Monopole hierarchy in transitions out of a Dirac spin liquid
MMonopole hierarchy in transitions out of a Dirac spinliquid
Éric Dupuis a , William Witczak-Krempa a,b a Département de physique, Université de Montréal, Montréal (Québec), H3C 3J7, Canada b Centre de Recherches Mathématiques, Université de Montréal; P.O. Box 6128, Centre-villeStation; Montréal (Québec), H3C 3J7, Canada
Abstract
Quantum spin liquids host novel emergent excitations, such as monopoles ofan emergent gauge field. Here, we study the hierarchy of monopole operatorsthat emerges at quantum critical points (QCPs) between a two-dimensionalDirac spin liquid and various ordered phases. This is described by a confine-ment transition of quantum electrodynamics in two spatial dimensions (QED Gross-Neveu theories). Focusing on a spin ordering transition, we get the scal-ing dimension of monopoles at leading order in a large- N expansion, where 2 N is the number of Dirac fermions, as a function of the monopole’s total mag-netic spin. Monopoles with a maximal spin have the smallest scaling dimensionwhile monopoles with a vanishing magnetic spin have the largest one, the sameas in pure QED . The organization of monopoles in multiplets of the QCP’ssymmetry group SU(2) × SU( N ) is shown for general N. Contents1 Introduction 22 Model 53 Scaling dimension with fixed spin 7 /N . . . . . . . . . . . . . . . . 83.3 Solving the gap equations . . . . . . . . . . . . . . . . . . . . . . 113.4 Scaling dimension and spin-Hall mass . . . . . . . . . . . . . . . 12 q
155 Hierarchy as degeneracy lifting 17 N = 2 . . . . . . . . . . . . . . . . . . 175.2 Multiplets at the QCP for general N . . . . . . . . . . . . . . . . 185.2.1 Degeneracy in each magnetic spin sector . . . . . . . . . . 205.2.2 Reduction for N = 3 . . . . . . . . . . . . . . . . . . . . . 22 Preprint submitted to Elsevier February 10, 2021 a r X i v : . [ c ond - m a t . s t r- e l ] F e b Conclusion 22Appendix A Holonomy of the gauge field 23Appendix B General spin-Hall mass 24
Appendix B.1 Gap equations . . . . . . . . . . . . . . . . . . . . . 26Appendix B.2 Analytical solutions for ϑ ∈ { , π/ , π } . . . . . . . 26Appendix B.3 Numerical study for 0 < ϑ < π . . . . . . . . . . . 27 Appendix C Gap equations and µ = 0 q = 1 / monopoles for N = 2 Appendix E.1 SU(2) × SU(2) generators . . . . . . . . . . . . . . . 32Appendix E.2 Rotation of spin monopoles . . . . . . . . . . . . . . 32Appendix E.3 Computing the spin-Hall energy . . . . . . . . . . . 34
Appendix F General reduction problem 34
Appendix F.1 A relation with the permutation group . . . . . . . 34Appendix F.2 Clebsch-Gordan coefficients of the sign irrep . . . . 37Appendix F.3 Dimensions of the reduced irreps . . . . . . . . . . 38Appendix F.4 Monopoles with larger magnetic charges . . . . . . 40
1. Introduction
Anderson first proposed the idea of a quantum spin liquid, an insulator stateemerging in frustrated quantum magnets [1]. To formulate this, he used the ideaof a resonating valence bond theory which describes a highly entangled state. Itwas later realized that these kinds of systems can indeed host exotic phases ofmatter with fractional excitations and emergent gauge fields that evade Landauparadigms. These states motivated the study of gauge theories in a condensedmatter context. An important example is the Dirac spin liquid (DSL) which isdescribed by quantum electrodynamics in 2 + 1 dimensions (QED ) with 2 N flavors of gapless Dirac fermions, with typically N = 2 Dirac cones in quantummagnets.The formulation of the QED model is rather simple, i.e. an abelian gaugefield coupled to fermionic matter. Nevertheless, it is a strongly coupled theorywith a non-trivial IR limit. The model flows to an interacting fixed point for N large enough while its exhibits a chiral symmetry breaking below some finitenumber of fermion flavors [2; 3; 4; 5]. Many recent investigations still explorethis dynamical mass generation as well as other aspects of QED [6; 7; 8; 9; 10;11; 12; 13]. When the UV divergences of QED are regularized by a lattice, asit is naturally the case in condensed matter systems, one also has to account for2he compact nature of the U(1) gauge field. This aspect implies the existenceof topological disorder operators called monopole or instanton operators [14].While monopoles confine the gauge field in a pure gauge theory [15], a suffi-cient number of massless matter flavors screening the monopoles prevents theirproliferation. The scaling dimensions of monopoles operators determine whetheror not monopoles are relevant and destabilize the QED phase. Computations atleading order [14] and subleading order [16] in 1 /N using the state-operator cor-respondence indicate a stable theory for 2 N ≥
12. This result was recently con-firmed with Monte Carlo in the non-compact QED where monopoles are probedusing the background field method [17; 18]. Conformal bootstrap bounds relat-ing simply and doubly charged monopoles also yield coherent results [19; 20].To characterize the stability of a DSL, it should also be known which monopolecharges are allowed by lattice symmetries. The first results in this regard wereobtained on the square and Kagome lattices [21; 22] and were followed by acomprehensive analysis of the monopole transformation properties under manylattice symmetries [23; 24].Monopole operators have been studied in many other contexts with and with-out supersymmetry, including non-abelian gauge theories and Chern-SimonsMatter theories [25; 26; 27; 28; 20; 29]. In fact, monopole operators were firststudied in a bosonic theory, the CP model. This is the prototype model fordeconfined criticality [30; 31; 32; 33], i.e. QCPs with emergent enlarged sym-metry and fractionalized excitations separating classical Landau phases. Themonopoles in this model describe a VBS order. Their scaling dimension havebeen obtained at leading order [34; 35] and subleading order [36; 37] in 1 /N b , atorder O ( q N b ) in the 1 /q and 1 /N b expansion [38], and with numerical methodson various lattices [39; 40; 41]. Another interesting aspect of this model is itsconjectured duality with the QED -Gross-Neveu model (QED -GN) with N = 2fermion flavors [42]. The non-compact realization of this latter model has beenstudied in Refs. [43; 44; 45; 46; 47; 48]. In the compact version of QED -GN,the scaling dimension of monopole operators at leading order in 1 /N were foundto be the same as in QED [49; 50].The QED -GN model also underlies an important aspect of the DSL, whichis that this phase has been described as the parent state for many spin liq-uids [51; 23; 24]. Indeed, the QED -GN model describes the QPT from theDSL to a chiral spin liquid which is induced by tuning a flavor symmetry pre-serving a Gross-Neveu interaction in QED [52]. By tuning other Gross-Neveulike interactions in QED , it is also possible to describe transitions to confinedphases. As a flavor-dependent fermion self-interaction is tuned, fermions becomegapped and their screening effect is lost, letting monopoles proliferate [53; 54]. This leads to an interesting scenario in the Kagome Heisenberg antiferromag- Throughout the text, the theories we describe are assumed to be compact unless statedotherwise. This scheme does not apply to QED -GN where the symmetric fermion mass generates aChern-Simons term which prevents monopole proliferation. N = 2 valleys[55]. The confinement of this QSL to a coplanar antiferromagnet is describedby QED with a chiral Heisenberg Gross-Neveu interaction (QED -cHGN). Inthis model, a spin-Hall mass is condensed which in turn drives the condensationof monopoles with spin quantum numbers yielding the antiferromagnetic order[22; 54].Monopole scaling dimensions in QED -cHGN were obtained at leading orderin 1 /N in Ref. [49]. The minimal monopole scaling dimension obtained is lowerthan in QED . A hierarchy among monopole operators with different quantumnumbers was also found, but a proper complete treatment is still lacking. Theobjective of this paper is to put the hierarchy on a formal footing both quali-tatively and quantitatively. An improved characterization of monopole scalingdimensions in QED -cHGN will yield further analytical results which offer moretesting ground for experimental and numerical explorations of this system. Sim-ilarly, a hierarchy of monopole operators was described in Chern-Simons Mattertheories for monopoles with varying Lorentz spins. At leading order in 1 /N ,these operators share the same scaling dimension, but the degeneracy is liftedby higher order corrections. This effect was also seen in the conformal boot-strap [20]. The hierarchy considered in this work is instead among monopoleswith different flavor quantum numbers. The degeneracy lifting is natural as theQED flavor symmetry is partially broken at the QCP, i.e. in the QED -cHGNmodel.The paper is organized as follows. In the following section, we define theQED -cHGN theory, and review the role of monopole operators and how dif-ferent monopole types are distinguished by the fermionic self-interaction in thismodel. In Sec. 3, we obtain the scaling dimensions of monopoles as a func-tion of their total magnetic spin to put the monopole hierarchy on a formalfooting. In Sec. 4, the scaling dimensions of monopoles are computed with ananalytical approximation valid for large values of the magnetic charge. We in-terpret these results in Sec. 5 as a degeneracy lifting of monopoles in QED as we organize monopoles in multiplets of the reduced flavor symmetry groupdescribing QED -cHGN. In Sec. 6, we summarize our results. In Appendix Aand Appendix B, more general forms of the gap equations appearing in Sec. 3are studied to justify the restricted analysis presented in the main text. InAppendix C, it is shown why a particular region of parameters spaces does notyield solutions of the gap equations. In Appendix D, the diverging sums appear-ing in Sec. 3 are regularized. In Appendix E, the representation of monopoleswith minimal magnetic charge for the N = 2 QCP is explicitly constructed. InAppendix F, we show the detailed computations yielding the symmetry reduc-tion shown in Sec. 5. We also discuss how the analysis in this section may beextended to q > / Monopole operators are not necessarily Lorentz scalars in Ref. [20] as opposed to thestricter definition provided in Ref. [14]. . Model Let us consider 2 N flavors of massless two-component Dirac fermions, ψ A where A = 1 , , . . . , N . In a condensed matter language, these degrees offreedom correspond to the magnetic spin ↑ , ↓ and N nodes in momentum space,typically N = 2 in quantum magnets. These fermions correspond to the spinons,the spin-1 / − -cHGN model, whose action in Euclidean signatureis given by S = Z d x h − ¯Ψ /D a Ψ − h (cid:0) ¯Ψ σ Ψ (cid:1) i + . . . . (1)The fermions are organized in a flavor spinor, Ψ = (cid:0) ψ , ψ , . . . ψ N (cid:1) (cid:124) . Thegauge covariant derivative /D a acting on fermions is given by /D a = γ µ ( ∂ µ − ia µ ) , (2)where a µ is a U(1) gauge field. The Dirac matrices γ µ act on Lorentz spinor com-ponents and may be written in terms of Pauli matrices τ i as γ µ = ( τ , τ , − τ ).The cHGN interaction term has a coupling strength h and is defined with aPauli matrix vector σ acting on the SU(2) magnetic spin subspace. The ellipsisdenotes the Maxwell free action and the contribution from monopole operators M † q .These topological disorder operators owe their existence to the compact na-ture of the U(1) gauge field which implies a 2 π quantization of the magneticflux. Monopole operators insert integer multiples of the quantum flux 4 πq where2 q ∈ Z . Formally, the charge q may defined by the action of the magnetic cur-rent operator j µ top ( x ) = π (cid:15) µνρ ∂ ν a ρ ( x ) on the monopole operator M † q that canbe developed with the operator product expansion [14] j µ top ( x ) M † q (0) ∼ q π x µ | x | M † q (0) + · · · , (3)where the ellipsis denotes less singular terms as | x | →
0. The prefactor corre-sponds to the magnetic field of a Dirac monopole with charge q .We can first begin the description of the quantum phase transition (QPT)by analyzing the non-compact QED -cHGN model. For a sufficiently strongcoupling strength h > h c , a spin-Hall mass is condensed ˆ n · h ¯Ψ σ Ψ i >
0. Byusing a zeta regularization to find the critical coupling, the effective action atthe quantum critical point (QCP) is given by [49] S c eff = − N ln det (cid:0) /D a + φ · σ (cid:1) , (4)where φ is an auxiliary vector boson decoupling the cHGN interaction.5owever, this picture is incomplete. Even if the monopole operators at theQCP are irrelevant and a U(1) top global symmetry emerges in the infrared,the monopole operators will be dangerously irrelevant. While gapless mat-ter in QED may screen the monopoles [14] and prevent the confinement ob-served in a pure U(1) gauge theory [57], the situation changes as fermions aregapped. Their screening effect is lost following the condensation of the spin-Hallmass which in turn drives the proliferation of spin-polarized monopoles [53; 54].This confines the fermions, or in the context of quantum magnets, recombinesthe spinons that are fractionalized excitations of the underlying spin system.Monopole operators are thus an essential ingredient to properly understand thisconfinement-deconfinement transition.A monopole at the QCP is characterized by its scaling dimension ∆ M q thatcontrols the scaling behaviour of the two-point correlation function hM q ( x ) M † q (0) i ∼ | x | M q . (5)Since the model at the QCP is a conformal field theory, the state-operatorcorrespondence can be used to obtain this critical exponent [58]. More precisely,the correspondence implies that within the set of 4 πq flux operators, the minimalscaling dimension ∆ q is equal to the ground state of an alternate theory, namelythe QED -GN model defined on S × R with a magnetic flux Φ = 4 πq piercingthe two-sphere. The external gauge field sourcing this flux Φ = R d A q may bewritten as A q ( x ) = q (1 − cos θ )d φ , (6)or A qφ = q (1 − cos θ ) / sin θ in components notation. The singularity at θ = π can be compensated by a Dirac string which imposes the Dirac condition onthe magnetic charge 2 q ∈ Z as the string must remain invisible. By computingthe free energy of this alternate theory, the scaling dimension ∆ q was computedat leading order in 1 /N [49]. This scaling dimension obtained is smaller thanin QED and, specifically for the minimal magnetic charge, is given by ∆ / =2 N × .
