Unifying Description of Competing Orders in Two Dimensional Quantum Magnets

Quantum magnets provide the simplest example of strongly interacting quantum matter, yet they continue to resist a comprehensive understanding above one spatial dimension (1D). In 1D, a key ingredient to progress is Luttinger liquid theory which provides a unified description. Here we explore a promising analogous framework in two dimensions, the Dirac spin liquid (DSL), which can be constructed on several different lattices. The DSL is a version of Quantum Electrodynamics ( QED$_3$) with four flavors of Dirac fermions coupled to photons. Importantly, its excitations also include magnetic monopoles that drive confinement. By calculating the complete action of symmetries on monopoles on the square, honeycomb, triangular and kagom\`e lattices, we answer previously open key questions. We find that the stability of the DSL is enhanced on the triangular and kagom\`e lattices as compared to the bipartite (square and honeycomb) lattices. We obtain the universal signatures of the DSL on the triangular and kagom\`e lattices, including those that result from monopole excitations, which serve as a guide to numerics and to experiments on existing materials. Interestingly, the familiar 120 degree magnetic orders on these lattices can be obtained from monopole proliferation. Even when unstable, the Dirac spin liquid unifies multiple ordered states which could help organize the plethora of phases observed in strongly correlated two-dimensional materials.


I. INTRODUCTION
In recent years, gauge theories have been increasingly used to describe quantum magnets, particularly when geometric frustration leads to an enhancement of quantum fluctuations [1,2]. In these situations, classical descriptions are usually inadequate and entirely new 'quantum spin liquid' phases can emerge, described by deconfined charges of the gauge theory. Even when ordered states appear, the quantum interference between different orders has no classical analog, but can be captured by a gauge theory [3,4]. Progress in understanding quantum magnets will have ramifications well beyond the insulating state and could explain nearby conducting phases obtained on doping [5,6], including the high temperature superconductors seen in diverse systems from the copper oxide materials to the recently realized twisted bilayer graphene [7].
Previous gauge theory based approaches have pursued different approaches for bipartite and non-bipartite lattices. For example, starting with a Schwinger boson based representation of spins, a Z 2 gapped spin liquid is the natural 'mother' state for describing non-bipartite lattices [8][9][10][11][12][13], while the similar procedure for the bipartite case indicated Neel and valence bond crystal phases for bipartite lattices, separated by a deconfined quantum critical point [3,4,14]. These paradigms which are ultimately based on quantum disordering an initial classical ordered state (non-colinear versus colinear order on the non-bipartite and bipartite lattices respectively) represent significant progress towards a synthesis. However, here we will argue that an essentially quantum parent * To whom correspondence should be addressed, e-mail: avish-wanath@g.harvard.edu. state, the U(1) Dirac spin liquid can capture these insights and can further bridge bipartite and non-bipartite lattices using a common framework. Although there is a considerable theoretical literature on the U(1) Dirac spin liquid, the properties of a crucial class of excitation, the magnetic monopoles has, until now, not been systematically explored on different lattices. Here we compute the monopole symmetry quantum numbers, which then allows us to make considerable progress in understanding these remarkable states. The magnetic monopoles are 'instanton' excitations, that occur at points in the 2+1D spacetime. When they can be ignored (i.e. when irrelevant in the RG sense), the Dirac spin liquid can be stabilized, resulting in a significantly enlarged symmetry. In addition to the conformal invariance of the fixed point, a defining feature of the DSL is the emergence of a ∼ U (4) symmetry at low energies, which incorporates both spin and lattice symmetries.
A useful analog in one lower dimension is the Luttinger liquid, also a gapless phase that describes quantum liquids in 1+1D. There too, the stability of the phase is threatened by instanton excitations in the form of vortex tunneling events. Symmetry transformation properties of the instanton insertion operators play a key role in determining both stability and the nature of the ordered phases which result following instanton proliferation. The phase diagram of quantum spin chains [1,15] and the superfluid Mott transition of one dimensional bosons [16] can be understood in these terms. When instantons are irrelevant, the gapless Luttinger liquid is stabilized, but when they proliferate, a gapped phase such as the valence bond crystal or Mott insulator is obtained. In fact, the Luttinger liquid theory can be reformulated in terms of Dirac fermions coupled to a U (1) gauge field, and can be hence be viewed as the one-dimensional version of U (1) Dirac spin liquid.
Similarly, our computation of monopole quantum numbers sheds light on key issues such as: Can the Dirac spin liquid be a stable ground state and if so what are its key experimental signatures? If unstable, what are the likely alternate phases that are stabilized in its place? What is the underlying difference between bipartite and non-bipartite lattices?
Previously, early work on the square lattice quantum antiferromagnet, inspired by the copper-oxide materials, studied the staggered flux and π-flux mean field theories [17][18][19][20] within the fermionic representations of spins. Renewed interest emerged when analogous states on the kagomè lattice were introduced in [21][22][23][24][25]. The effect of fluctuations were studied in several works [26,27] and a dictionary relating fermion bilinears to local operators and the enlarged symmetry of the Dirac spin liquid were emphasized in [23,28]. Recently, the Dirac spin liquid on the triangular lattice has been studied [29][30][31][32]. However, most works have ignored the monopole excitations and their symmetry properties, with a few exceptions [23,[33][34][35][36]. In Ref [37], the conceptual framework to study monopoles with fermion zero modes was introduced. The important role of the Dirac sea Berry phase for spatial symmetries was invoked in [33], while numerical calculations of projected wavefunctions revealed properties of monopoles in [23,34]. A discussion of monopole symmetry properties on the square lattice and a numerical evaluation was reported in [35]. A trivial monopole was found, which appeared to be in conflict with the Lieb-Schultz-Mattis-Oshikawa-Hastings theorem (LSMOH) [38][39][40]. It was understood more recently that even with a trivial monopole, the QED 3 theory with N f = 4 still possesses a symmetry anomaly that forbids a trivial vacuum, in agreement with LSMOH theorem [41].
Here, we calculate symmetry transformation of monopoles paying special attention to the subtle 'Dirac sea' contributions, which arise from the Berry's phase acquired by monopoles on moving around gauge charges of the filled Dirac sea. The lattice provides a short distance cutoff that allows us to calculate this contribution, which proves crucial to the physics. We discuss previous works in light of our results. For example, we show that a trivial monopole appears in the square lattice DSL , consistent with duality based arguments [41] and earlier calculations [35] and does not contradict the LSMOH theorem. We then extend these arguments to other lat-tices where the arguments and methods of [35,41] do not apply. Physical consequences of these calculations for numerical simulations and experiments are then discussed.

