 0). Therefore, additionof the γ mass besides constituting a skyrmion texture,pushes the zero-modes bound to a vortex to finite ener-gies. As a direct consequence of the spectral asymmetry,the core of the skyrmion becomes electrically charged ofcharge + e or − e (depending on the sign of m ). Thecorresponding operator is Q elec = I , where I n is an n -dimensional identity matrix, which is the product of fivemutually anticommuting matrices appearing in H skyr Q skyr = Q elec = γ γ γ γ γ = I , (9)where Q skyr is the generalized charge of a skyrmion [17].For single-flavored and valley-degenerate QBT sys-tems, the Hamiltonian operator in the presence of a back-ground skyrmion of three mutually anticommuting massorders can be cast as direct or orthogonal sum of twoand four copies of H skyr , respectively. Such decompo-sition allows skyrmion to acquire chiral charges , whilebeing electrically neutral. Note that any Hermitian op-erator that commutes with the noninteracting Hamilto-nian ( H SF0 and H BLG0 ), such as (cid:126)S , generates the chiral symmetry of the system, and qualifies as a chiral chargeof the skyrmion. On the other hand, any mass opera-tor that anticommutes with H skyr can develop a finiteexpectation value in the core of a skyrmion. IV. REAL CLIFFORD ALGEBRA ANDCOMPETING ORDERS
In this section, we derive the internal algebra of com-peting orders in the core of topological defects using thereal representation of Clifford algebra. In order to de-scribe various insulating and superconducting mass gapswithin a unified representation, it is useful to double thenumber of fermionic components (Nambu doubling), andinclude both particle and hole in the spinor representa-tion. The resulting massive Nambu Hamiltonian is H Nam m ( k ) = H Nam0 ( k ) + mM. (10)The kinetic energy part of H Nam m ( k ) is given by H Nam0 ( k ) = H ( k ) ⊕ H (cid:62) ( − k ) ≡ (cid:88) j =1 , Γ j d j ( k ) , (11)where Γ j s, M are eight and sixteen dimensional Her-mitian matrices for single-flavored and valley-degenerateQBT systems, respectively. Here H ( k ) = β d ( k ) + β d ( k ) , (12)and β j are mutually anticommuting four and eight di-mensional matrices for these two systems. The Hermitianmatrix M represents a mass order, when it satisfies theanticommutation relation { Γ j , M } = 0. The fully gappedspectra of H Nam m ( k ), namely ± (cid:112) [ k / (2 m ∗ )] + m , thenextend over positive and negative energies.By construction the Nambu Hamiltonian H Nam m ( k )preserves the particle-hole symmetry, generated bythe antiunitary operator I ph = ( σ ⊗ I n ) K , and { H Nam m ( k ) , I ph } = 0, with n = 4 and 8 for single-flavoredand valley-degenerate QBT systems, respectively, and σ is the real off-diagonal Pauli matrix. Since, I ph = +1, itis always possible to find a representation, known as ‘Ma-jorana representation’, in which I ph = K , and H Nam m ( k )is purely imaginary [16, 72]. In real space representation,the operators d j ( k → − i ∇ ) is real . Thus two matricesappearing in the kinetic energy (Γ and Γ ), as well asany mass matrix ( M ) are imaginary . This is strikinglydifferent from the Dirac system, where due to the lin-ear dependence of d j ( k ) ∼ k j on spatial components ofmomentum, Γ j ’s are real . A. Single-flavored QBT
For single-flavored QBT the Nambu Hamiltonian inEq. (11) is eight dimensional, and i Γ j are purely real. Since iM is also real, we first seek to answer the follow-ing questions. What is the maximal number of q , so thatfor p ≥ eight , and mutually anticommuting p + q matrices satisfythe Clifford algebra C ( p, q )? The answer is seven . Theyconstitute C (0 ,
7) Clifford algebra. Two of them, namelyΓ and Γ , be two imaginary kinetic energy matrices,and M j are five mutually anticommuting mass matrices,with j = 1 , · · · , i Γ Γ that anticommutes with the ki-netic energy and satisfies the requisite criteria of a massmatrix. But, i Γ Γ commutes with five other mass ma-trices. Therefore, single-flavored QBT system altogethersupports six mass matrices, which we show explicitly inTable I. The i Γ Γ mass can be identified as the QAHI.Next we consider topological defects in such a system.