195 + O (1 /N ).As mentioned earlier, spin-polarized monopoles are favored by the spin-Hallmass condensation and yield the order parameter. It is useful to compare howdifferent types of monopole operators behave at the QCP. We first precise what ismeant by “types” of monopoles. While all monopole operators with a magneticcharge q share the same magnetic properties, they are distinguished by thedifferent possible fermion modes dressings that define supplementary quantumnumbers. The fermion occupation also determines Lorentz and gauge properties.For example, a flux operator with a vanishing fermion number is constructed byfilling half of the fermion modes [14; 49]. Among these fermion modes, there are4 | q | N special fermion zero modes which owe their existence to the topologicalcharge q of the flux operators [59]. By filling all negative energy modes andhalf of the zero modes, a flux operator with vanishing fermion number and a minimal scaling dimension is obtained. The zero modes occupation can be6urther constrained to select only operators with a vanishing Lorentz spin. Thismeans that varying which zero modes are dressed defines a set of distinct 4 πq flux operators which are Lorentz scalars, and have equal and minimal scalingdimensions. Those are the monopole operators of QED [14].The situation is slightly different in QED -cHGN. In the mean field the-ory, monopoles at the QCP are described by a non-vanishing spin-Hall mass h φ i A q ,S × R = M q ˆ n coming from the cHGN interaction. This parameter mostnotably confers a non-vanishing energy to the zero modes. This affects the scal-ing dimension of a monopole which is lowered by anti-aligning the spin-Hall massand the monopole magnetic spin polarization. Monopoles with different “zero”modes occupation may then have different scaling dimensions, i.e. there is amonopole hierarchy in QED -cHGN. While the hierarchy was partly exploredthough this prism in Ref. [49], here we provide a more complete and accuratediscussion on this point. Notably, here the spin-Hall mass is not fixed by themonopole with the lowest scaling dimension, it can instead vary in amplitudeand orientation for each type of monopole. In this manner, we characterize thehierarchy among monopoles operators by obtaining the scaling dimension as afunction of the monopole magnetic spin s in Sec. 3. We then show in Sec. 5 howmonopoles are organized as irreducible representations of the QCP symmetrygroup SU(2) × SU( N ) and how this makes contact with the scaling dimensionresults.
3. Scaling dimension with fixed spin
We now determine the scaling dimension of monopole operators M q ; s withmagnetic charge q and with magnetic spin s . The minimal scaling dimension inthis sector, ∆ q ; s , is obtained through the state-operator correspondence∆ q ; s = lim β →∞ F q ; s ≡ − lim β →∞ β − ln Z s [ A q ] , (7)where F q ; s is the rescaled free energy. The partition function and free energydefine a ground state for an alternate version of the QED -cHGN action at theQCP (4). As discussed in the previous section, this alternate model is definedon a compactified spacetime S × S β , where the “time” direction is also takencompact as it is regularized on a “thermal” circle with radius β . Additionally,an external magnetic field A q (6) is added to encode the magnetic flux of themonopole operator. Finally, the magnetic spin s of the monopole operator is Since these modes do not have a vanishing energy but nevertheless keep their topologicalorigin and their chiral aspect, we refer to them as “zero” modes. Here and throughout the text, we put quotes to emphasize that this is not the originaltime direction on R but rather the real direction obtained with conformal transformation R → S × R . µ S in the action S cS × S β [ A q ] + βµ S (cid:16) s ( s + 1) − S (cid:17) , (8)where we introduced the total spin operator (averaged over time) S = β − Z S × S β ¯Ψ γ σ Ψ . (9)More explicitly, this operator may be written as S = β − R S × S β Ψ † σ Ψ. Thelagrange multiplier equation yields the constraint h S i = s ( s + 1) which setsthe magnetic spin of the monopole operator.Since a monopole operator is dressed with half of the 4 | q | N fermion “zero”modes, its maximal spin is s max = | q | N . This corresponds to configurationswhere only “zero” modes with the same spin-1 / s min = 0 is obtained by dressingan equal proportion of “zero” modes with opposite spins. The magnetic spinof a monopole obtained by filling the Dirac sea and half of the zero modes isthus bounded as 0 ≤ s ≤ | q | N . (10) /N The interaction added in Eq. (8) to the QED -cHGN action to constrain thetotal monopole spin is quartic in fermions and can be decoupled with an auxil-iary boson field χ . As the spin-squared interaction is not diagonal in spacetime S = β − ( R x ¯Ψ γ σ Ψ · R y ¯Ψ γ σ Ψ), we should in principle introduce the bo-son in the same way. However, as we only seek to describe the free energy atleading order, we may replace the spin interaction with a diagonal formulation S → V β − R x ( ¯Ψ γ σ Ψ) , where V = R S √ gd x is the area of the two-sphere S . This will not affect the results as the expectation value is taken to behomogeneous. The auxiliary boson may then be introduced as the followingresolution of the identity Z D χ exp (cid:26) − Z χ V (cid:27) = Z D χ exp (cid:26) − Z V (cid:16) χ − V p µ S (cid:0) ¯Ψ γ σ Ψ (cid:1)(cid:17) (cid:27) . (11)For later convenience, we note that the equation of motion for χ relates theexpectation value of the boson to the spin polarization as h χ i√ µ S = V h ¯Ψ γ σ Ψ i . (12) In fact, this is not possible for odd N as one “zero” mode always remains unmatched, i.e. s min = 1 /
2. However, this effect is subleading in 1 /N and we only focus on the leading order. χ (and the boson φ decoupling the cHGN interactionin the original model QED -cHGN), fermions in the action (8) can be integratedout, yielding the following effective action S c eff = − N ln det (cid:18) /D a,A q − q µ S χ · γ σ + φ · σ (cid:19) + βµ S s ( s + 1) + Z χ V . (13)We can now compute the free energy (7) using a large- N expansion correspond-ing to a saddle point expansion of the effective action. The leading order willbe obtained by computing the saddle point value of the effective action F L . O .q ; s = N F (0) q ; s = 1 β S c eff (cid:12)(cid:12) Saddle pt. . (14)We take the saddle point configurations to be homogeneous. The two aux-iliary bosons are SU(2) Spin vectors and one of them may be oriented along ˆ z without loss of generality. We orient the first auxiliary boson φ along a generalunit vector ˆ n , h φ i = M q ˆ n , (15)where M q > The second boson can then be writtensimply as h χ i = P z ˆ z = p µ S m s ˆ z , (16)where the spin polarization m s (12) can be positive, negative or zero. As for thedynamical gauge field, gauge invariance requires its expectation value to vanish h a µ i = 0. These mean field ansatz are inserted in the effective action. The free en-ergy can be further simplified by recognizing that the spin variables scale as N since their magnitude can be formulated as a fraction of the total number offermion “zero” modes (10). With this in mind, a subleading term in the totalspin charge s ( s + 1) = s + O ( N ) is dropped and the total spin and spin polar-ization s, m s → N s, N m s are rescaled. The rescaled total spin is thus boundedfrom above by s max = q , i.e. s max = 1 / F (0) q ; s = f q ; s + µ m s (cid:0) s + m s (cid:1) , (17) By SU(2)
Spin vectors, we only refer to the spin flavor group without specifying transfor-mation properties under time reversal. The spin-Hall mass could be written as M q ; s since its value will depend on the magneticspin s . We omit this index for simplicity. In fact, the gauge field may have a non-trivial holonomy on the “thermal” circle R β dτ h a i 6 = 0. However, in the zero “temperature” limit, it is sufficient to take h a i = 0. SeeAppendix A f q ; s is the determinant operator f q ; s = − β − ln det (cid:0) /D − iµ σ z ,A q + M q ˆ n · σ (cid:1) , (18)and where we defined µ = µ S m s . (19)The saddle point parameters must solve the gap equations and minimize thefree energy. Before determining them, we need to reexpress the free energy bydeveloping the determinant operator. In QED -cHGN [49], the basis of spinormonopole harmonics can be used to diagonalize the determinant operator. Theprocedure is a simple generalization of the pure QED case [25; 16]. Here,the formulation is a bit more involved since the spin-Hall mass and the spinpolarization are not necessarily along the same axis. In what follows, we willsuppose this is the case and write M q = M q ˆ z , where we recall that M q > ϑ = 0 and ϑ = π are found in what follows. These twosolutions yield opposite spin polarizations m s , which supports the idea thatauxiliary bosons should anti-align. Another analytical solution exists for ϑ = π/