II. GAUGE THEORY DESCRIPTION OF QUANTUM SPIN SYSTEMS AND MONOPOLES
We will be interested in spin-1/2 systems on various two dimensional lattices. Let us briefly review the fermionic spinon decomposition of spin-1/2 operators on the lattice which will lead us to the desired gauge theory description. We first decompose the spin operator, where f i,α is a fermion (spinon) on site i with spin α ∈ {↑ , ↓} and σ are Pauli matrices. This re-writing is exact if we implement the constraint α f † i,α f i,α = 1. To make progress, we consider a mean field approximation that only imposes the constraint on average, followed by a discussion of fluctuations (see Ref. [2]), There is a gauge redundancy f i → e iαi f in the parton decomposition Eq. (1), which results in the emergence of a dynamical U (1) gauge field a µ that couples to the fermions f , i.e. t ij → t ij e iaij . 1 Next, we arrange the hopping term t ij such that the hopping model leads to a pair of Dirac nodes, per spin, at half filling. Such Dirac dispersions, with four flavors of Dirac fermions (with two spin and two 'valley' labels) can be realized on the honeycomb lattice with only nearest hopping, as well as on other lattices (square, kagomè, and triangular lattice) with appropriate choice of t ij as shown in Fig. 1. We note that the mean-field Hamiltonian actually breaks lattice symmetry, but the spin liquid state has all the lattice symmetry after we incorporate the gauge constraint. For example, the triangular lattice ansatz can be C 6 invariant if we supplement space group operation with an SU (2) gauge transformation In the low energy, long wavelength or infrared (IR) 1 The non-bipartite nature (second-neighbor hopping on bi-partite lattice) is needed to make sure that the gauge group is U (1) rather than SU (2) [2].
limit, the theory reduces to the following Lagrangian: 2 where ψ i is a two-component Dirac fermion with four 2 The staggered flux state on square lattice has velocity Mean-field ansatz of Dirac spin liquid on the square, triangular and kagomè lattice. Mean-field Hamiltonian has only nearest hopping with a flux in each plaquette. (b) Band structure of the square lattice π-flux mean field ansatz with two Dirac cones at the momentum points (k 1 , k 2 ) = (±π/2, π/2).
flavors labeled by i, and a µ is a dynamical U (1) gauge field. We choose (γ 0 , γ 1 , γ 2 ) = (iµ 2 , µ 3 , µ 1 ) where µ are Pauli matrices. This theory is also known as Quantum Electrodynamics in three space-time dimensions, QED 3 with four fermion flavors N f = 4. The theory as written implicitly assumes that the U (1) gauge flux, i.e. the total flux of the magnetic field, j µ = 1 2π µνλ ∂ ν a λ is conserved. This theory, sometimes referred to as noncompact N f = 4 QED 3 , flows to a stable critical fixed point in the IR, as supported by recent numerical studies [42].
However, it is clear that this conservation of flux cannot be the consequence of a microscopic symmetry. Our model has no corresponding U (1) symmetry at the microscopic level which is an artifact of a topological conservation law in the low energy model (hence we refer to this as U (1) top ). This is remedied by allowing for quantum tunneling between vacuaa of different total flux. The tunneling events, instantons, occur at space-time points and the corresponding operators that create (or destroy) 2π flux quanta and are termed monopole (antimonopole) operators. Unlike most other physical operators, the monopole cannot be expressed as a polynomial of fermion or gauge fields. Nevertheless, it is important to note that these operators are local: they modify the magnetic field locally and the inserted 2π flux is invisible at large distances. As such they should be included in our Lagrangian Eq. (3). A key question will be whether physical symmetries might restrict which monopole operators are allowed. For this, we need to take a more careful look at the monopole operators.

III. MONOPOLES AND ZERO MODES
Previous calculations of monopole quantum numbers in gauge theories of quantum magnets have largely focused on bosonic QED 3 where the spinions are bosons, such as in the CP 1 models of quantum magnetism anisotropy, but it is irrelevant at large N limit [28]. [3,43,44]. There monopoles play a key role in descriptions of the Neel to VBS transition in square lattice quantum antiferromagnets. The case of fermionic QED 3 is in many ways richer, one of which is the presence of monopoles zero modes. We will see that this allows for a wider description of physical phenomena. For example, both collinear and non-collinear magnetic orders can be captured within a single theory, in contrast gauge theories of bosonic spinons capture one or the other, depending on the nature of the gauge group [3,45,46].
In a theory with massless Dirac fermions, monopoles occur along with fermion zero modes. Recall, massless Dirac fermions in a magnetic field form Landau levels, in particular a zero energy Landau level with a degeneracy equal to the number of flux quanta Φ/2π. Thus, addition of 2π flux creates a fermion zero mode for each Dirac fermion flavor, hence with N f = 4 we expect four zero modes. To maintain neutrality of gauge charge, we must fill half these modes, which can be done in C 4 2 = 6 ways. So a monopole operator can be schematically written as where f † i creates a fermion in the zero-mode associated with ψ i , and M bare creates a "bare" flux quanta without filling any zero mode. These operator can be more precisely defined, through state-operator correspondence, as states of the QED 3 theory defined on a two-sphere S 2 with a 2π background flux [37]. The Dirac zero-modes have zero angular momenta on the sphere in a 2π background flux. This implies that the monopoles are scalars under the Lorentz group. In terms of the SU (4) flavor symmetry, they form a vector of SO(6) = SU (4)/Z 2 . Note, the monopoles are in contrast to all other gauge invariant operators, such as fermion bilinears only transform under SU (4)/Z 4 without carrying the U (1) top charge. Thus, all physical operators are invariant under the combined action of the center of SO(6) and a π-rotation in U (1) top . So the precise global symmetry group is: together with the Lorentz group and discrete symmetries C 0 , T 0 , R 0 of the QED 3 Lagrangian (3). One can certainly consider 2π-monopoles in higher representations of SO (6), but in this work we will assume that the leading monopoles (with lowest scaling dimension) are the ones that form an SO(6) vector -this is physically reasonable and can be justified in large-N f limit. Instead of working with the explicit definition of monopoles from Eq. (4), we shall simply think of the monopoles as six operators {Φ 1 , ...Φ 6 } that carries unit charge under U (1) top and transform as a vector under SO (6): Clearly the lattice spin Hamiltonians does not have such a large symmetry -typically we only have spin rotation, lattice symmetries (lattice translation, rotation and reflection) and timereversal symmetries. The enlarged symmetry (such as SO(6) × U (1) top /Z 2 ) will emerge at low energy if terms breaking this symmetry down to the microscopic symmetries are irrelevant. We will discuss this in detail in Sec. IV.
A key question addressed in this paper is: given a U (1) Dirac spin liquid realized on a lattice, how do monopoles transform under the microscopic symmetries? In other words, how are the microscopic symmetries embedded into the enlarged symmetry group? Clearly spinrotation can only be embedded as an SO(3) subgroup of the SO(6) flavor group, meaning that three of the six monopoles form a spin-1 vector, and the other three are spin singlets. Denote the three singlet monopoles as V 1,2,3 = Φ 1,2,3 and the three spin-1 monopoles as S 1,2,3 = Φ 4,5,6 . In terms of filling zero modes these operators can be written as: where refers to the 2×2 antisymmetric matrix, σ, τ corresponds to the spin and valley index and we have split the subscript in f i to valley indices α, β and spin s, s . An important observation here is that since monopoles are local operators, they transform as linear representations of the symmetry group, in contrast to gauge charged fermions that transform under a projective symmetry group. Thus, for example, the monopoles transform as integer spin representations, unlike the spinons which carry spin one half. Other discrete symmetries can be realized, in general, as combinations of certain SO(6) rotations followed by a nontrivial U (1) top rotation, and possibly some combinations of C 0 , T 0 , R 0 . (Remember that Lorentz group acts trivially on the 2π-monopoles.) Many of these group elements can be fixed from the symmetry transformations of the Dirac fermions ψ i . For example, if the symmetry operation acts on ψ as ψ → U ψ with a nontrivial U ∈ SU (4), then we know that the monopoles should also be multiplied by an SO(6) matrix O that corresponds to U . This SO(6) matrix O can be uniquely identified up to an overall sign, which can also be viewed as a π-rotation in U (1) top . The same logic applies to other operations including C, T , R. The only exception is the flux symmetry U (1) top : there is no information regarding U (1) top in the symmetry transformation properties of the low energy Dirac fermions. Fixing the possible U (1) top rotations in the implementations of the microscopic lattice symmetries, and exploring their consequences, is our main task.
Since this is a key point, let us expand on it. Recall that via the state-operator correspondence, the monopole operators can be constructed by filling the Dirac sea together with half of the zero modes. Under a lattice symmetry, the transformation of zero modes corresponds to the SO(6) matrix, while the phase factor of U (1) top can be attributed to a Berry phase contribution, arising from the filled Dirac sea of the gauge charged partons. Physically, the U (1) top rotation arises from moving the monopole operator around a closed path that encloses gauge charge. This also inspires the numerical extraction of monopole quantum numbers as we will describe in the following [33,35].