1. Vortex
If we construct a vortex out of two mutually anticom-muting masses, say M and M , according to M vor ( x ) = | m ( r ) | [ M C φ + M S φ ] . (13)Then the vortex Hamiltonian, defined as H Namvor = H Nam0 ( k → − i ∇ ) + M vor ( x ) , (14)supports four zero-energy modes. Note that four mu-tually anticommuting matrices in H Namvor close a C (4 , H Namvor can bedecomposed as orthogonal sum of two identical copies ofthe four-dimensional Hamiltonian H vor , shown in Eq. (6).Each of them hosts two zero energy states [51]. Conse-quently, the eight dimensional vortex Hamiltonian H Namvor supports 2 × H Namvor canacquire finite expectation value inside the vortex core bysplitting the zero-energy manifold. The number of suchmatrices is only three , and they are M , M , M , whichtogether close an SU(2) algebra. They can be placed atthree vertices of a triangle, see Fig. 1. Three generatorsof the SU(2) rotations are { E , E , E } , where E jk = iM j M k . Also note that E jk commute with H Namvor , thusgenerating its chiral symmetry.
2. Skyrmion
Next we proceed to construct a skyrmion out of threemutually anticommuting masses, say M , M and M , The Clifford algebra C ( p, q ) defines a set of p + q mutually anti-commuting matrices, where p ( q ) of them squares to +1( − M M M M E E E E E E ( a ) M M M M EM EM E E E E M ( b ) M M M M EM EM E E E E M ( c ) M M M M EM EM E E E E M ( d ) M M M M EM EM E E E E M ( e ) FIG. 2: Geometric representation of five SO(4) subgroups [Eq. (20) and Appendix B], resulting from the generators of SO(5)rotations among ten masses that anticommute with the vortex Hamiltonian H Namvor in a valley-degenerate QBT system. Fourmasses (in red) belonging to each SO(4) group reside at the four vertices of a square. Each arm and diagonal of a square standfor the U(1) rotation between two mutually anticommuting masses. The generators of SO(4) subgroups are depicted in blue. described by the single-particle Hamiltonian H Namskyr = H Nam0 ( k → − i ∇ ) + (cid:88) j =1 m j ( x ) M j (15)where m ( x ) is given in Eq. (8). Now only one of thethree generators of the SU(2) symmetry of H Namvor , namely E , commutes with H Namskyr , and generates the chiralsymmetry. This matrix rotates between remaining twomasses M and M that anticommute with H Namskyr . There-fore, the core of the skyrmion supports these two massesand E represents its unique generalized charge. It isstraightforward to show that Q skyr = E = Γ Γ M M M , (16)the product of five mutually anticommuting matrices ap-pearing in H Namskyr . In this case, total five mutually anti-commuting matrices, yield the right number of matricesin d = 2 to support a WZW term [59, 60]. B. Valley-degenerate QBT
We initiate the discussion on competing phases invalley-degenerate QBT systems by asking the followingquestion. What is the maximal number of q , so that for p ≥ six-teen , and mutually anticommuting p + q matrices satisfy a C ( p, q ) Clifford algebra? The answer is eight . They con-stitute C (0 ,
8) Clifford algebra [57]. Two of them can beused to define the noninteracting Hamiltonian in termsof the imaginary matrices Γ and Γ . The remaining sixmatrices are the mutually anticommuting mass matrices M j with j = 1 , · · · ,
6. Altogether one can construct 28imaginary mass matrices, see Table II. There exists one This is so because seven mutually anticommuting matrices satisfythe constraint i Γ Γ M M M M M ∝ I . imaginary mass, namely i Γ Γ , which anticommutes withthe noninteracting Hamiltonian, but commutes with restof the masses. It is identified as the QAHI, and does notplay any role in the forthcoming discussion.