2, but it yields a larger scaling dimension and can thus be discarded as itdoes not correspond to a global minimum of the free energy. The intutionthat the auxiliary bosons should anti-align is confirmed in Appendix B as noother solutions are found when taking a general orientation of the spin-Hall mass M q = M q (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ ). We thus proceed with the simplification M q = M q ˆ z .The determinant operator (18) can be diagonalized by introducing spinormonopole harmonics S ± q,‘,m . These functions diagonalize generalized total spinoroperator J q → j ± ( j ± + 1), where j ± = ‘ ± /
2. The azimuthal and magneticquantum numbers, respectively ‘ ∈ {| q | , | q | + 1 , . . . } and m ∈ {− ‘, − ‘ + 1 , . . . ‘ } ,define the eigenvalues of L q and L zq which are diagonalized by monopole harmon-ics [60] which serve as components of the spinor monopole harmonics. For mini-mal angular momentum ‘ = | q | , only the S ± q, | q | ,m spinor exists and it correspondsto a zero mode of the Dirac operator. In the j = ‘ − / S + q ; ‘ − ,m , S − q ; ‘,m ) (cid:124) ,the Dirac operator becomes a matrix with c-number entries [14]. As for thespin-Hall mass, its contribution is diagonal in this basis as noted in Ref. [49].The resulting diagonal determinant operator therein is adapted by shifting theMatsubara frequency to account for the presence of the spin chemical potential f q ; s = − β X σ = ± X n ∈ Z (cid:20) d q ln (cid:0) ω n − iµ σ + iσM q (cid:1) + ∞ X ‘ = q +1 d ‘ ln (cid:0) ( ω n − iµ σ ) + ε ‘ (cid:1)(cid:21) , (20)where ε ‘ is the energy and d ‘ is the degeneracy ε ‘ = q ‘ − q + M q , d ‘ = 2 ‘ , (21)10nd ω n are the fermionic Mastubara frequencies ω n = 2 πβ ( n + 1 / , n ∈ Z . (22)Note that the energy ε ‘ is dimensionless as we choose units where the radiusof the two-sphere p V / (4 π ) is equal to one. Also, we have supposed that themagnetic charge is positive q >
0. In the end, the monopole scaling dimen-sion is independent of the sign of the charge. Taking the sum over Mastubarafrequencies , Eq. (20) is simplified to f q ; s = − β − (cid:20) d q ln (cid:0) β ( M q − µ ))] (cid:1) + ∞ X ‘ = q +1 d ‘ ln (cid:0) βε ‘ ) + cosh( βµ )] (cid:1)(cid:21) . (23) We now obtain the gap equations by varying the free energy with respect tothe original saddle point parameters M q , µ S , P z ∂ M q f q ; s = 0 , (24) m s ∂ µ f q ; s + s = 0 , (25) q µ S ∂ µ f q ; s + P z = 0 . (26)The last gap equation could be solved with µ S = P z = 0, but this yields un-physical results (see Appendix C). Instead, if we take µ S = 0 and P z = 0, thethird gap equation can be written as ∂ µ f q ; s = − m s , (27)where we used Eq. (16). The LHS can be developed explicitly as ∂ µ f q ; s = d q (cid:18) sinh( β ( M q − µ ))1 + cosh( β ( M q − µ )) (cid:19) − ∞ X ‘ = q +1 d ‘ (cid:18) sinh( βµ )cosh( βε ‘ ) + cosh( βµ ) (cid:19) . (28)By taking µ as µ = M q + β − ln (cid:18) m s /d q − m s /d q (cid:19) , (29) We may define ˜ M q = M q − µ in the first term and ˜ µ = µσ in the second term, anddirectly read results from Ref. [49] /β since µ < ε q +1 while thefirst term in Eq. (28) yields the required result (27). Inserting this in the secondgap equation (25), we obtain m s = s . (30)This means that the polarization is maximized. We turn to the remaining gapequation for M q . The derivative of the determinant operator with respect to M q is given by ∂ M q f q ; s = − d q (cid:18) sinh( β ( M q − µ ))1 + cosh( β ( M q − µ )) (cid:19) − M q ∞ X ‘ = q +1 d ‘ ε − ‘ sinh( βε ‘ )cosh( βε ‘ ) + cosh( βµ ) . (31)Inserting in this expression the result for µ (29), the first gap equation (24)becomes 2 m s − M q P ‘ d ‘ (cid:15) − ‘ = 0. A positive solution for M q can only be foundfor m s <
0. This shows, as mentioned above, that the spin polarization and thespin-Hall mass should be anti-aligned. Taking m s = − s , the gap equation thenbecomes − s − M q ∞ X ‘ = q +1 d ‘ (cid:15) − ‘ = 0 . (32)This equation can be solved for any allowed spin 0 ≤ s ≤ d q /
2. We discuss itssolutions later on.We first turn to the computation of the free energy and the scaling dimension.Using the solution for µ (29), the determinant operator (18) at leading orderin 1 /β is given by − P ‘ d ‘ ε ‘ . The rest of the free energy (17) is reexpressedusing µ = M q + O (1 /β ) and m s = − s . Taking the zero “temperature” limitof the resulting free energy, we obtain the scaling dimension at leading order in1 /N ∆ q ; s = − N (cid:18) sM q + ∞ X ‘ = q +1 d ‘ ε ‘ (cid:19) + O (1 /N ) . (33) The last result (33) shows how monopoles with the largest spin s have aminimal contribution of the spin-Hall mass to their scaling dimension. The onlyremaining parameter is the spin-Hall mass M q as other fields were evaluated.This mass can be obtained by solving numerically a regularized version of the Combining Eqs. (16) and (19), we may note that P z = p µ m s , which, since µ > m s <
0, is imaginary. As χ is an auxiliary boson introduced as a resolution of the identity(11), an imaginary expectation value poses no problem: A gaussian integral shifted in thecomplex plane yields the same result. s is found. Both regularized expressionsare shown in Appendix D. Here, we simply show their solutions for multiplevalues of s .The spin-Hall mass M q and the scaling dimension ∆ q ; s for the minimalmagnetic charge q = 1 / s in Fig. 1. For the maximal spin s max = d q /
2, the gap equation and the mini- Δ / /( ) M / / ( N / ) Figure 1: Scaling dimension ∆ q ; s per number of fermion flavors 2 N and spin-Hall mass M q as a function of the monopole spin s for a minimal magnetic charge q = 1 /
2. The spin isexpressed as a fraction of its maximal value s max = N/
2. Here, the spin s is not rescaled by N . mal scaling dimension in the 4 πq sector of QED -cHGN found in Ref. [49] areretrieved, that is ∆ q ; s max = ∆ QED -cHGN q . More explicitly, the scaling dimensionis given by ∆ q ; s max = − N (cid:16) d q M q + 2 X ‘ d ‘ ε ‘ (cid:17)(cid:12)(cid:12)(cid:12) Saddle pt. + O ( N ) , (34)where the spin-Hall mass is evaluated at its saddle point value found by solvingthe gap equation (32) for s = s max . The last expression for ∆ q ; s max yields theminimal scaling dimension ∆ QED -cHGN q found in Ref. [49]. For a minimal spin s min = 0, the mass at saddle point vanishes. In turn, this means that the leadingorder scaling dimension corresponds to the monopole scaling dimension in pure13ED , ∆ q ; s min = ∆ QED q , or, more explicitly,∆ q ;0 = − N X ‘ d ‘ ε ‘ | M q =0 + O ( N ) . (35)The scaling dimension ranges between these values for intermediate values ofthe spin s min < s < s max ∆ QED -cHGN q ≤ ∆ QED -cHGN M q ≤ ∆ QED q , L.O. in 1 /N . (36)This result includes the effect of fermion occupation on the mass M q . Thisaspect was neglected in Ref. [49] where the mass was considered fixed in orien-tation and amplitude, defined as to yield the smallest possible lower bound forscaling dimensions in the 4 πq sector. The operator corresponding to this mini-mal scaling dimension was dubbed “spin down” monopole and other monopoleswere obtained by modifying its “zero” modes occupation. Here, by finding anoptimal spin-Hall mass parameter in each spin sector, a smaller upper boundaryon monopole scaling dimensions in QED -cHGN (36) is found. This is schema-tized in Fig. 2 for the case q = 1 / N = 2. E (a) E (b)Figure 2: Schematic representation of the fermion modes dressing of monopole operators with N = 2 valleys v = L, R and a minimal magnetic charge q = 1 /
2. (a) The s = 1 , m s = − h ¯Ψ σ Ψ i ∝ M q ˆ z where M q >
0. (b)A spin singlet monopole ( s = m s = 0) is described by a vanishing spin-Hall mass M q = 0.Here, the monopole polarized along the L valley is shown. This hierarchy is characterized explicitly for various values of spin and mag-netic charge. Monopole scaling dimensions obtained numerically are shown inFig. 3. Every q = 1 / q . This is not necessarily the case for monopoleswith larger magnetic charges, e.g. ∆ s min < ∆ s max . Analytical approxima-tions for the scaling dimensions obtained with a large- q expansion and shown inTable 1 are also plotted in Fig. 3. There is a good agreement with the numericalresults even for small values of the magnetic charge.14 / s max / / Δ q ; s / ( N ) / Figure 3: Scaling dimension ∆ q ; s per number of fermion flavors 2 N as a function of q for s/s max = { , / , / , } where s max = qN . The lines correspond to analytical approxima-tions of the scaling dimensions obtained through a large- q expansion and shown in Table 1.