A. A Numerical Calculation of Monopole Berry Phase
We adopt the following approach to calculate the monopole quantum numbers. For concreteness, consider the π flux theory on an L × L square lattice on the torus and uniformly spread the monopole flux of 2πq, so that each plaquette contains an additional flux of 2πq/L 2 . We numerically verify that the energy spectrum has a finite size gap with exactly 4q zero modes. This is consistent with the theoretical expectation and justifies constructing monopole operators on a torus. We now need to decide which zero modes to fill before proceeding with the monopole quantum number calculation. Let us now specialize to the case of a single monopole,i.e. q = 1. Note, the following calculation will extract both the contribution from the zero modes, as well as the more elusive Dirac sea contribution (see appendix B for precise Berry phase definition). Since the latter is common to all flavors of monopoles, it suffices to consider just the monopole which corresponds to filling the whole Dirac sea plus the two spin-up zero modes, while leaving the two spin down zero modes empty. This monopole preserves the largest set of lattice symmetries and corresponds to the monopole operator Φ † 4 − iΦ † 5 . We then numerically calculate its lattice quantum numbers by evaluating ψ|G R · R|ψ , where G R is a gauge transformation that keeps the Hamiltonian invariant after the lattice symmetry transformation R.
These quantum numbers are not all independent: they should satisfy the algebraic relations of the space group and time-reversal symmetry T [23,33,35]. This greatly reduces the number of U (1) top phase factors to determine, and moreover it constrains the quantum numbers of monopoles to a discrete set of allowed solutions. For , (c) To extract the lattice momentum, we calculate ψ|G T1 · T 1 |ψ = ρe ik1 , and find k 1 = 2π/3 on the triangular lattice and k 1 = π on the square lattice. ρ is close to 1 for a large system size, which indicates that the translation symmetry is approximately preserved.
example, the momentum of the monopole in the kagomè Dirac spin liquid must be 0. coming from the symmetry relation Since monopole charge does not change under rotation or translation in this case, we can assign the Berry phase θ C , θ 1/2 with C 6 , T 1/2 respectively. Furthermore, from the fermion PSG which control fermion zero mode transformation, (appendix A) one knows the spin triplet monopoles S i 's stay invariant up to Berry phases under these symmetries and hence transform as: combining with Eq. (7) immediately yields θ 1 = 0. Similarly one gets vanishing Berry phase for all translations. An intuitive way to see this is that the S i 's momentum should stay invariant under C 6 and the only C 6 invariant point in Kagome Brillouin zone is the Γ point. 3 The nontrivial phase factor of the kagomè Dirac spin liquid is associated with the C 6 rotation about the plaquette center [23] and we use the aforementioned numerical scheme to directly read off this rotation quantum number. We list the possible Berry phases compatible with algebraic relations of symmetry group on 4 lattices in appendix A and below is the numerically found Berry phase which indeed falls among one of the possible choices constrained by algebraic relations. On the triangular lattice, the lattice momentum of the monopole Φ † 4 is constrained, by point group symmetries, to be one of 0, ±2π/3. We determine its precise value here by evaluating ψ|G T1,2 · T 1,2 |ψ , i.e. combining translation T 1,2 with the appropriate gauge transformation G T1,2 to leave the mean field ansatz invariant. Again, we consider filling the Dirac sea and the two spin up zero modes as our ground state |ψ . Unfortunately, one cannot resort to reading off the relevant quantum numbers since the Hamiltonian on a torus with the added flux is not translation invariant. To see this, note that the phase factors of Wilson loops e C a·dl , along two adjacent columns (rows) will always differ by 2π/L (there are L unit cells between them). Such translation symmetry breaking will be invisible in the thermodynamic limit, since the phase difference of adjacent Wilson loops 2π/L → 0 as L increases. We expect that translation symmetry to be recovered for large enough system sizes. Indeed, this is what we observe in Fig. 2, which shows our numerical results for ψ|G T1 · T 1 |ψ = ρe ik1 . For a large L, the momentum k 1 is perfectly quantized to 2π/3. Moreover, the amplitude ρ approaches unity, indicating the restoration of translation symmetry. As a comparison, we also calculate the monopole's momentum on the square lattice, and we get k 1 = π. Therefore, the momentum of the monopole Φ † 4 on the π flux square lattice is (π, π), on the triangular lattice it is quantized to (−2π/3, 2π/3) and C 6 equivalents. We also numerically verify that under translations on honeycomb and kagomè lattices, monopoles has zero Berry phase dictated by algebraic relations. For triangular lattice, since reflection involves charge conjugation, monopole charge does not change and numerically we find the Berry phase under reflection to be 0.
We note that our result for the square lattice is consistent with an earlier calculation [35] which used a cylinder geometry to select from the discrete set of possibilities allowed by crystal symmetries. The cylinder geometry is particularly convenient since translation symmetry in one direction can be preserved even in the presence of flux. Unfortunately, we caution that for other lattices, the cylinder geometry gives an incorrect answer (i.e. π even for the triangular lattice where the only consistent momenta are 0, ±2π/3.). We believe this problem potentially arises from the presence of edge states, which can be circumvented by adopting the torus geometry as we have done here.
Finally, we remark that our numerical method cannot determine the precise transformation of the timereversal symmetry T and reflections. This, however, can be calculated analytically as we will describe elsewhere [47], which also confirm the results here for other symmetries. It turns out that for all the Dirac spin liquid states we consider, the spin-singlet monopoles are even under time-reversal Φ † 1,2,3 → Φ 1,2,3 , while the spin triplet monopoles are odd under the time-reversal Φ † 4,5,6 → −Φ 4,5,6 . It contrast, Ref. [23] conjectured that all monopoles in the kagomè Dirac spin liquid are odd under T .