1. Vortex
First we focus on vortex constituted by two mutu-ally anticommuting mass matrices M and M , followingthe protocol in Eqs. (13) and (14). But, now all matri-ces (Γ , Γ , M , M ) are sixteen-dimensional. Since thesefour matrices satisfy C (4 ,
0) algebra, H Namvor can be castas orthogonal sum of four copies of H vor , see Eq. (6).Consequently, H Namvor supports eight zero-energy modes.Any operator, say X , that anti-commutes with H Namvor can acquire a finite expectation value by splitting the sub-space of the zero-energy states. To establish the internalstructure of such competing orders we need to search forall imaginary matrices X that satisfy the anticommuta-tion relations { X, Γ j } = { X, M j } = 0, for j = 1 ,
2. Onecan immediately find at least four candidates for X : M , M , M , and M . However, they do not exhaust all pos-sibilities. In terms of four imaginary matrices appearingin H Namvor , we can define another Hermitian matrix E = Γ Γ M M . (17)Even though { H Namvor , E } = 0, E by construction is areal. So, E is not a mass. Nevertheless, we can definethe following six imaginary Hermitian matrices M jk = iEM j M k , (18)where 3 ≤ j, k ≤
6, but with j (cid:54) = k and j > k , whichanticommute with H Namvor . Hence, altogether there are ten masses, anicommuting with H Namvor . Any one of themcan acquire finite expectation value by splitting the eight-dimensional subspace of zero energy states.In order to demonstrate the two-fold degeneracy ofthe zero-energy manifold, we search for all possible can-didates for the sixteen-dimensional unitary operator U , M M M M M M E E E E E E IA II AM M M M M M E E E E E E A IV A ( a ) M M M M M M E E E M E M E M E M V A VI AM M M M M M E E E M E M E E A III B ( b ) M M M M M M E E E M E M E M E M A IX AM M M M M M E E E M E M E E B II B ( c ) M M M M M M E E E M E M E M E M X A IX BM M M M M M E E E M E M E E B IV B ( d ) M M M M M M E E E M E M E M E M X B V BM M M M M M E E E M E M E E B IB ( e ) E E E E E E E E E E E E EM EM E EM EM E EM EM E EM EM E EM EM E EM EM E M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M I C II C III C IV C V C VI C VII C VIII C IX C X C
FIG. 3: (a)-(e): Four SU(2) subgroups (each represented by a triangles), resulting from the corresponding SO(4) subgroup [seeFig. 2]. Each arm of a triangle represents a U(1) rotation between two mutually anticommuting masses (red). The correspondinggenerator is shown in blue. Each set of SU(2) generators j (= I, · · · , X ) [see Eq. (21)] rotate between three distinct flavors ofthree mutually anticommuting masses, occupying the vertices of these triangles jα , with α = A, B and C , yielding the color orflavor degeneracy among the competing chiral triplet masses in the vortex core. The SU(2) triangles jC are shown in ( f ). such that we can define the pseudo time-reversal operator J K = U K , satisfying J K = − H V , J K ] = 0. SinceΓ , Γ , M , M are imaginary and J K = −
1, the imagi-nary unitary operator U must satisfy { H V , U } = 0. Dueto the enlarged dimensionality of H Namvor , in fact there areten possible choices of U , given by U ∈ (cid:8) M , M , M , M , M , M , M , M , M , M (cid:9) . Therefore, any one of the ten masses that anticommuteswith H Namvor can be a candidate for U . Since all massmatrices are Hermitian and imaginary J K = −
1. Ifone of them, say M , acquires local expectation value( m ) near the vortex core, there are still six candidatesfor U , namely M j and M j with j = 4 , ,
6, such that (cid:2) J K , H Namvor + m M (cid:3) = 0. Therefore, split zero energymodes continue to enjoy the two-fold degeneracy.A question arises quite naturally. What is the in-ternal algebra among these 10 competing masses? No-tice each member of the set of 10 masses matrices,say M , anticommutes with 6 other masses (namely, M , M , M , M , M , M ), and commutes with 3 othermasses (namely, M , M , M ). Such an algebra is thedefining property of an SO(5) group, constituted by prod-uct matrices. Therefore, 10 masses that can develop fi-nite expectation value within the zero-energy subspaceclose an SO(5) algebra. By contrast, in a Dirac system(such as MLG) six mass orders in the vortex core sat-isfy SU(2) ⊗ SU(2) algebra [16], which is isomorphic to SO(4). The 10 generators of SO(5) rotations (each ofthem causing U(1) rotation between two specific mutu-ally anticommuting masses) are given by