4. Scaling dimensions for large- q The scaling dimension of monopole operators may also be approximatedby an analytical expression obtained with a large- q expansion. The expansionwas presented in Ref. [49] for the monopole with maximal spin and minimalscaling dimension. Building on this result, analytical approximations were alsoproposed for other types of monopoles, but these results again neglected thebackreaction of the “zero“ modes occupation on the spin-Hall mass. Here, weshow the proper analysis for any magnetic spin s .To start this computation, we may read the unregularized free energy (at zerotemperature) off the scaling dimension (33) with the relation ∆ q ; s = N F (0) q ; s + . . . By changing the summation index ‘ → ‘ + q + 1, the free energy becomes F (0) q − qM q ˜ s − ∞ X ‘ =0 ( ‘ + q + 1) q ( ‘ + q + 1) − q + M q , (37)where we introduced ˜ s ≡ s/s max in order to factorize the dependence on q . The15addle point equation is then given by − q ˜ s − M q ∞ X ‘ =0 ‘ + q + 1 q ( ‘ + q + 1) + M q − q = 0 . (38)Inserting the following mass squared ansatz M q = 2 χ q + χ + O ( q − ) , (39)the free energy (37) and the gap equation (38) can be expanded in powers of1 /q , respectively12 F (0) q = − √ q / (cid:16) ζ − / + ˜ sχ / (cid:17) − q / √ (cid:18)(cid:0) χ + χ (cid:1) ζ / − χ ζ − / + 5 ζ − / + ˜ sχ χ − / (cid:19) + O ( q − / ) . (40)and 0 = 4 q / (cid:16) ζ / + ˜ sχ − / (cid:17) − q − / (cid:16) (cid:0) χ + χ (cid:1) ζ / + 2 χ ζ / − ζ − / + ˜ sχ χ − / (cid:17) + O ( q − / ) , (41)where ζ n ≡ ζ ( n, χ ) is the zeta function used to regularize P l =0 ( l + (1 + χ )) − n .The gap equation at leading order yields a transcendental condition defining χ while χ is determined by a linear condition at next-to-leading order ζ / + ˜ sχ − / = 0 , (42) χ + 2 χ / (cid:0) χ ζ / + 2 χ ζ / − ζ − / (cid:1) ˜ s + χ / ζ / = 0 . (43)We solve these equations by fixing s and by finding numerically the values of χ and χ . These coefficients yield the value of the spin-Hall mass and are insertedin the free energy (40). The resulting monopole scaling dimensions at leadingorder in 1 /N for various magnetic spin s are shown in Table 1. The lines inFig. 3 giving the monopole scaling dimensions against q are plotted using theseanalytic approximations.Note that the absence of the O ( q ) term in ∆ q ; s is expected. In 2 + 1-dimensional CFTs with a U(1) global charge q , the q contribution in the large- q expansion of the scaling dimension is predicted to be universal. It must thusbe independent of the number of fermion flavors 2 N defining our specific model[61; 38], i.e. the O ( q N ) in the large- N and large- q expansion of ∆ q ; s mustvanish. As noted in last section, the gap equation (38) for s = 0 is solved for a vanishing spin-Hallmass M q = 0. In this case, we must simply take χ = χ = 0. able 1: Analytical approximation of the monopole scaling dimensions for s/s max = { , / , / , } obtained in the large- q expansion. ss max ∆ q ; s N + O ( q − / ) + O ( N − )1 0 . q / + 0 . q / / . q / + 0 . q / / . q / + 0 . q / . q / + 0 . q /
5. Hierarchy as degeneracy lifting
The hierarchy shown in Eq. (36) is now analyzed from the point of viewof symmetry. Our non-perturbative analysis does not depend on a large- N expansion. The multiplet organization of monopoles at the QCP is obtained byshowing how monopoles in the DSL reorganize as the flavor symmetry of QED is broken to the flavor symmetry of QED -cHGN. To make contact with thescaling dimensions obtained in the last section, it is important to recognize thatthe spin-Hall mass is a SU(2) Spin vector.The cHGN interaction δ L ∼ ( ¯Ψ σ Ψ) inducing the confinement-deconfinementtransition breaks down the flavor symmetry of QED asSU(2 N ) → SU(2) × SU( N ) . (44)While monopoles in QED are all related by SU(2 N ) rotations and share thesame scaling dimension, this is not the case in QED − cHGN. The hierarchyof monopole operators in QED − cHGN observed in the previous section maybe explained as a degeneracy lifting of monopoles in QED .Monopoles in QED are organized as irreducible representations (irreps) ofthe flavor symmetry group SU(2 N ). We focus our attention of monopoles witha minimal magnetic charge q = 1 /
2. This is the simplest case as monopoleoperators are then automatically Lorentz scalars [14]. The case with a largermagnetic charge is briefly discussed in Appendix F. It is useful to first definea bare monopole operator M † Bare creating a 2 π magnetic flux background andfilled only with negative energy modes. A monopole operator is then obtainedby adding in half of the 2 N zero modes creation operators c † I i M † I ...I N = c † I . . . c † I N M † Bare , I i ∈ { , , . . . , N } . (45)Given the antisymmetric commutation relations between the fermionic creationoperators, the expression above clearly shows how q = 1 / form the rank- N completely antisymmetric irrep of SU(2 N ). N = 2We first discuss the monopole hierarchy for a finite N situation to providesome intuition. While the discussion is centered on symmetries and not dynam-ics, we do need to assume that the QCP still exists at finite N . We focus on17he case with N = 2 valleys v = L, R which is the most relevant to quantummagnets. Monopole operators then have two zero modes creation operators andmay be expressed in the following form c † A ( c † ) (cid:124) M † Bare , (46)where A acts on vectors in flavor space c † = (cid:0) c †↑ ,L , c †↑ ,R , c †↓ ,L , c †↓ ,R (cid:1) . At theQED -cHGN QCP, monopoles are organized as triplets [22; 23] M † Spin = c † ( σ y σ ⊗ µ y ) ( c † ) (cid:124) M † Bare , (47) M † Nodal = c † ( σ y ⊗ µ y µ ) ( c † ) (cid:124) M † Bare , (48)where σ and µ are Pauli matrices vectors respectively acting on magnetic spinand nodal subspaces (in our notation, µ z = | L i h L | − | R i h R | and σ z has theusual definition |↑i h↑| − |↓i h↓| ). In particular, the spin triplet forms the orderparameter ˆ n · h i M † Spin i for coplanar antiferromagnetic phase of the Kagomeheisenberg antiferromagnet [22; 54].More formally, monopoles in QED (46) form the rank-2 completely anti-symmetric irrep, that we note by its dimension . Following the symmetryreduction (44), this irrep is reduced in irreps ( m , n ) with dimension m × n ofthe QCP subgroup SU(2) Spin × SU(2)
Nodal → ( , ) ⊕ ( , ) . (49)The spin and nodal triplet, respectively ( , ) and ( , ), have total magneticspin given by S = 1 and S = 0. They are the finite- N analogues of q = 1 / s max = 1 and s min = 0, respectively, described in Sec. 3 M † / s max → ( , ) , M † / s min → ( , ) . (50)As we perform a SU(2) Spin rotation, the “zero” modes occupation is modifiedand so does the orientation of the spin-Hall mass. For example, a spin flip ex-changes spin down and spin up “zero” modes, changing the polarization sign ofa monopole, i.e. M † q ; s,m s → M † q ; s, − m s . With this transformation, the sign ofthe spin-Hall mass also changes h ¯Ψ σ z Ψ i → −h ¯Ψ σ z Ψ i , which leaves the scalingdimension unchanged. The spin flip is schematically shown in Fig. 4. In Ap-pendix E, we explore the explicit realization of monopoles suggested in Eqs. (47-48) and we also obtain a representation of the SU(2) × SU(2) generators. Actingwith these on the monopoles, it can be seen that these operators indeed forma reducible representation of this group. We also build explicitly the rotationshown in Fig. 4, and the rotation that sends the quantization axis in the x − y plane. N We now give a description of the monopole hierarchy for general N . Again,our starting point is the organization of monopoles in QED . As noted above,18 (a) (b)Figure 4: Schematic representation of the fermion zero modes dressing of monopole operatorswith N = 2 valleys v = L, R and a minimal magnetic charge q = 1 /
2. (a) The s = 1 , m s = − h ¯Ψ σ Ψ i ∝ M q ˆ z can be rotated to (b) the s = 1 , m s = − h ¯Ψ σ Ψ i ∝ − M q ˆ z . these monopoles form the rank- N antisymmetric tensor of SU(2 N ) (45). Interms of a Young tableau, this may be written as a single column of N boxes N ( ... . (51)The organization of monopole operators at the QED -cHGN QCP can then beunderstood by finding how this SU(2 N ) irrep reduces as a representation of thesubgroup SU(2) × SU( N ).In the last section, we discussed the case N = 2 where the rank-2 completelyantisymetric irrep of SU(4) is reduced to the representation ( , ) ⊕ ( , ) ofSU(2) × SU(2). The following Young tableaux schematize this reduction (49) → (cid:16) , (cid:17) ⊕ (cid:16) , (cid:17) , (52)where the bold subscripts indicate the respective irrep’s dimension. The re-duction can also be written explicitly for general N . The rank- N completelyantisymmetric irrep (51) is reduced as N ( ... → b N/ c M b =0 N − b z }| { ... ...... | {z } b , N − b ... ... (cid:27) b ... ! , (53)This is coherent with results in Ref. [62] where this reduction is obtained upto N = 8, or with the output of a recent symbolic computation packagefor representation theory [63]. This result was stated in Ref. [20] where theorganization of flux operators in Lorentz symmetry multiplets was discussed.We give the proof of this result (53) in Appendix F. Using the notation employed in Ref. [62], Eq. (53) can be reexpressed as (1 N ) → b N/ c M b =0 (cid:16) ( N − b, b ) , (2 b , N − b ) (cid:17) . N boxes, (Υ ν , Υ ˜ ν ), wherethe first diagram Υ ν has a maximum of two rows since it must be a SU(2)irrep. This plays a key role when proving Eq. (53). This total number of boxesalso corresponds to the number of fermion zero modes dressing the monopole.Indeed, the monopole may be noted as M † σ v ,σ v ,...,σ N v N . The SU(2) Spin irrepencodes symmetry relations between spin indices σ i , and the same goes for valleyindices v i and the SU( N ) Nodal irrep. In the simplest case where b = 0, the irrepbecomes (cid:18) ... | {z } N , N (cid:26) ... (cid:19) . (54)The corresponding monopole simply transforms symmetrically in its spin indicesand antisymmetrically in its valley indices. That is, the monopole is a valleysinglet and a spin multiplet with maximal spin. For other values of b , theirreps describe mixed symmetries between the indices. Every box represents aspin or valley index which is antisymmetrized with its column neighbours andsymmetrized with its row neighbours. The antisymmetry of Eq. (53)’s LHSis realized in each of the RHS irreps through the following prescription : i)Match pairs of antisymmetrized spins (columns with two boxes) with pairs ofsymmetrized valleys (rows with two boxes) ii) Match the remaining symmetrizedspins with the remaining antisymmetrized valleys.To gain a better physical intuition, it is useful to label the Young tableauxin Eq. (53) with the magnetic spin s instead of a number of boxes b . Thedimension of the SU(2) Spin irrep is N − b + 1. This number should naturally beidentified with 2 s + 1. Also, we can eliminate every column of two boxes for theSU(2) Spin irreps which correspond to antisymmetrized pairs of spins. Ignoringthose “bounded” spin indices by removing their corresponding boxes obscursthe underlying zero modes dressing, but it puts the SU(2)
Spin irreps in a morefamiliar form. The remaining boxes correspond to the “free” spin indices thatform the spin- s multiplet. With this, the monopole representation at the QCPcan be written as N/ M s =( N mod 2) / ... | {z } s , N + s ... ... (cid:27) N − s ... ! . (55)For the maximal spin s = N/
2, the valley irrep is a singlet, as noted earlier inEq. (54).