IV. MONOPOLE SYMMETRY TRANSFORMATION: STABILITY AND PROXIMATE ORDERS
The transformation properties of monopoles on various lattices are summarized in the three tables below. The fact that monopoles are local operators implies that they transform as linear (rather than projective) representations of the symmetry groups. Note, the bipartite lattices have a trivial monopole i.e. one that transforms as the identity representation under all symmetries. We elaborate on the consequences of this observation below.
In Table I, the monopole transformation properties for the honeycomb DSL and square lattice staggered flux DSL are presented. Results for the square lattice align with the results of M transformations of Ref [35] after making the identification For the special case where the flux on the square lattice is φ = π an additional unitary symmetry, charge conjugation, is present which sends Φ † 1/3/4/5/6 → Φ 1/3/4/5/6 , and Φ † 2 → −Φ 2 . Combining this with T 1/2 , C 4 , T gives the transformations for translations and rotations, time-reversal in this state.

A. Stability of the DSL
The stability of the U (1) Dirac spin liquid can be discussed in three stages. First, there is Eqn. (3), QED 3 with N f = 4 flavors, which neglects monopoles and has a global SU (4) flavor symmetry. The stability of this theory has been discussed both from the numerical (lattice gauge theory) perspective [42,48] as well as from the epsilon expansion, [49], all of which conclude that it flows to a stable fixed point. We therefore begin our discussion by analyzing the remaining two effects -that of four fermion interaction terms that break SU(4) symmetry, and that of magnetic monopoles.
Let us begin with the scaling dimension of the monopole operator. Within a large N f approximation, the scaling dimension is: , so setting N f = 4 yields ∆ 1 = 1.02 < 3, which implies that this operator is strongly relevant. While the true scaling dimension at N f = 4 could be different, this is unlikely to exceed 3. We will therefore assume that the single monopole operator is a relevant perturbation. For the bipartite lattices, the presence of a trivial monopole implies a single monopole insertion operator is allowed on symmetry grounds in the Lagrangian. Then, we do not expect the U (1) Dirac spin liquid to be a stable phase. What does it flow to? The most likely scenario is that chiral symmetry is broken, i.e. a mass term is developed by spontaneous symmetry breaking. This still leaves a gapless photon, which is removed by monopole proliferation [50]. We will argue below that this does not lead to additional symmetry breaking, and conclude that the colinear Neel order or common VBS orders on bipartite lattices are likely to be realized in this theory at the lowest energies.
On the other hand for the non-bipartite lattice DSLs considered here, i.e. the triangular and kagomè DSL, no such trivial monopole is present. This has a number of consequences. First, the QED 3 theory discussed here could potentially represent a stable phase, with an enlarged SU (4) × U (1)/Z 4 global symmetry which appears in the low energy limit. We discuss this and other possibilities below. First, let us discuss the issue of monopole operator scaling dimensions. For the triangular lattice, under translations, note that Φ 1,2 have k 1 = π/3 and Φ 3 has k 1 = −2π/3, and the lowest order invariant monopole terms are: Note, the mismatch in momentum with fermion bilinears, which only pick up phase factors that are multiples of π, implies that there is no invariant term with a smaller monopole charge. Within a large N f calculation [51], the scaling dimension of this triple monopole is , which makes it very likely to be an irrelevant perturbation at the SU(4) symmetric fixed point. The remaining operator to inspect is the four fermion that breaks SU(4) symmetry, that can be written as and Φ 4/5/6 are spin triplet monopoles. Symmetry operations T 1/2 , R x denote translation along two lattice vectors (for honeycomb T 1/2 direction has 2π/3 angle between them) and reflection along horizontal bonds, respectively. Rotation implies site centered 4-fold rotation for the square lattice and hexagon centered 6-fold rotation for the honeycomb. There is always a trivial monopole (highlighted in red) for DSLs on both these bipartite lattices. On proliferating the trivial monopole the emergent symmetry is reduced from U (1) top × SO(6) → SO (5), and the 15 SO(6) adjoint fermion bilinears spilt according to 5 + 10. The 5 fermion bilinears, which form an SO(5) vector, are now symmetry equivalent to 5 monopoles, as listed in the last column, which is relevant to the chiral symmetry breaking pattern described in Sec IV B 1 and Eq. (12). it is expected to be relevant to understanding the phase structure on the triangular lattice. In contrast, on the kagomè lattice an inspection of the monopole and mass term transformation laws imply (see Table III) the following two invariant terms: where M 0i ≡ ψτ i ψ. Note, the first term involves a combination of a single monopole insertion operator and a fermion bilinear, which may be regarded as the excited state of a monopole with larger scaling dimension, and the second term refers to doubled monopole insertion, which preserves symmetry if one considers the associated Lorentz singlet operators, details in appendix D. The scaling dimensions for these operators, estimated from large N f is ∆ 1 * = ∆ 1 + 2 √ 2 ∼ 3.84 and ∆ 2 = 0.673N f −0.194 ∼ 2.50. While these are nominally relevant, their closeness to 3 implies that we should leave open the possibility of a stable phase or critical point on the kagomè lattice described by a U(1) Dirac spin liquid. Regardless of stability, this difference in the nature of the monopoles from the bipartite case will have an important impact on proximate orders that we document below. In particular, relatively complex magnetic orders such as the 120 degree state and the 12 site VBS pattern on the triangular lattice are captured.

B. Chiral Symmetry Breaking, Monopole Proliferation and Ordered States
Now, we will be concerned with identifying ordered states that can be reached from the Dirac spin liquid, either as a result of an intrinsic instability, or because interactions are tuned to trigger a phase transition. The scenario that we will assume is that of a two step process with spontaneous mass generation occurring first, i.e. a fermion bilinear spontaneously acquires an expectation value by symmetry breaking, followed by the monopole proliferation and confinement [50]. The 16 fermion bilinears are classified as 1 ⊕ 15, a singlet and adjoint representation of SU(4)∼ SO (6). Depending on the symmetries of the interaction, a mass termψM ψ, with M being either the identity or a vector such as M = (M 01 , M 02 , M 03 ), can be generated. This is captured by the following Gross-Neveu type model: φ represents bosonic fields , which can either be a scalar field or a vector field depending on the type of generated massψM ψ.
The singlet mass is a quantum Hall mass termψψ, which breaks time reversal and parity symmetry. If spontaneously generated, it will lead to a chiral spin liquid, a gapped phase with topological order but gapless edge states and semion excitations. In this scenario, the Chern Simons term suppresses monopole proliferation.
The second scenario is when chiral symmetry is broken by the spontaneous generation of one of the 15 chiral mass terms, which are conveniently labeled in terms of Different from the quantum Hall (singlet) massψψ, the chiral mass does not lead to a Chern-Simons term. Therefore, we are left with a pure U (1) gauge theory, which may have a further instability to monopole proliferation and confinement. A key input is to identify which monopole is selected following the chiral symmetry breaking by one of the 15 mass terms. Operationally, this selection arises since the mass term splits the zero mode degeneracy in the monopole. For example, consider a 'quantum spin Hall' mass term M 30 =ψσ 3 ⊗ 1ψ, that associates S z spin density with magnetic flux. A magnetic monopole then has both the spin-down zero modes filled, corresponds to S † 1 + iS † 2 , which, when inserted into Eq.
bare , consistent with their filling down spin modes. A general relation between the mass terms and monopoles can be obtained by doing a SO(6) rotation on the familiar case of quantum spin Hall mass. (The mass terms are in the adjoint representation of SO(6) and the monopoles are the SO(6) vectors.) We find that in general, a mass term ±ψT ab ψ will lead to proliferation of the monopole ( n a ± n b ) · Φ, where T ab is the SO(6) generator that rotates in the plane spanned by the two orthogonal unit vectors { n a , n b }. Specifically, 1. The mass term ±M c0 will proliferate the S a ± iS b monopole, where (a, b, c) is an even permutation of (1, 2, 3); this leads to non-collinear magnetic order that fully breaks SO(3) spin , see sec IV B 2, IV B 3 and appendix E.
2. The mass term ±M 0c will proliferate the V a ± iV b monopole, where (a, b, c) is an even permutation equivalent of (1, 2, 3); this leads to valence bond solid(VBS) type order that breaks lattice symmetries, see sec IV B 1, IV B 2, IV B 3.
3. The mass term ±M ab will proliferate the S a ∓ iV b monopole, resulting in mixed order with collinear spin order along σ a direction and VBS order, see appendix E, with the exception of pure Neel order for mass M a2 , M a3 on square and honeycomb lattices, respectively, see sec IV B 1.
We now discuss in more detail the consequence of a trivial monopole on the bipartite lattices, where chiral symmetry breaking alone may determine the ordered states, in contrast to the triangular and kagomè cases where the monopole proliferation (confinement) leads to additional symmetry breaking.