G ∈ (cid:8) EM , EM , EM , EM ,E , E , E , E , E , E (cid:9) , (19)where E jk = iM j M k . Each generator anticommutes(commutes) with 6 (3) other generators, and they closean SO(5) algebra. See also Appendix B.An SO(5) group has five SO(4) subgroups. Betweenany two of them there exists three common generators,precisely the number of common planes between twofour-dimensional subspaces of a five-dimensional sphere.The generators of each SO(4) subgroups are shown inblue in Fig. 2, and in Appendix B we explicitly showthat each each of them satisfies SO(4) ∼ =SU(2) ⊗ SU(2) al-gebra. In addition, one can construct the following five‘four-tuplets’ of four mutually anticommuting masses be-longing to the SO(4) subgroups( a ) ≡ { M , M , M , M } , ( b ) ≡ { M , M , M , M } , ( c ) ≡ { M , M , M , M } , ( d ) ≡ { M , M , M , M } , ( e ) ≡ { M , M , M , M } . (20)In Fig. 2 masses are shown in red, and four masses fromeach SO(4) subgroup reside at the vertices of a square.Four masses belonging to any SO(4) subgroup are mu-tually anticommuting reconciles with the fact that themaximal number of mutually anticommuting mass ma-trices is six in BLG. Therefore, if the system chooses tosplit the zero-energy manifold by breaking SO(4) chiralsymmetry of H Namvor , it can be accomplished in five differ-ent patterns.On the other hand, an SO(5) group has ten SO(3) orSU(2) subgroups. Their generators are the following( I ) ≡ ( E , E , E ) , ( II ) ≡ ( E , E , E ) , ( III ) ≡ ( E , E , E ) , ( IV ) ≡ ( E , E , E ) , (21)( V ) ≡ ( EM , EM , E ) , ( V I ) ≡ ( EM , EM , E ) , ( V II ) ≡ ( EM , EM , E ) , ( V III ) ≡ ( EM , EM , E ) , ( IX ) ≡ ( EM , EM , E ) , ( X ) ≡ ( EM , EM , E ) . For any j = I, · · · , X , three SU(2) generators ( A α s) sat-isfy the group algebra [ A α , A β ] = i(cid:15) αβδ A δ , where (cid:15) αβδ isthe fully antisymmetric Levi-Civita symbol. As shown inFig. 3, each set of SU(2) generators can rotate betweenthree distinct flavors of three mutually anticommutingmasses, occupying the vertices of three triangles jα ,where α = A, B and C . Therefore, if the system choosesto split the zero-energy manifold by breaking its SU(2)chiral symmetry, there are ten choices ( j = I, · · · , X ).And each SU(2) chiral symmetry can be broken by threeflavors ( jA, jB and jC ) of chiral-triplet masses. Suchextra three-fold degeneracy among triplet mass orders istermed the flavor or color degeneracy. In Sec. V B 1 weshow its explicit examples.
2. Skyrmion
Now we focus on skyrmion [see Eq. (15)], with Γ , Γ , M , M and M as sixteen-dimensional imaginary Hermi-tian matrices. In the presence of an underlying skyrmionthere is no bound state at zero energy. But, the onesat finite energies still possess two-fold degeneracy, guar-anteed by the pseudo time-reversal operator J K , with U ∈ { M , M , M , M , M , M } . There are two sets ofthree mutually anticommuting masses that close SU(2)algebra and also anticommute with H Namskyr . They are { M , M , M } and { M , M , M } . and placed at three vertices of two triangles, see Fig. 4.The generators of SU(2) rotations ( E , E , E ) are,however, identical for two SU(2) triangles. In addition tothe intra-triangle SU(2) symmetry, there exist an inter-triangle U(1) symmetry, generated by Y = EM = Γ Γ M M M , (22) Ten SU(2) subgroups can be found in the following way. EachSO(4) subgroup yields two SU(2) subgroup, see Appendix B.Hence, five SO(4) subgroups give ten SU(2) subgroups. EM EM EM M M M M M M E E E E E E FIG. 4: Two sets of three mutually anticommuting masses(red) that also anticommute with H Namskyr and develop fi-nite expectation values in the skyrmion core of a valley-degenerate QBT system, are placed at three vertices of twotriangles. The SU(2) rotations in both triangles are generatedby { E , E , E } (represented by the arms of the trianglesand are shown in blue). The dotted lines represent the inter-triangle U(1) rotations, generated by EM , among three pairsof two identical vertices belonging to two triangles. that rotates between two masses residing at identical ver-tices of two triangles. The generator of the U(1) symme-try is the product of five mutually anticommuting matri-ces appearing in H Namskyr . As [ H Namskyr , E ij ] = [ H Namskyr , Y ] =0, and [ E ij , Y ] = 0, the bound states in the core of theskyrmion possesses SU(2) ⊗ U(1) chiral symmetry, whichgets broken by the induced masses. While Y deter-mines the generalized charge of the skyrmion ( Q skyr ),three generators