The degeneracy of monopoles at the QCP for each magnetic spin sector isfound in what follows. The do so, we compute the dimension of the reducedirreps in Eq. (55), which is given by the product of the spin and valley irreps’dimensions. The dimension of the spin irrep is the usual spin degeneracy 2 s + 1.20s for the valley irrep, it can first be observed that it is generated through atensor product of completely antisymmetric tensors N + s ...... ⊗ ... (cid:27) N − s = N + s ... ... (cid:27) N − s ... ⊕ N + s +1 ...... ⊗ ... o N − s − . (56)The valley irrep’s dimension can be obtained by computing the dimensions of theother representations in this relation. Each composite antisymmetric tensor hasa dimension given by a binomial factor. Specifically, the dimension of the SU( N )irrep corresponding to a single column of b boxes is (cid:0) Nb (cid:1) . The tensor product’sdimension is simply the product of these binomial factors. The dimension ofthe valley irrep is thendim N + s ... ... (cid:27) N − s ... = (cid:18) N N + s (cid:19)(cid:18) N N − s (cid:19) − (cid:18) N N + s + 1 (cid:19)(cid:18) N N − s − (cid:19) . (57)We can then obtain the total degeneracy of monopoles in each magnetic spinsectorΩ s = (2 s + 1) × (cid:20)(cid:18) N N + s (cid:19)(cid:18) N N − s (cid:19) − (cid:18) N N + s + 1 (cid:19)(cid:18) N N − s − (cid:19)(cid:21) . (58)In Appendix F, we confirm this result with a method appropriate for generalYoung tableaux. We also show that the total dimension of the reduced irreps isequal to the original antisymmetric irrep in SU(2 N ), that is P s Ω s = (cid:0) NN (cid:1) .We briefly reformulate the last result. The factor 2 s + 1 comes from thepossible spin polarizations of the monopole. As for the valley sector, we take asa starting point two antisymmetric tensors A v ,v ,...,v N/ s ˜ A v ,v ,...,v N/ − s thatassign valley indices to the N zero modes. This corresponds to the tensor prod-uct in the LHS of Eq. (56). The 2 s supplementary valley indices in the firsttensor can be attributed to the 2 s “free” spin indices which give the monopoleits magnetic polarization. Indeed, these “free” spin indices form a symmetricmultiplet and must be matched with antisymmetrized valley indices to yield anantisymmetric state. On the other hand, the remaining N − s “bounded” spinindices (the boxes eliminated in Eq. (55)) form antisymmetrized spin pairs thatshould be matched with symmetrized valley pairs. Thus, every configurationwith at least one pair of antisymmetrized valley indices matched to an anti-symmetrized spin pair should be removed. Such configurations can be writtenas A v ,v ,v ,...,v N/ s ˜ A v ,v ,...,v N/ − s , which is the tensor product in the RHS ofEq. (56). It then follows that distinct spin pairs are also related by antisym-metrized valley indices, which yields the overall antisymmetric state.21 .2.2. Reduction for N = 3Using the result in Eq. (53), we can retrieve the reduction of monopoles atthe QCP for N = 3 that was discussed in Ref. [50]. In this case, the flavorsymmetry breaking is SU(6) → SU(2)
Spin × SU(3)
Nodal . Monopoles in QED are organized as the rank-3 antisymmetric representation of SU(6). At theQCP, this is decomposed as → (cid:16) , (cid:17) ⊕ (cid:16) , (cid:17) , (59)which agrees with Ref. [62]. The total dimension of the reduced irreps is4 × × Spin spin is s max = 3 / q = 1 / N = 3 (see Eq. (10)). It corresponds to a spin quadruplet, the in Eq. (59). This irrep represents monopoles with the lowest scaling dimension.As noted before, the fact that N is odd yields a non-vanishing minimal with s min = 1 /
2, the in Eq. (59). As three “zero” modes must be filled, the lowestpolarization that can be obtained is by paring two modes in a singlet, leavingone remaining spin that forms the doublet.
6. Conclusion
We characterized the hierarchy of monopole operators in QED -cHGN. Us-ing the state-operator correspondence, we obtained the scaling dimensions ofmonopoles as a function of their magnetic spin at leading order in 1 /N . Thespin-Hall mass parameter generated by the critical fermion self-interaction al-lows to lower the scaling dimension of monopoles, and this effect is more pro-nounced for monopoles with larger magnetic spins. The minimal scaling dimen-sion identified in Ref. [49] corresponds to monopoles with a maximal magneticspin. Monopoles with a vanishing spin have the largest scaling dimension whichis the same as in QED . This hierarchy is natural from the point of view ofsymmetry as monopoles at the QCP are organized as irreps of SU(2) × SU( N )which are labeled by the magnetic spin. These irreps were obtained explic-itly for q = 1 / N ) → SU(2) × SU( N ) to reduce the monopole irrep formed in QED . Thisalso allowed to obtain the degeneracies remaining at the QCP following the de-generacy lifting. It would be interesting to improve the analysis for q > / with Monte Carlo[17; 64; 65; 18] and conformal bootstrap [19; 20], it would be interesting to seesimilar investigations in the QED -GN-like models. Acknowledgements
We thank Jaume Gomis and Sergueï Tchoumakov for useful discussions. É.D.was funded by an Alexander Graham Bell CGS from NSERC. W.W.-K. wasfunded by a Discovery Grant from NSERC, a Canada Research Chair, a grant22rom the Fondation Courtois, and a “Établissement de nouveaux chercheurs etde nouvelles chercheuses universitaires” grant from FRQNT.
Appendix A. Holonomy of the gauge field
Let us also consider the holonomy of the gauge field on the “thermal” circle. α = 1 V Z S × S β d x √ g a . (A.1)In our mean field ansatz, we now leave open the possibility of a non-trivialexpectation value of the gauge field h a i = β − α . (A.2)Modifying Eq. (20) accordingly, the determinant operator becomes f q ; s = − β − X σ = ± X n ∈ Z (cid:20) d q ln (cid:0) ω n + β − α − iµ σ + iσM q (cid:1) + ∞ X ‘ = q +1 d ‘ ln (cid:16)(cid:0) ω n + β − α − iµ σ (cid:1) + ε ‘ (cid:17)(cid:21) . (A.3)Taking the sum over the mastubara frequencies, the same logic in passing fromEq. (20) to Eq. (23) is used f q ; s = − β − (cid:20) d q ln (cid:0) α ) + cosh( β ( M q − µ ))] (cid:1) + ∞ X ‘ = q +1 X σ = ± d ‘ ln (cid:0) (cid:2) cosh( βε ‘ ) + cosh (cid:0) β ( σµ + iβ − α ) (cid:1)(cid:3)(cid:1)(cid:21) . (A.4)The gap equation for a is0 = β ∂f q ; s ∂α = − d q sin( α )cos( α ) + cosh( β ( M q − µ )) (A.5)+ ∞ X ‘ = q +1 X σ = ± i sinh (cid:0) β ( σµ + iβ − α ) (cid:1) cosh( βε ‘ ) + cosh( β ( σµ + iβ − α )) . (A.6)This vanishes for α = 0 , π . Inserting this solution in the determinant operator,we obtain f ± q ; s = − β − (cid:20) d q ln (cid:0) ± β ( M q − µ ))] (cid:1) + ∞ X ‘ = q +1 d ‘ ln (cid:0) βε ‘ ) ± cosh( βµ )] (cid:1)(cid:21) . (A.7)23ith this, we can proceed to solve the remaining gap equations to find the othermean field parameters in these two cases for M ± q , µ ± S , Σ ± . At leading order in1 /β , the effective action evaluated at both these saddle points is the same. Thepartition function obtained by summing over these two saddle points then yieldsan extra factor of 2, Z ≈ e − βNF (0) q ; s where F (0) q ; s was found in the main text. Wecould directly ignore this as ln 2 = O ( β N ) and the factor does not contributeto the leading order results Z s [ A q ] = exp n − βN F (0) q ; s + O ( β N , N β ) o . (A.8)However, it is more interesting to remark that this factor is cancelled with aproper normalization. When computing the scaling dimension ∆ q ; s , we shouldactually use the normalized partition function∆ q ; s = − β − lim β →∞ ln (cid:18) Z s [ A q ] Z [0] (cid:19) . (A.9)Normally, this vacuum parition function is not mentioned, as the free energyusually vanishes, leaving a trivial normalization factor Z [0] = 1. However, onthe “thermal” circle, the non-trivial holonomy also contributes to this partitionfunction. By setting q = 0 (and µ = 0) in the gap equation for a , the twosolutions for holonomies α = 0 , π remain. In this case, Z [0] = 2, which exactlycancels the extra factor in Z s [ A q ]. Appendix B. General spin-Hall mass
In this section, we find the monopole scaling dimension using a more generalansatz than the one employed in Sec. 3. We let the auxiliary bosons havedifferent orientations. We keep χ along ˆ z while the spin-Hall mass is orientedmore generally as M q = M q ˆ n . The more general determinant operator inEq. (18) then becomes f q ; s = − β − X n ∈ Z " d q ln det [ − i ( ω − iµ σ z ) + M q · σ ]+ ∞ X ‘ = q +1 d ‘ ln det [ − i O q,‘ ( ω − iµ σ z + i P q,‘ ) + M q · σ ] , (B.1)where the matrices O q,‘ and P q,‘ are given by [14; 49] O q,‘ = 1 ‘ (cid:18) − q − p ‘ − q − p ‘ − q q (cid:19) , P q,‘ = p ‘ − q ‘ (cid:18)p ‘ − q − q − q − p ‘ − q (cid:19) . (B.2)24he spin-Hall mass M q may be parameterized by its norm M q and two angles( ϑ, ϕ ) for its orientation ˆ nM q ˆ n = M q (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ ) . (B.3)The determinant operator can be diagonalized in the magnetic spin subspace f q ; s = − β − d q ln (cid:20) (cid:18) (cid:18) β q M q sin ϑ + ( M q cos ϑ − µ ) (cid:19)(cid:19)(cid:21) + 2 X σ = ± ∞ X ‘ = q +1 d ‘ ln (cid:20) (cid:18) β (cid:15) ‘,ϑ,σ (cid:19)(cid:21) ! , (B.4)where (cid:15) ‘,ϑ,σ = r(cid:16)q ‘ − q + M q cos ϑ + σµ (cid:17) + M q sin ϑ . (B.5)For convienience, we may write this as (cid:15) ‘,ϑ,σ ≡ q ( ε ‘,ϑ + σµ ) + M q sin ϑ , (B.6)where ε ‘,ϑ = q ‘ − q + M q cos ϑ . (B.7)For ϑ = 0 , π , this corresponds to the eigenvalue ε ‘ defined in the main text.We note that the determinant operator (B.4) is independent of ϕ whichindicates an azimutal symmetry. By setting ϑ = 0, the determinant operatorused in the main text (23) is retrieved. As shown in Eq. (17), the full free energyexpression is given by F (0) q ; s = f q ; s + µ m s (cid:0) s + m s (cid:1) . Inserting Eq. (B.4) in thislast expression, it is found that the free energy is invariant under ϑ → π − ϑ , m s → − m s , (B.8)where the last transformation also implies µ → − µ (19). This means that ϑ = π and m s = s is a solution with the same free energy as the ϑ = 0 and m s = − s solution found in the main text. For this second solution, the spinpolarization and the spin-Hall mass are still anti-aligned.For later convenience, we write explicitly the free energy. We may alreadytake the large- β limit for the non-zero modes as ‘ − q is order O ( β ) whichlets us take log [2 cosh ( β(cid:15) ‘,ϑ,σ / → β(cid:15) ‘,ϑ,σ /