Bipartite lattices
We first highlight the existence of a trivial monopole in the DSL on bipartite lattices shown in table I. A trivial monopole, by our definition, stays invariant (or goes to its conjugate) under translations, rotations,reflections and time-reversal symmetry. (Note that time-reversal reverses the monopole charge, so by invariant we mean The presence of the trivial monopole in the Lagrangian will presumably drive the theory to strong coupling. The emergent symmetry is broken from SO(6) × U (1)/Z 2 to SO(5) by this trivial monopole. The remaining SO(5) still possess a symmetry anomaly [41], so the theory cannot flow to a trivially gapped symmetric state. A natural assumption is that in the absence of further fine tuning, this will lead to chiral symmetry breaking. But the 15 adjoint masses no longer stand on equal footing: There is a term allowed in Lagrangian coupling fermion bilinears and monopoles that is invariant under SO(6) × U (1) top /Z 2 and other discrete symmetries.
Once the trivial monopole V j condenses, the equation above picks out the terms involving V ( †) j which then becomes more relevant than other mass terms -hence the masses M 0i (i = j), M aj (a = 1, 2, 3) are more likely to be generated in chiral symmetry breaking. These five mass terms form a vector under the SO(5) flavor symmetry that remains unbroken by the V j condensation, while the remaining (less relevant) ten mass terms transform as a rank-2 symmetric traceless tensor.
For example, on the square lattice, there is a symmetry trivial monopole iV 2 − iV † 2 = 2 Im V 2 . Its proliferation will lead to the spontaneous generation of a mass term M i ∈ {ψτ 2 ⊗ σ i ψ,ψτ 1/3 ψ} per discussion above, yielding spontaneous symmetry breaking. Indeed, the chiral symmetry breaking states are the familiar Neel or columnar VBS states. First, we note that the mass termsψσ i ⊗ τ 2 ψ have the same symmetries as the familiar Neel order along the σ i direction, while mass terms ψτ 1,3 ψ have the same symmetry transformation properties as the columnar VBS order parameter (table IV in appendix for mass transformation). According to our previous discussion, the generation of a mass term will further lead to monopole proliferation. This however does not further break symmetry. From Eq. (12) it is clear that in the presence of the trivial monopole V j , the coupling between SO(5)-vector mass terms and the five nontrivial monopoles becomes effectively linear (since V j = 0), which makes the two sets of operators essentially identical from symmetry point of view. Therefore monopoles will not further break any symmetry after an SO(5)-vector mass condensate is established. For example, on the square lattice the mass termψτ 2 σ 3 ψ will lead to the monopole condensation Im (V 2 +iS 3 ) = 0, which does not break further symmetries. Similarly on honeycomb lattice, the five masses {ψτ 3 σ i ψ,ψτ 1/2 ψ} leads to proliferation of monopoles V 3 + iS i , V 2/1 + iV 3 , respectively, which result in Neel/Kekule VBS. These orders also align with the corresponding masses (table IV for  mass transformation).
Potentially, on tuning the balance between Neel and VBS orders, a deconfined critical point may be accessed [4,14,41] (the transition may also be first order, depending on the exact long distance fate of the theory). The possible emergent SO(5) symmetry [52] at the Neel-VBS transition is nothing but the unbroken flavor symmetry of QED 3 . It is important to note that the SO(5) emergent symmetry of the deconfined critical point is only a subgroup of the ∼ U (4) symmetry of the DSL.
Previously, there has been debate regarding the ground state associate with the staggered flux spin liquid state on the square lattice. Based on the monopole quantum numbers reported here (and in ref [35]) and the discussion above, we conclude that it describes an ordered state. Although the precise nature of the order is determined by microscopic interactions, it is certainly compatible with the commonly observed colinear Neel order.
If rotation symmetry is broken (e.g., from square to rectangle lattices), monopole momenta are not enforced to be quantized by algebraic relations and generally we expect all elementary monopoles are forbidden by symmetry. The stability of DSL on square and honeycomb lattices is hence enhanced.

Triangular Lattice
On the triangular lattice, we find the symmetry transformation properties detailed in Table II. We remark that the momenta of monopoles receive a contribution from the nontrivial Berry phase in U (1) top for translations.
As we have noted, magnetic monopoles correspond to local operators, and their symmetry transformation allows us to interpret them as order parameters, which, if condensed, break symmetry in particular ways.
Magnetic Order: For example, the spin triplet monopoles S 1,2,3 are the order parameter of the 120 o non-collinear magnetic order. If the monopoles condense, we will have a spin ordering pattern S r = S(n 1 cos(Q · r) + n 2 sin(Q · r)) To establish that this order is the 120 o non-collinear magnetic order, we need to further show that monopoles condense in a channel that satisfies n 1 · n 2 = 0. Recall that there are two steps to generate an order. First, a fermion mass is spontaneously generated through chiral symmetry breaking. Next, the mass term will pick one monopole to condense. This two-step mechanism guarantees n 1 · n 2 = 0. For instance, the massψσ 3 ψ will have the monopoles condensing in the channel S 1 + iS 2 = 0, S 1 − iS 2 = 0, and S 3 = 0, which satisfies the constraint n 1 · n 2 = 0. This eventually yields the 120 o magnetic order with the magnetic moments lying in the S x S y plane, with the specific chirality shown in Fig 3(4). VBS order: The general principle for determining the VBS order induced by a specific chiral symmetry breaking scenario involves the following single rule: match only the preserved symmetries. By this we exclude symmetries under which the mass/monopoles obtain a nontrivial phase (i.e., only when they stay strictly invariant do the symmetries count as being preserved). When a valley-Hall mass M 0i leads to the condensation of spin singlet monopoles V j +iV k giving rise to VBS order, typically, the order parameter is a polynomial of the condensed monopole and the valley Hall mass. This polynomial retains only the common preserved symmetries of V i and the mass, but not any nontrivial transformation (or phases) (e.g., under original translations); hence so does the VBS order. For example, consider translations. It suffices for the VBS pattern to have a new (and usually enlarged) unit cell commensurate with both the fermion bilinear and the condensed monopole, i.e., nontrivial phases under translations on the original lattice do not further constrain the VBS order 4 .
Following this logic, the new Bravais lattice for VBS on triangular lattice is set by the momenta of Φ 1/2/3 andψτ i ψ's. So from Fig 3(1), the reduced Brillouin zone has an area which is 1 12 of the original one resulting from the Berry phase attached to the monopoles, corresponding exactly to the √ 12 × √ 12 VBS order with a 12-site unit cell. A generic combination ofψτ i ψ induces monopole condensation that breaks all spatial symmetries except for translations commensurate with the new unit cell. Following the previous discussion, there are no further symmetry constraints on the VBS order pattern inside the enlarged unit cell. In the following, we display some typical patterns that could descend from the DSL. Shown in fig 3(2) is a generic pattern used in Ref [53] as a ground state candidate for a quantum dimer model on triangular lattice, with maximal flippable plaquettes. The plaquette VBS shown in Fig 3(3)) has an additional C 3 symmetry and results from a particular mass M 01 + M 02 − M 03 with the corresponding proliferation, both of which stay invariant under C 3 around the marked triangle center (equivalently T 1 C 2 6 ) according to table II.