of the SU(2) rotations correspond to its isospin . The notion of the generalized charge is also ger-mane in Dirac [17] and single-flavored QBT systems [seeSec. IV A 2]. But, ‘isospin’ of the skyrmion is unique tovalley-degenerate QBT systems.Notice that the U(1) charge of the skyrmion rotatesbetween three pairs of distinct induced masses, residingat any two equivalent vertices of two triangles, while eachgenerator of isospin SU(2) symmetry rotates between twocopies distinct induced masses, residing at the end ofidentical arms of two triangles, see Fig. 4. Therefore,the induced U(1) or SU(2) quantum number of skyrmiongives rise to the color degeneracy among the competingorders in its core, which we exemplify in Sec. V B 2.In terms of the masses related by the U(1) rotation,generated by Q skyr = Y , we can define three copies offive-tuplet of mutually anticommuting masses T charge1 = { M , M , M , M , M } , T charge2 = { M , M , M , M , M } , T charge3 = { M , M , M , M , M } . (23)The existence of five mutually anticommuting massesgives the right number to support WZW term in d =2 [59, 60]. Here such a topological term is named charge-0WZW. Due to the flavor degeneracy, the same charge-WSM term can arise for three copies of induced masses.One can also construct isospin-WZW term from the fol-lowing two five tuplets of anticommuting masses T isospin , = { M , M , M , M , M } , T isospin , = { M , M , M , M , M } , (24)where the U(1) rotation between the induced masses,namely ( M , M ) and ( M , M ), is generated by oneof the generators of isospin SU(2) symmetry, namely E . Therefore, same isospin-WZW term can arise fortwo copies of five tuplets of masses. Isospin-WZW termscan also be derived for the induced masses, related viaU(1) rotation generated by E and E . The WZWterm in believed to be responsible for continuous (andpossibly deconfined) quantum phase transition. There-fore, in valley-flavored QBT systems such an unconven-tional quantum phase transition can take the system toa variety of competing broken symmetry phases, due totheir flavor degeneracy in the skyrmion core, a detailedanalysis of which is left for a future investigation. V. EXAMPLES
Upon establishing the internal algebra of competingorders in the core of topological defects of mass orders,we now discuss some physically pertinent examples forboth single-flavored and valley-degenerate QBT systems.
A. single-flavored QBT
To this end we refer to Table I for all masses in thissystems and Sec. II A for the corresponding definition ofeight-component spinor. First, we consider a vortex ofeasy-plane components of QSHI. Therefore, in Eq. (13)( M , M ) = τ ⊗ ( σ , σ ) ⊗ α , and three mutually anticommuting masses are the easy-axis QSHI, and real and imaginary components of thesinglet s -wave pairing. Any one of these three masses cansplit the zero-energy manifold of H Namvor [see Eq. (14)], andacquire a finite expectation value. On the other hand, avortex inside the s -wave paired state, i.e. when( M , M ) = ( τ , τ ) ⊗ σ α , supports all three components of the QSHI inside its core.Next we consider a skyrmion of QSHI. The generalizedcharge of such a skyrmion Q skyr = τ ⊗ σ ⊗ α , (25)is the standard electric charge Q elec in the Nambu ba-sis. Note that Q skyr generates a U(1) rotation between the real and imaginary components of s -wave pairing.Therefore, core of the skyrmion of QSHI supports s -wavepairing, and a vortex of s -wave superconductor allows thelocal formation of QSHI. B. Valley-degenerate QBT
Next we discuss the competing phases in the core ofvortex and skyrmion in Bernal BLG. Readers should con-sult Table II for sixteen-dimensional representation of allmasses, and Sec. II B for the definition of the spinor basis.
1. Vortex
Due to a large number of masses, one can construct amyriad of vortices out of any two mutually anticommut-ing masses shown in Table II. However, we restrict thediscussion to some physically pertinent situations.(1) Vortex of spin-singlet Kekule currents ( M =K E , M = K O ): Ten masses that anticommute with K E and K O are the layer polarized (LP) and layer antiferro-magnet ( (cid:126) N) states, and spin-triplet f -wave superconduc-tor ( (cid:126) F , (cid:126) F ). Four-tuplets of masses forming the SO(4)subgroups of mutually anticommuting masses are (cid:8) LP , F , F , F (cid:9) , (cid:8) LP , F , F , F (cid:9) , (cid:8) N , N , F , F (cid:9) , (cid:8) N , N , F , F (cid:9) , (cid:8) N , N , F , F (cid:9) . If the zero-energy manifold gets split by spontaneouslybreaking the SU(2) spin rotational symmetry ( (cid:126)S ), it canbe accomplished by nucleating either the N´eel layer anti-ferromagnet ( (cid:126)