2. In this limit, the free energy isgiven by F (0) q ; s = − β − d q ln (cid:20) (cid:18) (cid:18) β q M q sin ϑ + ( M q cos ϑ − µ ) (cid:19)(cid:19)(cid:21) + µ m s (cid:0) s + m s (cid:1) − X σ = ± ∞ X ‘ = q +1 d ‘ (cid:15) ‘,ϑ,σ . (B.9)25 ppendix B.1. Gap equations The gap equations are obtained by varying F (0) q ; s with respect to the origi-nal saddle point parameters M q , µ S , P z , ϑ . In this more general case, the gapequations for µ S and P z , m s ∂ µ f q ; s + s = 0 , (B.10) q µ S ( ∂ µ f q ; s + 2 m s ) = 0 , (B.11)can still be combined to yield the condition m s = s if µ s = 0. We are then leftwith a system of three gap equations ∂ M q f q ; s = 0 , (B.12) ∂ µ f q ; s − m s = 0 , (B.13) ∂ ϑ f q ; s = 0 , (B.14)where the second equation is the gap equation for P z divided by p µ S /
2. Theexplicit expression for the gap equations is given by − (cid:18) M q − µ cos ϑM q cos ϑ − µ (cid:19) C − M q X σ X ‘ d ‘ (cid:18) ε ‘,ϑ + cos ϑµ σε ‘,ϑ (cid:15) ‘,ϑ,σ (cid:19) = 0 , (B.15) C − m s − X σ X ‘ d ‘ σ (cid:18) ε ‘,ϑ + µ σ(cid:15) ‘,ϑ,σ (cid:19) = 0 , (B.16) − (cid:18) µ M q sin ϑM q cos ϑ − µ (cid:19) C + M q µ sin ϑ cos ϑ X σ X ‘ d ‘ σε ‘,ϑ (cid:15) ‘,ϑ,σ = 0 , (B.17)where C is defined as C = d q ( M q cos ϑ − µ ) tanh (cid:16) β q µ − µ M q cos ϑ + M q (cid:17)q µ − µ M q cos ϑ + M q . (B.18) Appendix B.2. Analytical solutions for ϑ ∈ { , π/ , π } We first focus on the cases ϑ ∈ { , π/ , π } . For these angles, the sum overnon-“zero” modes in the ϑ gap equation (B.17) does not contribute as it isproportional to sin ϑ cos ϑ →
0. The sum in the second gap equation (B.16) alsovanishes if we suppose that µ < ε ‘ for ϑ = 0 , π and µ ∼ O (1 /β ) for ϑ = π/ ε ‘,ϑ + µ σ ) /(cid:15) ‘,ϑ,σ → σ canceleach other. Later on, we see that this assumption does allow to find a solution.The gap equations then become − (cid:18) M q − µ cos ϑM q cos ϑ − µ (cid:19) C − M q X σ X ‘ d ‘ (cid:18) ε ‘,ϑ + cos ϑµ σε ‘,ϑ (cid:15) ‘,ϑ,σ (cid:19) = 0 , (B.19) C − m s = 0 , (B.20) − (cid:18) µ M q sin ϑM q cos ϑ − µ (cid:19) C = 0 . (B.21)26he gap equation for ϑ (B.21) is solved trivially for ϑ = 0 , π . If ϑ = π/
2, itrequires M q = 0. As for the second equation derived from ∂F (0) q /∂P z (B.20), itis solved with the following µ µ = M q cos ϑ + β − ln (cid:18) m s /d q − m s /d q (cid:19) . (B.22)Using all previous results, the remaining gap equation becomes − ϑ ) m s − M q X ‘ d ‘ ε − ‘ = 0 , (B.23)where sgn(0) = 0. For ϑ = π/
2, the first term vanishes and the gap equationis solved since it was established that M q = 0 for this angle. For ϑ = 0 , π and m s = − sgn(cos ϑ ) s , we retrieve the same gap equation as in the main text(32). Putting together all the previous results, we can also find back the scalingdimension in Eq. (33). As for ϑ = π/
2, the only term that contributes to thefree energy (17) at leading order in 1 /β is the sum over non-“zero” modes. Thus,the free energy becomes F (0) q ; s (cid:12)(cid:12) ϑ = π/ = − X ‘ d ‘ ε ‘ | M q =0 + O (1 /β ) . (B.24)This is the free energy we would obtain in QED . The scaling dimension for ϑ = π/ ϑ = 0 , π . This solution should thus be discardedas it is not a global minimum. Appendix B.3. Numerical study for < ϑ < π We found two minima of free energy for ϑ = 0 , π and a maximum for ϑ = π/ ϑ cannot be solved so simply. This is because the sum over non-“zero” modes contributions doesn’t simplify like it does for ϑ ∈ { , π/ , π } . Inthis case, we resort to solving the gap equations numerically.We search the root of the three gap equations (B.15 - B.17) yielding thesolution for M q , µ and ϑ (the solution for the polarization is already known, m s = − s sgn(cos ϑ )). Had we not taken the large- β limit for non-“zero” modesstarting from Eq. (B.9), the sums would also contain a factor tanh ( β(cid:15) ‘,ϑ,σ / q = 1 / s = s max /
2, but the situation is similar for other magneticcharges and magnetic spins.To solve the gap equations, we first seek a solution of the first two gapequations with a fixed value for ϑ . We then insert the solutions for M q and µ and the fixed value of ϑ in the last gap equation to see if it is satisfied, i.e. ∂F (0) q /∂ϑ should vanish. As shown in Fig. 5(a), the only would-be solution inthe range 0 < ϑ < π/ µ = 0. However, this contradicts theassumption that µ S = 0 unless we take m s = 0. The full treatment of the fourgap equations, without assuming µ S = 0, yields the same solution. In the casewhere µ = 0, the gap equation for µ S yields the condition m s cos ϑ = − s /d q .27s we enforced m s = − s in the range 0 < ϑ < π/
2, the numerical solutionindeed occurs at ϑ = arccos(2 s/d q ), as shown in Fig. 5(a). However, as arguedin the main text and further in App. Appendix C, this solution has divergencesin the second derivatives of the free energy and should be discarded.A closer look at the situation near ϑ = 0 is shown in Fig. 5(b). The so-lution obtained in the main text is recovered in this limit as M q → .
14 andexp { β ( µ − M q cos ϑ ) } → / q = 1 / s = s max / m s = − s ). ● ∂ F q ( ) / ∂ϑ ■ M q ◆ μ ' ϑ = ArcCos ( / d q ) - ϑ (a) ● ∂ F q ( ) / ∂ϑ ■ M q ◆ Exp { β ( μ ' - M q Cos ( ϑ ))} - ϑ (b)Figure B.5: Numerical investigation of possible solutions to the gap equations for a) 0 <ϑ < π/ ϑ = 0. We set β = 10 . Values of M q and µ are found by solvingEqs. (B.15-B.16) at fixed ϑ . ∂F (0) q /∂ϑ which yields the LHS of the remaining gap equationbecomes 0 for ϑ = 0 and ϑ = arccos(2 s/d q ).
28s for the ϑ = π/ π/ ϑ = π/ M q →
0, and, again, exp { β ( µ − M q cos ϑ ) } → / ● ∂ F q ( ) / ∂ϑ ■ M q ◆ Exp { β ( μ ' - M q Cos ( ϑ ))} ϑ - - - - Figure B.6: Numerical investigation of possible solutions to the gap equations for near ϑ = π/
2. Values of M q and µ are found by solving Eqs. (B.15-B.16) at fixed ϑ . We set β = 10 .The solution at π/ ∂F (0) q /∂ϑ → M q → We found no other solutions numerically. This justifies our assumption inthe main text where we worked only with ϑ = 0. Appendix C. Gap equations and µ = 0 In the main text, we obtained three gap equations (24-26) for M q , µ S and P z , respectively. The third one can be solved for µ S = P z = 0. This also meansthat µ = 0. The first gap equation then simply becomes − d q − M q ∞ X ‘ = q +1 d ‘ ε − ‘ = 0 , (C.1)which can be written as − s max − M q ∞ X ‘ = q +1 d ‘ ε − ‘ = 0 . (C.2)This has the same form as the gap equation found in the main text (32) yieldsas a solution the maximal possible spin-Hall mass. As for the free energy, it alsotakes the form obtained in the main text for a maximal spin F (0) q ; s = − M q s max − X ‘ = q +1 d ‘ ε ‘ + O ( β − ) . (C.3)29his solution with µ = 0 thus seems to indicate a smaller scaling dimensionthat the one proposed in the main text where s max → s . However, by inspectingthe second derivatives of the free energy at this saddle point, divergences arefound. The third gap equation can be derived with respect to µ S ∂ F (0) q ; s ∂µ S ∂P z = 12 √ µ S ∂ µ f q ; s + 12 m s r µ S ∂ µ ∂ µ f q ; s . (C.4)Developing this and taking the large- β limit, this becomes ∂ F (0) q ; s ∂µ S ∂P z = 12 √ µ S , (C.5)which is singular. This shows how this solution has a bad behavior and shouldbe ignored. Appendix D. Regularized gap equation and scaling dimension
The free energy used to find the monopole scaling dimension has a divergingsum P ∞ ‘ = q +1 d ‘ ε ‘ , where the degeneracy and energy are defined in Eq. (21). Byobtaining the first two orders in the 1 /‘ expansion of the summand d ‘ ε ‘ = 2 ‘ + ( M q − q ) + O ( ‘ − ) ≡ d ‘ ε div ‘ + O ( ‘ − ) , (D.1)the diverging sum can be rewritten as ∞ X ‘ = q +1 d ‘ ε ‘ = ∞ X ‘ = q +1 d ‘ ( ε ‘ − ε div ‘ ) + ∞ X ‘ = q +1 d ‘ ε div ‘ . (D.2)In this expression, the first sum is convergent ∞ X ‘ = q +1 d ‘ ( ε ‘ − ε div ‘ ) = ∞ X ‘ = q +1 (cid:20) d ‘ ε ‘ − d ‘ − ( M q − q ) (cid:21) , (D.3)while the second sum is divergent ∞ X ‘ = q +1 d ‘ ε div ‘ = 2 ∞ X ‘ = q +1 (cid:20) ‘ − s ) + (cid:16) − s (cid:17) (cid:0) M q − q (cid:1) ‘ − s (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) s =0 . (D.4)This divergent sum may be continued analytically to the Hurwitz zeta function P ∞ k =0 ( k + a ) − s = ζ ( s, a ) [66] ∞ X ‘ = q +1 d ‘ ε div ‘ = 2 ∞ X ‘ =0 ( ‘ + ( q + 1)) + (cid:0) M q − q (cid:1) ∞ X ‘ =0 ( ‘ + ( q + 1)) = 2 ζ ( − , q + 1) + (cid:0) M q − q (cid:1) ζ (0 , q + 1) (D.5)30eplacing the zeta functions with their polynomial expressions, we obtain ∞ X ‘ = q +1 d ‘ ε div ‘ = 12 (2 q + 1) (cid:18) q ( q − − M q (cid:19) . (D.6)Using these results, we obtain the regularized version of the gap equation (33)∆ q ; s N = − sd q M q + (2 q + 1) (cid:18) M q − q ( q − (cid:19) − ∞ X ‘ = q +1 h d ‘ ε ‘ − d ‘ − (cid:0) M q − q (cid:1) i , (D.7)where the spin-Hall mass M q is determined by the regularized version of thegap equation (32) − sd q + 2 M q (cid:18) q + 1 − ∞ X ‘ = q +1 (cid:2) d ‘ ε − ‘ − (cid:3)(cid:19) = 0 . (D.8) Appendix E. Representation of q = 1 / N = 2 We can generally write a monopole operator as M † I = D † I M † Bare , (E.1)where D † I is an operator that creates half of all zero modes available. For N = 2 and q = 1 / D † I corresponds to a zero modes creation operators bilin-ear c † A I ( c † ) (cid:124) as in Eq. (46). Taking the z axis as the spin quantization axis,monopoles in the helicity basis are written as follows D †↓ = 12 c † (cid:20) − σ z ⊗ iµ y (cid:21) ( c † ) (cid:124) = 12 (cid:16) c † L ↓ c † R ↓ − c † R ↓ c † L ↓ (cid:17) , (E.2) D †↑ = 12 c † (cid:20) σ z ⊗ iµ y (cid:21) ( c † ) (cid:124) = 12 (cid:16) c † L ↑ c † R ↑ − c † R ↑ c † L ↑ (cid:17) , (E.3) D †↑↓ = 12 c † (cid:20) σ x √ ⊗ iµ y (cid:21) ( c † ) (cid:124) = 12 (cid:16) c † L ↑ c † R ↓ + c † L ↓ c † R ↑ (cid:17) √ − ( L ↔ R ) . (E.4)For example, the spin down monopole acting on the vacuum D †↓ M † Bare | i mayschematically be represented as shown in Fig. E.7.We may reorganize the monopoles in the real vector basis used in the maintext in Eqs. (47) and (48) M ↓ − M ↑ − i ( M ↓ + M ↑ ) √ M ↑↓ = 12 c † [ σ y σ ⊗ µ y ] ( c † ) (cid:124) M † Bare . (E.5)The same can be done for monopoles in the nodal triplet by simply exchangingspin and valley indices ↑ , ↓ ↔ L, R . 31 igure E.7: Schematic representation of the spin down monopole with the fermion zero modesoccupation.