Kagomè Lattice
Magnetic Order: On the kagomè lattice, the situation is very similar to the triangular lattice, where the Berry phase of Dirac sea for C 6 rotation is 2π 3 by numerical calculation (appendix C), which is consistent with the translationally invariant ('q = 0') 120 o magnetic order under six-fold rotation. The results are summarized in Table III. VBS Order: Masses M 0i result in VBS order. Generally, since the momenta of masses and monopoles are located at points like (0, π) in the Brillouin Zone (leftmost panel of fig 4), the VBS pattern has an enlarged unit cell 4 times as big as the original one.
As before, we construct VBS order parameters from chiral symmetry breaking by looking at only the symmetries strictly preserved by both the mass and the condensed monopole. Shown in fig 4 are examples which reproduce previously found VBS patterns on the kagome lattice [23,54,55]. Specifically, Fig 4(3) results from mass M 02 with V † 1 + iV † 3 = e −i π 4 , which preserves R y but breaks C 6 according to table III; the bond patterns have the same symmetry group (translations and R y ) as that found in a recent DMRG study proximate the spin liquid phase (panel C of Fig. 2 in Ref. [54]). Further, Fig. 4(1)(2) result from a C 6 invariant mass M 01 + M 02 − M 03 and the associated monopole = 1, i, respectively. They are C 6 invariant, while (1) preserves R y and (2) breaks R y , owing to Φ † kag → Φ kag under R y , leading to the preservation of R y if Φ † kag = 1 in 1. Fig 4(1) reproduces Fig.4 in Ref. [21] or Fig. 5 in Ref. [23], where the 12 bonds around a unit cell are enhanced (or weakened an in our convention). Moreover symmetries of figs 17 and 18 in Ref. [23] both align with the real part of V i 's. We remark that the specific expectation values of monopoles discussed above sit at the  1. R y , C 6 denotes reflection with respect to y axis and six-fold rotation around center of hexagon. The 6-fold rotation symmetry acting on monopoles cannot be incorporated into the vector representation of SO(6) owing to the nontrivial Berry phase, which is in line with the magnetic pattern expected on the kagome lattice.
extremum of Eq. (10) and hence these more symmetric patterns optimize the Landau-potential given by the two-fold monopole on kagomè lattice.

V. EXPERIMENTS, NUMERICS AND DISCUSSION
Our calculation of symmetry quantum numbers of monopole excitations in the QED 3 Dirac spin liquid theory on different 2D lattices indicates that the DSL on triangular and kagomè lattices may be a stable phase. This would represent a remarkable state of matter, with enhanced symmetries including a ∼ U (4) symmetry combining spin rotation and discrete spatial symmetries, as well as invariance under conformal (including scaling) transformations. We emphasize that the low energy excitations of the DSL includes both fermionic spinons and the magnetic monopole continuum. These two types of excitations have different characteristic signatures in the spectral function, originating from the different scaling dimensions of the operators. They are also typically located at different high symmetry points of the Brillouin zone. The low energy spin-triplet excitation arising from pairs of spinons are located near the M points of the Brillouin zone for both the triangular and kagomè lattice. In contrast, the spin-triplet monopole excitations appear at the K points for the triangular lattice, and at the Γ point condensing to Φ † kag = 1, i, respectively. Pattern 3 results from mass M 02 with V † 1 + iV † 3 = e −i π 4 , which preserves R y but breaks C 6 . Pattern 1 is symmetry equivalent to the Hastings VBS of Ref. [21], and Pattern 3 is symmetry equivalent to the VBS found by DMRG proximate to the spin liquid phase in Ref. [54] (panel C of Fig 2).
for the kagomè lattice 5 These readily measurable characteristics should help in the empirical search for these phases.
Comparing with numerical studies, the kagomè Heisenberg antiferromagnet (KHA) was recently argued to be consistent with a Dirac spin liquid [25,56,57]. Indeed, on increasing the next neighbor coupling J 2 > 0, a 120 o (the so called q = 0) ordered phase is observed on exiting the spin liquid. Furthermore, a diamond VBS pattern ( Fig. 4(4)) was found to be proximate the spin liquid of KHA [54], consistent with our identification of proximate orders. A third piece of evidence is that a chiral spin liquid (CSL) is observed by increasing the J 2 and J 3 interactions [58][59][60], or by adding a small spin chirality term J χ S i · (S j × S k ) [61]. This CSL can be understood as the DSL with a singlet mass mψψ.
It should however be noted that gapped spin liquids have also been proposed as the ground state in this parameter regime [54,[62][63][64]. In Herbertsmithite, a material realization of the KHA, a spin liquid ground state is reported. Although low energy excitations are observed, an analysis that includes disorder points to a spin liquid with a small spin gap of ∼ J/20 [65,66]. Nevertheless, a comparison of the low energy spectral weight in kagome