N), real ( (cid:126) F ) or imaginary ( (cid:126) F ) componentsof the spin-triplet f -wave pairing, representing the colordegeneracy of competing orders in the vortex core.(2) Vortex of easy-plane layer anti-ferromagnet ( M =N , M = N ) supports singlet Kekule currents (K E and K O ), Kekule pair-density-waves (sK , sK , pK andpK ), and the easy-axis component of QSHI (SH ), layeranti-ferromagnet (N ) and f -wave pairing (F and F ).Five SO(4) subgroups of competing masses are (cid:8) N , K E , sK , sK (cid:9) , (cid:8) N , K O , pK , pK (cid:9) , (cid:8) F , F , K O , K E (cid:9) , (cid:8) SH , F , pK , sK (cid:9) , (cid:8) SH , F , pK , sK (cid:9) . In contrast to a Dirac system (such as MLG), the vortexcore of easy-plane N´eel layer antiferromagnet supportsspin-singlet pair-density-waves in BLG.(3) A vortex in the easy-plane of QSHI ( M = SH , M = SH ) supports easy-axis layer anti-ferromagnet(N ), QSHI (SH ) and Kekule spin-currents (K andK ), s -wave pairing (S and S ), and Kekule pair-density-waves (sK , sK , pK and pK ). The associatedfour-tuplets of mutually anticommuting masses are (cid:8) SH , S , pK , sK (cid:9) , (cid:8) SH , S , pK , sK (cid:9) , (cid:8) N , K , , pK (cid:9) , (cid:8) N , K , sK , sK (cid:9) , (cid:8) K , K , S , S (cid:9) . In contrast to a similar situation in MLG, where the vor-tex zero modes only support the s -wave pairing, in BernalBLG they can also accommodate translational symmetrybreaking Kekule pairings. So far, we discussed vortex invarious insulating phases of BLG, discerning sufficientdifferences with their counterparts in MLG. Next we dis-cuss vortex of superconducting masses in this system.(4) First we consider a vortex of s -wave pairing, with M = S and M = S . It supports layer polarizedstate (LP), QSHI ( (cid:126) SH), and Kekule spin-currents ( (cid:126) K E and (cid:126) K O ). The four-tuplet of masses are (cid:8) SH , SH , K , K (cid:9) , (cid:8) SH , SH , K , K (cid:9) , (cid:8) SH , SH , K , K (cid:9) , (cid:8) LP , K , K , K (cid:9) , (cid:8) LP , K , K , K (cid:9) . If the zero-energy manifold gets split by lifting the SU(2)spin rotational symmetry, it can be accompanied byQSHI ( (cid:126)
SH) or two spin-triplet Kekule currents ( (cid:126) K E and (cid:126) K O ), manifesting the flavor degeneracy.(5) In the vortex core of spin-singlet s -Kekule pairing( M = sK , M = sK ), one can find layer antiferromag-net ( (cid:126) N), QSHI ( (cid:126)
SH), and specific components of spin-singlet (K E ) and spin-triplet ( (cid:126) K O ) Kekule currents. Thecorresponding four-tuplet of masses are (cid:8) SH , SH , N , K (cid:9) , (cid:8) SH , SH , N , K (cid:9) , (cid:8) SH , SH , N , K (cid:9) , (cid:8) N , N , N , K E (cid:9) , (cid:8) K , K , K , K E (cid:9) . On the other hand, the SU(2) spin rotational symmetryof the zero modes can be lifted by layer antiferromagnet( (cid:126)
N), QSHI ( (cid:126)
SH) or spin Kekule current ( (cid:126) K O ), manifest-ing the color degeneracy among the competing orders.A similar algebra among ten masses in the vortex of p -Kekule superconductor can be constructed after takingK O → K E and (cid:126) K O → (cid:126) K E . Therefore, vortex core of allspin-singlet superconductors (the s -wave and two Kekuleones) supports topological QSHI.(6) Finally, we focus on the vortex phase of the spin-triplet f -wave pairing. For concreteness, we orient thesuperconducting order parameter along the z -direction(easy-axis), i.e., M = F , M = F . Inside the vor-tex core, one then finds layer polarized state (LP), easy-plane components of layer anti-ferromagnet (N , N ) andspin-triplet Kekule currents (K , K , K , K ), easy-axisQSHI (SH ), and singlet Kekule currents (K E and K O ).The five sets of four mutually anticommuting masses are (cid:8) N , SH , K , K (cid:9) , (cid:8) N , SH , K , K (cid:9) , (cid:8) K E , K O , N , N (cid:9) , (cid:8) LP , K E , K , K (cid:9) , (cid:8) LP , K O , K , K (cid:9) . Therefore, all four gapped supercondcutors supportQSHI and some translational symmetry breaking massesin the vortex core. It is also interesting to notice that thevortex phase of pair-density waves additionally supportsthe N´eel layer antiferromagnet.