Appendix E.1.
SU(2) × SU(2) generators
Generators T a of the SU(4) algebra may be realized with one zero modecreation operator and one zero mode destruction operator T a = c † O a c . (E.6)For example, we may identify the raising spin operator, which is part of theSU(2) Spin ⊗ SU(2)
Nodal subalgebra of SU(4), as s + = 12 c † (( σ x + iσ y ) ⊗ ) c = c † L ↑ c L ↓ + c † R ↑ c R ↓ . (E.7)Under the action of this operator, the spin down monopole transforms in othermonopoles in spin triplet while the monopoles in the nodal triplet are annhiliated s + s + M ↓ = √ s + M ↑↓ = 2 M ↑ , (E.8) s + M R = s + M LR = s + M L = 0 . (E.9)The factors involved are the usual total magnetic spin eigenvalue p s ( s + 1).One may also define the lowering spin and the azimutal spin number operators,respectively given by s − = s † + , s z = 12 c † ( σ z ⊗ ) c = 12 ( c † L ↑ c L ↑ − c † L ↓ c L ↓ + ( L ↔ R )) . (E.10)It is simple to show that these operators obey the SU(2) algebra commutation re-lations [ s + , s − ] = 2 s z . Again, analogous valley operators v may be constructedby the exchange ↑ , ↓ ↔ L, R . The action of these SU(2) × SU(2) generators { s − , s z , s + , v − , v z , v + } on the monopole operators is shown in Table E.2. It isreadily seen that the monopole spin and nodal triplets form a reducible repre-sentation of SU(2) × SU(2).
Appendix E.2. Rotation of spin monopoles
We may explicitly show the rotation of spin monopoles mentioned in themain text. To do so, we first rexpress monopoles in the real vector representation(E.5) in the following basis = 1 √ M ↓ − M ↑ ) , = − i √ M ↓ + M ↑ ) , = M ↑↓ . (E.11)32 able E.2: The monopoles form a reducible representation ( , ) ⊕ ( , ) of SU(2) × SU(2)(helicity basis). M ↓ M ↑ M ↑↓ M R M L M LR √ s + M ↑↓ M ↑ s z −M ↓ M ↑ √ s − M ↑↓ M ↓ √ v + M LR M L v z −M R M L √ v + M LR M R In this real basis, the angular momentum operators take the form ( J I ) jk = (cid:15) Ijk .Considering an SU(2) transformation of the fermionsΨ → e i θ · σ Ψ , (E.12)we can find the transformation of the vector representations. The fermion bi-linears and the monopoles transform a bit differently¯Ψ σ i Ψ → ¯Ψ e − i ϑ · σ σ i e i ϑ · σ Ψ = R ij ¯Ψ σ j Ψ , (E.13) c † ( iσ y σ i )( c † ) (cid:124) → c † ( iσ y ) e i ϑ · σ (cid:124) σ i e − i ϑ · σ (cid:124) ( c † ) (cid:124) = ˜ R ij c † ( iσ y σ j )( c † ) (cid:124) , (E.14)where ˜ R ij = R ij | ϑ x,z →− ϑ x,z and we used the fact that ( ϑ · σ ) σ y = − σ y ( ϑ · σ (cid:124) ).We will perform these rotations explicitly on the spin-Hall mass ¯Ψ σ z Ψ and thespin down monopole M ↓ . In the real vector basis, they are expressed as¯Ψ σ z Ψ = , M ↓ = 1 √ − i . (E.15)We can compare how these vectors rotate along the y axis. In this case, theyare transformed by the same rotation matrix R ϑ y = ˜ R ϑ y = cos ϑ y ϑ y − sin ϑ y ϑ y . (E.16)We first consider a rotation ϑ y = π/ x − y plane. The spin-Hall mass is rotated to ¯Ψ σ x Ψ R ϑ y = π/ (cid:0) ¯Ψ σ z Ψ (cid:1) = − = = ¯Ψ σ x Ψ . (E.17)As for the spin down monopole, it is rotated to a combination of all the monopoles(in the helicity basis) of the spin triplet R ϑ y = π/ M ↓ = 1 √ − − i = 1 √ i = 12 ( M ↓ + M ↑ ) − √ M ↑↓ . (E.18)33his is an eigenstate of the angular momentum oriented along ˆ x , ( J x ) ij = (cid:15) ij .Indeed, J x R π/ M ↓ = − R π/ M ↓ . This monopole operator now creates a statewith S x = − σ x Ψ.The ϑ y = π rotation is more simple. In this case R ϑ y = π ¯Ψ σ z Ψ = − − = − ¯Ψ σ z Ψ , (E.19) R ϑ y = π M ↓ = − − √ − i = 1 √ i = M ↑ . (E.20)Thus, the mass is shifted ¯Ψ σ z Ψ → − ¯Ψ σ z Ψ while the spin down monopole isrotated to the spin up monopole M ↓ → M ↑ , just as expected. This situationwas shown in Fig. 4 Appendix E.3. Computing the spin-Hall energy
We may also explicitly compute the energy of the spin-Hall mass term forthe state | ψ I i = M † I | i in the three situations considered above. i) For the spindown monopole, the state is M †↓ | i = |↓↓i ≡ | s = 1 , m s = − i . The energy ofthe spin-Hall mass term oriented along ˆ z for this state is M q ( σ z ⊗ + ⊗ σ z ) |↓↓i = − M q |↓↓i . (E.21)ii) After the π/ R ϑ y = π/ M †↓ | i = |↓↓i + |↑↑i − |↓↑i − |↑↓i , (E.22)while the spin-Hall mass becomes oriented along ˆ x . Its energy contribution tothis state remains − M q M q ( σ x ⊗ + ⊗ σ x ) (cid:16) R ϑ y = π/ M †↓ | i (cid:17) (E.23)= M q [( |↑↓i + |↓↑i − |↑↑i − |↓↓i ) + ( |↓↑i + |↑↓i − |↓↓i − |↑↑i )]2= − M q (cid:16) R ϑ y = π/ M †↓ | i (cid:17) . (E.24)iii) Finally, after the π rotation, the state and the action of the spin-Hall massare as expected R ϑ y = π M †↑ | i = |↑↑i , − M q ( σ z ⊗ + ⊗ σ z ) |↑↑i = − M q |↑↑i . (E.25) Appendix F. General reduction problem
Appendix F.1. A relation with the permutation group
There is an ambiguity when discussing the reduction SU(2 N ) → SU(2) × SU( N ) as the SU(2) × SU( N ) subgroup of SU(2 N ) is not uniquely defined. The34ubgroup SU(2) × SU( N ) describing the QED -cHGN model is not the same asthe one obtained by the chain SU(2 N ) ⊃ SU( N ) × SU( N ) ⊃ SU(2) × SU( N ).The two subgroups have different branching rules. In this regard, it is useful toconsider a more general reduction problemSU( M N ) → SU( M ) × SU( N ) , (F.1)where M and N are integers, and our case corresponds to M = 2 and N = N .We note in passing that this embedding in a larger symmetry group is alsoused to find the multiplicity of flux operators that transform as Lorentz scalars.For magnetic charges larger than the minimum q = 1 /
2, the number of zeromodes dressing a monopole (45) is 4 | q | N rather than 2 N . However, starting withrequirement that half of zero modes should be filled, it is natural to first builda rank-2 | q | N completely antisymmetric irrep of SU(4 | q | N ). As an intermediatestep to reduce this to the real symmetry group SU(2) × SU(2 N ), one can considerthe reduction SU(4 | q | N ) → SU(2 | q | ) × SU(2 N ) [14; 27; 20] which correspondsto setting M = 2 | q | and N = 2 N .The generators of SU( M N ) may be parameterized by taking Kronecker prod-ucts of the SU( M ) and SU( N ) generatorsSU( M N ) : T MNa ∈ { T Ma ⊗ , ⊗ T Na , T Ma ⊗ T Na } , (F.2)where a ∈ { , . . . , ( M N ) − } , a ∈ { , . . . , ( M ) − } and a ∈ { , . . . , ( N ) − } .The subgroup SU( M ) × SU( N ) is completely defined by the set of unbrokengenerators, which areSU( M ) × SU( N ) : { T Ma ⊗ , ⊗ T Na } . (F.3)This represents the reduced symmetry group of the QED -cHGN QCP. Whilethe spin-Hall term ¯Ψ σ Ψ transforms as a vector under the first generators T Ma ⊗ ,the cHGN interaction term ( ¯Ψ σ Ψ) is a scalar built from this vector and thustransforms trivially. Once the unbroken generators are specified, it follows thatthe fundamental representation of SU( M N ) simply reduces as the fundamentalof SU( M ) and SU( N ) → ( , ) . (F.4)This may be understood with rank-1 tensors as a SU( M N ) index α ∈ { , . . . , M N } can be decomposed in one SU( M ) index σ ∈ { , . . . , M } and one SU( N ) index v ∈ { , . . . , N } h α = h ( σv ) ≡ f σ g v . (F.5)We may also define rank-2 tensors in this manner, h αβ = f ση g vw . Definingsymmetrized and anti-symmetrized tensor respectively as t { i,j } = ( t ij + t ji )and t [ i,j ] = ( t ij − t ji ), we may write explicitly the RHS of (52) as a tensorwhose simplified form corresponds to the LHS of (52) f { σ,η } g [ v,w ] + f [ σ,η ] g { v,w } = 12 ( f ση g vw − f ησ g wv ) = h [ α,β ] . (F.6)35his method is difficult to implement as the rank of the tensors is increased, i.e.the number of boxes in the Young diagrams is increased. However, this specificexample has the merit of showcasing an important property of the reduction ofinterest (F.1): The decomposition is independent of M and N . Indeed, the onlyinformation needed to show Eq. (F.6) is the SU( M N ) index decomposition (F.5)which characterizes the reduction studied (F.1 , F.3). This can be understoodby inspecting the relation between the reduction SU(
M N ) → SU( M ) × SU( N )and the finite permutation group of f objects S f [62].This independence also manifests by the fact that general reductions ofSU( M N ) → SU( M ) × SU( N ) may be built by taking products of the fundamen-tal representation [62]. For example, by taking the product of the fundamentalrepresentation reduction (F.4) with itself ⊗ → ( ⊗ , ⊗ ) , (F.7)one finds can write the reduction for (1) ⊗ (1) = (2) ⊕ (1 ). By also developingthe RHS, the reduction in Eq. (52) can be found. Given that this is the reductionof a reducible representation (2) ⊕ (1 ), the associations made between the LHSand RHS of (F.7) are not straightforward. However, this shows again thatthis decomposition is independent of the indices M and N as they were notinvolved in the computation With this procedure, it is clear that a Youngtableau in the SU(