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< l a t e x i t s h a 1 _ b a s e 6 4 = " s d l v T Z D 4 Z 2 z b 4 l i l U q 3 a g H / U C t c = " > A A A C L X i c b V D L S g N B E J z 1 b X x F P X o Z D I I I y m 4 u e h T 1 4 F H B q J A N o X e 2 E w d n Z t e Z X j E s + S E v / o o I H h T x 6 m 8 4 i T n 4 K m g o q r q Y 6 U p y J R 2 F 4 U s w N j 4 x O T U 9 M 1 u Z m 1 9 Y X K o u r 5 y 7 r L A C G y J T m b 1 M w K G S B h s k S e F l b h F 0 o v A i u T 4 c + B e 3 a J 3 M z B n 1 c m x p 6 B r Z k Q L I S + 3 q U Z x g V 5 o S b w x Y C 7 2 t f p y A j X M n B 8 O 3 Y 8 I 7 s r o 8 k p r 3 e Z y i I m h H M Z r 0 W 6 R d r Y U 7 4 R D 8 L 4 l G p M Z G O
G l X n + I 0 E 4 V G Q 0 K B c 8 0 o z K l V g i U p F P Y r c e E w B 3 E N X W x 6 a k C j a 5 X D a / t 8 w y s p 7 2 T W j y E + V L 8 n S t D O 9 X T i N z X Q l f v t D c T / v G Z B n b 1 W K U 1 e E B r x 9 V C n U J w y P q i O p 9 K i I N X z B I S V / q 9 c X I E F Q b 7 g i i 8 h + n 3 y X 3 J e 3 4 k 8 P 6 3 X 9 g 9 G d c y w N b b O N l n E d t k + O 2 Y n r M E E u 2 e P 7 I W 9 B g / B c / A W v H + t j g W j z C r 7 g e D j E / 8 F q c U = < / l a t e x i t > For the S=1/2 triangular lattice antiferromagnet, numerical diagonalization and density matrix renormalization group (DMRG) studies have revealed a spin disordered state on adding a small second neighbor coupling 0.07 < J 2 /J 1 < 0.15 [67]. Variational Monte Carlo calculations have concluded that the Dirac spin liquid [30] is a very competitive ground state. A further piece of evidence is obtained on adding an explicit but small spin chirality interaction J χ which is found to immediately lead to a CSL [68,69] in this range of parameters. This is consistent with perturbing the Dirac spin liquid with an explicit mass term mψψ, which is immediately generated on breaking time reversal and parity symmetry, leading to a chiral spin liquid. Outside this parameter range, the CSL is also obtained, but only on adding a finite value of J χ .
Given the relatively small values of J 2 involved, even the nearest neighbor Heisenberg model should display aspects of spin liquid physics at intermediate scales which can be studied in future experiments and numerics. There are relatively few experimental candidates for the triangular lattice S=1/2 materials that are undistorted. Two recently studied candidates are Ba 3 Co Sb 2 O 9 and Ba 8 CoNb 6 O 24 . The latter compound fails to order even at the lowest temperatures measured ∼ J/25, [70] and is a promising spin liquid candidate. Although the former compound orders at low temperatures, its excitation spectrum [71] is hard to account for within spin wave theory. We note that at a qualitative level, the discrepancies from spin wave theory are connected to low energy spectral weight at the M points, which are recovered within the QED 3 theory, where they arise from fermion bilinears. It has been pointed out that additional terms beyond the Heisenberg interaction may be present in these materials [72] which help stabilize a spin liquid state. The transition metal dichalcogenide 1T-TaS 2 has also been proposed as a quantum spin liquid where the spin degrees of freedom reside on clusters that form a triangular lattice [73]. The possibility of realizing the Dirac spin liquid state on the triangular lattice should give additional impetus to novel physical realizations of S=1/2 triangular lattice magnets for example, in ultracold atomic lattices and in twisted bilayers of transition metal dichalcogenides which remain to be experimentally realized [74].
In Fig 5 we show the momenta and other spatial symmetry quantum numbers of the 15 fermion bilinears (adjoint masses with identical scaling dimension) and 1 singlet mass, and those of the six monopoles. These should help guide the search for Dirac Spin Liquids in X-ray (sensitive to the spin singlet excitations) and neutron (which probe both singlet and triplet excitations) scattering experiments.
In addition to a potentially stable spin liquid phase, which would represent a remarkable new state of matter, the DSL provides a unified picture to describe competing orders which have already been observed either in experiments or in numerical calculations, on different lattices.
These range from the colinear Neel states on bipartite lattices to the 120 o degree ordered states on the triangular and kagomè lattices, and to spin singlet valence bond crystal states. It is hoped that such a unified picture of two dimensional magnetism will deepen our understanding of some of the most interesting correlated electronic materials.

Honeycomb
On honeycomb lattice, with mean-field ansatz of uniform fermion hopping, one could similarly work out the PSG and the constraints on monopole quantum numbers. The Dirac points stay at momenta Q = ( 2π 3 , 2π 3 ), Q = −Q. The physical symmetries act as where T 1/2 is the translation along two basis vectors with 2π/3 angle between them,C 6 is π/3 rotation around a center of a honeycomb plaquette, and R denotes reflection along the direction of the unit cell. From the transformation of Dirac fermions, one gets transformation of fermion masses. The algebraic relations between symmetries constrain the Berry phase of the Dirac sea as follows: (berry phase for translation T 1/2 : θ 1/2 , for C 6 : θ C ) where C 3 = RC −1 6 R −1 C 6 is the 3-fold rotation around a A sub-lattice site. Together they stipulate that Note it's the relations involving translation and rotation such as C 3 T 2 = T 1 C 3 that enforces the vanishing berry phase under translations.

Triangular lattice
There's a "staggered π flux" configuration of t ij on the triangular lattice. We choose a particular gauge of t ij to realize this mean field as in Fig 1. Under appropriate basis the low-energy Hamiltonian reads as the standard form with 4 gapless Dirac fermions. In the new basis, the matrices in the Dirac equation are For later purposes, the charge conjugate operation is given here as The PSG for all the symmetry operations transform as the following: There are the following defining relations of the symmetry group which gives constraints on Berry phase Applying the above algebraic relation to monopole transformations, we find the following constraints on the Berry phase for translation T 1/2 : θ 1/2 and reflection R : θ R aswe could fix other phase factors to be and numerically we find θ 1 = 2π 3 , θ R = 0.

Kagome
On kagome lattice, similar to triangular case, Hermele et al calculated the kagome DSL with staggered flux meanfield ansatz, with three gamma matrices as γ ν = (µ 3 , µ 2 , −µ ! ), and we have for the PSG of Dirac fermions as The algebraic relations used to fix translation and rotation Berry phase read: which fix the Berry phase for translation to be zero θ 1 = θ 2 = 0 (notice this is the result of relations like C 6 T 1 = T 2 C 6 , which means rotation selects discrete values for momenta); the phase for rotation θ C = nπ/3 while we numerically find θ C = 2π/3.