2. Skyrmion
Now we consider skyrmion of triplet masses in BLG.(1) N´eel layer anti-ferromagnet, with M j = N j for j =1 , ,
3. The six mass matrices that anticommute with H Namskyr [see Eq. (15)] are Kekule currents (K E and K O )and spin singlet Kekule superconductors (sK , sK , pK and pK ). The generalized U(1) charge of the skyrmion Q N´eel skyr = τ ⊗ σ ⊗ η ⊗ α , (26)is the chiral or valley charge ( Q ch ), which changes signbetween two valley. The valley charge rotates betweenthe following three pairs of masses, (1) K E and K O , (2)sK and pK , and (3) sK and pK , manifesting the flavordegeneracy of competing order. One generator of theSU(2) isospin is the electric charge Q elec = τ ⊗ σ ⊗ η ⊗ α , which rotates between the real and imaginarycomponents of the s -Kekule (sK and sK ) and p -Kekule(pK and pK ) pair-density-waves. Therefore, skyrmioncore of N´eel order can become charged by nucleating aspecific Kekule superconductor. (2) QSHI, with M j = SH j for j = 1 , ,
3. Six com-peting masses are the real and imaginary components of s -wave (S , S ), s -Kekule (sK , sK ) and p -Kekule (pK ,pK ) pairings. The U(1) charge of this skyrmion Q QSHIskyr = τ ⊗ σ ⊗ η ⊗ α , (27)is the regular electric charge ( Q elec ), which rotates be-tween the real and imaginary components of three sin-glet pairings. One generator of SU(2) isospin is the chiralcharge Q ch . Hence, the core of a skyrmion of QSHI canhost three different types of spin-singlet superconductors,leading to the notion of the color degeneracy among com-peting orders. In contrast, only the s -wave pairing canbe realized in the skyrmion core of QSHI in MLG [11]. VI. SUMMARY AND DISCUSSION
To summarize, here we unveil the internal algebra ofcompeting orders inside the core of the topological de-fects, such as vortex and skyrmion, of various the or-dered phases in two-dimensional fermionic systems thatin the normal phase are described by biquadratic touch-ing of the valence and conduction bands. We con-sider two realizations of such systems, describing singled-flavored and valley-degenerate QBTs, respectively real-ized on the checkerboard or Kagome lattice [58] andBernal BLG [23]. In the former system, four zero-energy In Dirac system, such as MLG, a skyrmion of the N´eel order doesnot permit any superconducting mass in its core [12]. s -wavesuperconductor get split by the QSHI, while a skyrmionof QSHI becomes electrically charged and sustains s -wavepairing in its core.The internal algebra of competing orders in valley-degenerate QBT systems is much richer. For example,eight zero-energy vortex modes can supports ten massesthat close an SO(5) algebra. While there are five possiblepatterns for splitting the zero modes by lifting its SO(4)chiral symmetry, they can also be split by spontaneouslybreaking the SU(2) chiral symmetry in ten different ways.Most interestingly, each SU(2) symmetry can be brokenby three distinct set of chiral triplet masses, giving riseto the notion of color or flavor degeneracy of competingorders inside the vortex core. As a concrete example ofsuch flavor degeneracy, we note that zero modes bound tothe vortex of Kekule current orders can be split by spon-taneously breaking the SU(2) spin rotational symmetryby either N´eel layer antiferromagnet or the real and imag-inary components of the spin-triplet f -wave pairing.On the other hand, a skyrmion composed of three mu-tually anticommuting masses possesses an SU(2) ⊗ U(1)chiral symmetry, and therefore supports a generalizedU(1) charge and SU(2) isospin. While the U(1) chargerotates between three distinct pairs of masses, each gen-erator of SU(2) isospin symmetry rotates between twodistinct pairs of masses, once again yielding the coloror flavor degeneracy among competing orders within theskyrmion core. As a concrete outcome of such a rich alge-braic structure, we note that skyrmions of QSHI and N´eelantiferromagnet supports singlet Kekule pairings, whilethe vortex phase of spin-singlet pair-density-waves (s-Kekule and p-Kekule) supports both insulating masses.A question of practical importance arise quite natu-rally. How to stabilize a real space vortex in an or-dered phase? Notice that an easy-plane configurationof N´eel layer antiferromagnet or topological QSHI canbe realized in the presence of an in-plane external mag-netic field [36], which only couples to the spin of electrons(Zeeman coupling) without causing the Landau quanti-zation [73], restricting these two orderparameters withinthe easy-plane, thereby providing the requisite U(1) sym-metry to support a vortex. Superconducting vortex canbe realized in BLG by bringing a type-II superconductor,such as Nb, to close proximity and applying a magneticfield such that H c < H (cid:28) H c . On the other hand,a two-component mass order, such as the spin-singletKekule current, is expected to support vortex defect deepinside the ordered phase.Deep inside a triplet ordered phase, such as layer an-tiferromagent and QSHI, skyrmions are expected to ap-pear naturally. Recently it has been shown that singlet s -wave pairing can be nucleated through the condensa-tion of skyrmions of QSHI in MLG [62]. Furthermore,continuous quantum phase transition between two dis- tinct broken symmetry phases in the presence of topo-logical WZW terms can now be demonstrated in quan-tum Monte Carlo simulations within the half-filled zerothLandau level of MLG, without encountering the infamous sign problem [63, 64]. These recent developments areencouraging, and should be applicable for Bernal BLG,where continuous phase transition driven by charge- andisospin-WZW terms can be tested numerically. Acknowledgments
This work was supported by the Startup grant of Bi-tan Roy from Lehigh University. Author thanks Igor F.Herbut for useful discussions and correspondences, andMax Planck Institute for the Physics of Complex Sys-tems, Dresden, Germany for hospitality.