M N ) side with f boxes generally reduces to Young tableauxof SU( M ) × SU( N ) with the same number f of boxes. Those properties hintat a the importance of the permutation group of f objects S f in this reductionproblem (F.1).If we define Υ λ as a certain diagram λ among Young tableaux with f boxes,then the general reduction (F.1) may be written asΥ λ → M ν,ρ c νρλ (Υ ν , Υ ρ ) , (F.8)where c νρλ is a coefficient of fractional parentage (CFP). The relation with S f manifests itself through these CFPs: For each Young tableau Υ λ , there is acorresponding irrep D λ of S f , and the CFP of Υ λ in (Υ ν , Υ ρ ) is the Clebsch-Gordan coefficient (CGC) of D λ in the decomposition of the direct productrepresentation D ν ⊗ D ρ [62] D ν ⊗ D ρ = M λ c νρλ D λ . (F.9)Since the characters χ of a direct product is the product of the characters χ ν ⊗ ρ = χ ν χ ρ , (F.10) This general procedure may however include diagrams with too many rows and that arenot allowed in either SU( M ) or SU( N ). These must simply be removed. c νρλ = 1dim( S f ) X r p r χ ν ( C r ) χ ρ ( C r ) χ ∗ λ ( C r ) , (F.11)where χ ν ( C r ) is the character of D ν and p r is the number of group elements ina conjugacy class C r . Appendix F.2. Clebsch-Gordan coefficients of the sign irrep
We seek to reduce the rank- N completely antisymmetric irrep of SU(2 N )which is a Young diagram with f = N boxes. More precisely, this irrep corre-sponds to a single column of N boxes (51). This diagram corresponds to thesign irrep of S N that we note D sign . To find out how the rank- N completely an-tisymmetric irrep of SU(2 N ) reduces, we must therefore find the CGCs c νρ sign that give the contribution of D sign in the reduction of D ν ⊗ D ρ .Let D ν be an irrep of S N and D ˜ ν its conjugate. Diagrammatically, theseYoung tableaux are the transposed of each other (e.g. the two diagrams in theRHS of Eq. 53). As the conjugate irrep D ˜ ν is simply the direct product ofthe irrep ν with the sign irrep, D ˜ ν = D sign ⊗ D ν , its character is simply theproduct of their characters (F.10), i.e. χ ˜ ν = χ sign χ ν . Using this, the product ofcharacters that define c νρ sign (F.11) may be rewritten as χ ν ( C r ) χ ρ ( C r ) χ ∗ sign ( C r ) = χ ν ( C r ) χ trivial ( C r ) χ ∗ ˜ ρ ( C r ) . (F.12)To obtain this, we also used that, for any equivalence class C r , the characters ofthe sign irrep are ±
1, implying that | χ sign | = 1 = χ trivial , and more generallythat the characters of the permutation group are real, χ ˜ ρ = χ ∗ ˜ ρ . This relation(F.12) implies that c νρ sign = c ν trivial ˜ ρ . The latter CGC gives the decompositionof D ˜ ρ in D ν ⊗ D trivial = D ν . Obviously, the coefficient is only non-“zero” if ˜ ρ = ν ,which in turn means that c νρ sign = δ ρ, ˜ ν . More explicitly, this means that onlypairs of conjugate irreps have a contribution from the sign irrep in their directproduct decomposition D ν ⊗ D ρ = δ ρ, ˜ ν D sign ⊕ . . . . (F.13)This result (F.13) implies that the CFPs (F.8) are equal to one for every pairof irrep and its conjugate Υ λ → M ν (Υ ν , Υ ˜ ν ) . (F.14)In the case of interest where M = 2 and N = N , we must exclude irreps ofSU(2) × SU( N ) where tableaux in the SU(2) side have more than two rowssince they are not include in SU(2). In the end, this corresponds exactly to thereduction we announced in Eq. (53). In Appendix F.3, we explicitly check thatthe dimensions of these diagrams match.37 ppendix F.3. Dimensions of the reduced irreps We explicitly check the dimensions of irreps in the reduction of monopolesshown in the main text (53) N ( ... ! SU(2 N ) → b N/ c M b =0 N − b z }| { ... ...... | {z } b ! SU(2) , N − b ... ... (cid:27) b ... ! SU( N ) . (F.15) Dimensions of the
SU(2) × SU( N ) irreps The SU(2) subdiagram simply has dimension N − b + 1dim N − b z }| { ... ...... | {z } b ! SU(2) = dim ... | {z } N − b ! SU(2) = N − b + 1 , (F.16)where we removed columns of two boxes which transform trivially in SU(2). TheSU( N ) diagrams requires more work. The dimension F/H of such a diagram isfound using the factor over hooks rule [67]. The SU( N ) Young tableau’s boxescan be labeled as N − b N N +1 ... ... N +1 − b N +2 − b bN − b ... b N − b . (F.17)Then, the numerator F is the product of all the box labels in the Young tableauabove. It can be decomposed as the product of labels in the first column andin the second F = F L × F R = ( N )! b ! × ( N + 1)!( N + 1 − b )! . (F.18)38s for the denominator H , the length of the hooks for each box must be multi-plied. It is useful to decompose it in three sections H = h A × h B × h C = ( N − b )! × ( N − b + 1)!( N − b + 1)! × b ! , N − b B C ... ...
B C (cid:27) bA ... A (cid:27) N − b . (F.19)Putting all together, this becomesdim N − b ... ... (cid:27) b ... ! SU( N ) = ( N )! b ! × ( N + 1)!( N − b + 1)!( N − b )! × ( N − b +1)!( N − b +1)! × b != ( N − b + 1) ( N + 1)( N − b + 1) × (cid:18) Nb (cid:19) . (F.20)The total dimension of the SU(2) × SU( N ) irrep isdim N − b z }| { ... ...... | {z } b ! SU(2) × dim N − b ... ... (cid:27) b ... ! SU( N ) = ( N − b + 1) ( N + 1)( N − b + 1) × (cid:18) Nb (cid:19) . (F.21)Replacing b with N/ − s , it can be verified that this result indeed correspondsto Eq. (58). Total dimension
The dimension of the SU(2 N ) irrep should match the dimension of theSU(2) × SU( N ) representation to which it is reduceddim N ( ... ! SU(2 N ) = b N/ c X b =0 dim N − b z }| { ... ...... | {z } b ! SU(2) × dim N − b ... ... (cid:27) b ... ! SU( N ) . (F.22)Summing over the index b , we obtain the RHS of Eq. (F.22). The LHS ofEq. (F.22) is the dimension of the rank- N antisymmetric irrep of SU(2 N ) whichis (cid:0) NN (cid:1) . It is left to show that (cid:18) NN (cid:19) = b N/ c X b =0 ( N − b + 1) ( N + 1)( N − b + 1) × (cid:18) Nb (cid:19) . (F.23)39he RHS of Eq. (F.23) with even N = 2 x with x ∈ Z + can be simplified to theexpected result x X b =0 ( N − b + 1) ( N + 1)( N − b + 1) × (cid:18) Nb (cid:19) (cid:12)(cid:12)(cid:12) N =2 x = π (2 x )! (cid:0) x − (cid:1) ! (cid:0) x − (cid:1) ! (cid:0) x − (cid:1) ! x ! Γ (cid:0) (cid:1) Γ (cid:0) (cid:1) = (4 x )!(2 x )! = (cid:18) NN (cid:19)(cid:12)(cid:12)(cid:12) N =2 x . (F.24)For odd N , we could not show (F.23) analytically, but it was confirmed numer-ically up to N = 10 + 1. Appendix F.4. Monopoles with larger magnetic charges
The results in the present section directly apply to the study of monopolewith a magnetic charge larger than the minimum q = 1 /
2. As we take a magneticcharge higher than the minimum, q > /
2, more fermion zero modes becomeavailable. However, filling half of those 2 d q N zero modes will generate monopolewith non vanishing Lorentz spins. This is still a good starting point: We define4 πq flux operators with vanishing fermion number by generalizing Eq. (45)Φ † I ...I N = c † I . . . c † I N M † Bare , I i ∈ { , , . . . , d q N } . (F.25)These flux operators form the rank- d q N completely antisymmetric tensor ofSU(2 d q N ). In QED , it reduces in irreps of the symmetry group SU(2) rot × SU(2 N ). Consider the following chainSU(2 d q N ) ⊃ SU( d q ) × SU(2 N ) ⊃ SU(2) × SU(2 N ) . (F.26)Then, a first step in reducing the antisymmetric irrep of SU(2 d q N ) is to firstconsider the reduction SU(2 d q N ) → SU( d q ) × SU(2 N ) . (F.27)This is just a subcase of the general reduction considered above (F.1) with M = d q and N = 2 N . The reduction is then simply(1 d q N ) → M ν (Υ qν , Υ q ˜ ν ) , (F.28)where Υ qν , Υ q ˜ ν are pairs of conjugate Young tableaux with d q N boxes, and Υ qν has at most d q rows. To obtain irreps in QED -cHGN, this should then bereduced as SU( d q ) × SU(2 N ) → SU(2) × SU(2 N ) . (F.29)Of course, we only need to know how the treat the reduction of the first sub-algebra SU( d q ) → SU(2). At this point, SU(2) singlets may be selected and40epresent the monopole operators, i.e. flux operators with vanishing fermionnumber and that transform as Lorentz scalarsSU(2 d q N ) → SU(2) × SU(2 N ) , (F.30)(1 d q N ) → M ν ∈ V Ω ν ( , Υ q ˜ ν ) , V = { ν | (Υ qν ) SU( d q ) → Ω ν SU(2) ⊕ . . . } , (F.31)where Ω ν is a degeneracy to be determined. Then, we can proceed to thereduction of the flavor symmetry irreps with ν ∈ V just as we did in the maintext for monopoles with q = 1 / N ) → SU(2) × SU( N ) , (F.32)Υ q ˜ ν → . . . , (F.33)where the ellipsis indicates the various diagrams involved in the reduction. Toour knowledge, there is no way to systematically obtain the reduction for general q like we did for general N and q = 1 /
2. We will simply examine the case N = 2.We consider the reduction given bySU(4 d q ) → SU( d q ) × SU(4) → SU( d q ) × SU(2) × SU(2) . (F.34)For the second smallest magnetic charge q = 1, the reduction is given bySU(8) → SU(2) × SU(4) , (F.35) → (cid:16) , (cid:17) ⊕ (cid:18) , (cid:19) ⊕ , ! . (F.36)Keeping only the Lorentz singlet, we the decompose the multiplet of SU(4)in irreps of SU(2) × SU(2)SU(4) → SU(2) × SU(2) , (F.37) → ( , ) ⊕ ( , ) ⊕ ( , ) ⊕ ( , ) . (F.38)The first two irreps shows there is accidental degeneracy. One could continuefor larger values of q , e.g. q = 3 / → SU(3) × SU(4) , (F.39) → (cid:18) , (cid:19) ⊕ (cid:18) , (cid:19) ⊕ (cid:18) , (cid:19) ⊕ , ! , (F.40)and so on. In these other cases, the first subalgebra must also be reduced toSU(2) in order to select Lorentz scalars.41 eferences [1] P. Anderson, Materials Research Bulletin , 153 (1973), ISSN0025-5408, URL .[2] R. D. Pisarski, Phys. Rev. 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