Appendix B: Sign ambiguity of Berry phase
Here we remark on the sign ambiguity of Berry phase. As stated in the main text, there is a Z 2 element in both SO(6),i.e., its center and U (1) top , i.e., −1 that act identically on all physical operators, i.e., trivially on fermion bilinears and giving a minus sign for all 6 monopoles. Berry phase, by definition, is the element in U (1) top under certain symmetry transformation that is embedded into the emergent symmetry group SO(6) × U (1) top /Z 2 ; therefore whether the Z 2 operation belongs to SO(6) or U (1) top ,i.e., Berry phase, is arbitrary and up to one's choice of convention. In other words, certain PSG ψ → W ψ where W is an SU (4) matrix can be changed to an equally good PSG ψ → iW ψ since iW is also an SU (4) matrix. While this additional i factor has no effect on fermion bilinears, it corresponds to the center in SO(6) which changes sign for all monopoles. The physical symmetry operation should stay invariant regardless this change in PSG meaning the Berry phase should change by π to compensate the center in SO(6).
In our numerics and Berry phase analysis by algebraic relations, we define the SO(6) element by the transformation of 6 fermion bilinearsψτ 1/2/3 ψ,ψσ 1/2/3 ψ to eliminate this sign ambiguity. This is possible because actually all physical symmetries act within the SO(3) valley × SO(3) spin subgroup of the SO(6), and the three valley(spin) hall massesψτ 1/2/3 ψ,ψσ 1/2/3 ψ, which are adjoint representation for SO(3) valley/spin , respectively, happen to constitute also the vector representation of SO(3) valley × SO(3) spin . Hence they can be used as the reference frame with regard to which the Berry phase is defined. x y a b c A 1 (r)  On the kagomè lattice, we numerically find ψ|G C6 · C 6 |ψ = e i2π/3 , hence the lattice angular momentum of spin triplet monopole is 2π/3. Specifically, we have a L × L kagomè lattice on a torus, and consider the parton mean-field ansatz of Dirac spin liquid with uniformly spreading 2π flux on the kagomè lattice (see Fig. 6), where the gauge fields A(r = (x, y)), x, y = 0, 1, · · · , L − 1, are We further diagonalize the single-particle Hamiltonian, and construct the monopole state |ψ by filling the Dirac sea as well as two spin-up zero modes. Finally we numerically obtain the lattice angular momentum, ψ|G C6 ·C 6 |ψ = e i2π/3 . Note, in this particular case we are able to retain symmetry in the state with a single monopole insertion.
Appendix D: Symmetry-allowed higher-order monopoles on Kagome and Triangular lattices We consider the transformations of 4π monopoles on Kagome lattice Dirac spin liquid. Under 4π magnetic fluc, each Dirac fermion bears 2 zero modes, carrying Lorenz spin−1/2, denoted by f k,s,± where ± associate to Lorentz index.
On kagome lattice, three gamma matrices are γ ν = (µ 3 , µ 2 , −µ 1 ) (acting in Lorenz index space), and we have for the PSG of Dirac fermions as where where τ, σ act in valley (k) and physical spin (s) spaces and zero modes transform as Dirac fermions. Leading 4π monopole consists of filling Dirac sea |4π and 4 out of 8 zero modes, giving 70 4π monopoles. Consider following three such monopolesΦ † i made of which are spin σ singlets, and p = ± denotes the lorentz index. These constitute Lorentz singlets and don't vanish even though zero modes are fermionic. Berry phases are twice forΦ i as for elementary Φ i and SO(3) valley part of the symmetry permutes the indices of τ 's (i.e., i inΦ i ) at most. which amounts to an excited lorentz singlet 2π monopole. The leading-order operator in this kind results from exciting one landau level n = −1 mode to n = 1 and has dimension ∆ 0 + 2 √ 2 ≈ 3.8 which is irrelevant. In the operator product expansion of the above term, the monopole carries vanishing lorentz spin and hence the coefficient of excited monopole from exciting n = 0 to n = 1 vanishes. We could also rule out the case with a lower dimension where a zero mode is excited to n = 1 Landau level since then, the monopole will inevitably carry lorentz spin 1 and won't be invariant under the lorentz group part Exp(iµ 3 π/3) for C 6 and iµ 1 for R in eq(D1) simultaneously.
For triangular lattice, the leading-order symmetry-allowed is 3−fold monopoles and the zero modes carry lorentz spin 1. Similarly, we construct lorentz singlet out of lorentz spin 1 zero modes and they transform formally as Φ 1 Φ 2 Φ 3 , i.e., the corresponding 6π monopole consists of where ±1, 0 labels lorentz spin and each factor labeled by valley index i = 1, 2, 3 creates two zero modes that carries total 0 lorentz spin (i.e., the lorentz index part amounts to |1, −1 −|0, 0 +|−1, 1 which is a singlet) and transform as the associated elementary monopole Φ i for the SO(6) flavor part. Hence we've got a lorentz singlet symmetry-allowed monopole on triangular latticesΦ 6π +Φ † 6π , dimension likely irrelevant.

Appendix E: Unconventional orders proximate to DSL
In this appendix we discuss the unconventional orders that could descend from DSL theory. We consider the "mixed" fermion mass ±ψσ i τ j ψ on different lattices and the quantum spin hall mass on bipartite lattices. The general strategy relating mass and monopole proliferation to symmetry-breaking orders is to find the microscopic spin operators (as simple as possible for practical purposes) that transform in the same way as the mass and monopoles (they do not have to be identical). The ordered states should have orders given by all of the microscopic operators.
With a "mixed" mass ±ψσ i τ j ψ, monopole V j ∓ iS i will proliferate depending on the sign of the mass. The conventional wisdom is that excluding square lattice case, mass and S i is time-reversal odd hence corresponding to spinful operators, simplest of which is a single spin operator (which we will show later does not suffice for triangular/square lattices); while V j is time-reversal even and does not carry spins, corresponding to valence bond operators S i · S j ; hence this scenario leads to a mixture of spin and valence bond order. One remarkable feature is that mass and monopoles all preserve spin rotation along σ i direction, leading to collinear spin orders along this direction, if any. Now we illustrate some simple or relatively symmetric order patterns resulting from ±ψσ i τ j ψ and monopoles. On triangular lattice, first consider a typical such mass, M 33 =ψσ 3 τ 3 ψ which favors V 3 + iS 3 . We consider the case where V 3 + iS 3 = 1, plotted as Fig 7(1). The spins should order along z direction. If we write down a trial association where Q = ( π 3 , 2π 3 ) is S 3 's momentum, we check that this operator has the same symmetry group as S 3 , i.e., aside from translations, S 3 preserves reflection along e 2 direction (C 2 6 R in main text notation) marked in fig 7(1). For V 3 , similarly we construct such correspondence In the case for quantum spin hall mass,ψσ i ψ and monopole S j + iS k results in spin order that fully breaks SO(3) spin . For quantum spin hall mass on square lattice, we use following spin operators (M 10 , M 20 , M 30 ) ∼ S spin−hall ≡ S 0 × r (−1) r1+r2 [S r × S r+e1+2e2 − S r × S r−2e2+e1 + S r × S r+2e1+e2 − S r × S r+2e1−e2 ).
(E6) and S 0 = 1 N r S r is the uniform component of lattice spin moments. This is odd under translations, reflections, rotations and time-reversal, and consistent with symmetries of quantum spin hall mass.
In the staggered flux phase, the monopoles do not have definite time reversal quantum numbers. The Neel order S neel from M i2 in the main text is related to Re[S i ]. The imaginary part of S i transforms as The real part of S i 's has the symmetry of Neel order S neel , and Hence the ordered state under the quantum spin hall mass is captured by 2 order parameters, S neel , S spin−hall that aligns along the Re[ S ], M 0i direction, respectively, orthogonal to each other. The co-existence of the neel order and products like S i × S j shows the quantum nature of the ordered state -the spins fluctuate in a coherent way while on average they order anti-ferromagnetically.
On honeycomb lattice, we have (M 10 , M 20 , M 30 ) ∼ S spin−hall = r S r × S r+ r e1 + S r × S r+ r e2 + S r × S r− r e1− r e2 (E8) where r = ±1 depending on which sublattice site r belong to and e 1/2 is the unit vector along translation T 1/2 with 2π/3 angle between them. This is rotation invariant with (0, 0) momentum and reflection odd. Similar to square lattice, the imaginary part of S i behave just like the Neel order parameter S neel along i direction while Note the spin order is collinear, since both the mass and S i 's are even under T C 6 . The resulting order is an inherently quantum "Neel" order, with order described by S neel , S spin−hall , identical to the chiral antiferromagnetic phase found in ref [11,12].