Appendix A: No doubling for QBT
Low energy excitations around a QBT point in a 2DBrillouin zone is described by the effective Hamiltonian H QBT ( k ) = α d ( k ) + α d ( k ) , (A1)where d j ( k ) are defined in Eq. (3), and α and α are mu-tually anti-commuting Hermitian matrices. But their di-mensionality remains unspecified for now. If there existsanother Hermitian matrix, say β , which anti-commuteswith both α and α , spectral symmetry of the energyeigenvalues is guaranteed. Next we ask the followingquestion. What is the minimum dimensionality of α i s,so that H QBT ( k ) is time-reversal invariant?Let us assume α i s are two-dimensional matrices.The maximal number of mutually anti-commuting two-dimensional Hermitian matrices is three , and they closea C (3 ,
0) algebra. Two of them are purely real, while theremaining one is purely imaginary. One can immediatelyidentify them as the Pauli matrices. Without any looseof generality, we can choose α and α to be purely real .The time reversal symmetry is represented by an anti-unitary operator I t = A K , where A is a unitary matrix.As we focus on the time-reversal symmetric system, I t H QBT ( k ) I − t = H (cid:63) QBT ( − k ) , (A2)since it describes the motion of spinless free fermions onreal space. Moreover, for spinless fermions one must have I t = +1 [66]. Note that d ( k ) , d ( k ) do not change singunder the reversal of time. Since we have taken α and α to be real , Eq. (A2) is satisfied when[ A, α ] = [ A, α ] = 0 . (A3)For two-dimensional matrices, there exist only one matrixwhich commutes with all the three mutually anticommut-ing Pauli matrices, the identity matrix ( σ ) with trace32. The time-reversal operator is, therefore, I t = K , and I t = +1, which is basis independent. Therefore, whenvalence and conduction band display quadratic touch-ing, the minimal representation of such a system can betwo-component, and therefore the system does not nec-essarily encounter the fermion doubling . On the otherhand, when d ( k ) = vk x , and d ( k ) = vk y , where v isthe Fermi velocity, the minimal representation of α and α is four-dimensional for spinless fermions in two dimen-sions [68], which leads to the notion of fermion doublingfor chiral Dirac fermions, such as in MLG, according tothe Nielsen-Ninomiya theorem [67]. Appendix B: Generators of SO(5), SO(4)
Ten generator of an SO(5) group can be labeled as J αβ ,where α, β = 2 , · · · ,
6, which satisfy J αβ = − J βα . Inaddition, they satisfy the following commutation relation[ J αβ , J µν ] = i (cid:2) δ βµ J αν + δ αν J βµ − δ βν J αµ − δ αµ J βν (cid:3) . (B1)In order to show that ten generator from Eq. (19), closean SO(5) algebra we write the first four entries of G as EM j = Γ Γ M M M j ≡ J j , (B2)such that J j = − J j for j = 3 , , ,
6. The rest of the sixentries from G can be expressed as E jk = 2 J jk for j =3 , , ,
6, and they also satisfy the antisymmetry property.Now it is straightforward to show that ten generatorsappearing in G , expressed as J αβ , where α, β = 2 , · · · , A = ( A , A , A ) , B = ( B , B , B ) , (B3)satisfying the commutation relations[ A j , A k ] = i(cid:15) jkl A l , [ B j , B k ] = i(cid:15) jkl B l , [ A j , B k ] = i(cid:15) jkl B l , (B4)for j, k, l = 1 , ,
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