Pinch-points to half-moons and up in the stars: the kagome skymap

Pinch point singularities, associated with flat band magnetic excitations, are tell-tale signatures of Coulomb spin liquids. While their properties in the presence of quantum fluctuations have been widely studied, the fate of the complementary non-analytic features -- shaped as half-moons and stars -- arising from adjacent shallow dispersive bands has remained unexplored. Here, we address this question for the spin $S=1/2$ Heisenberg antiferromagnet on the kagome lattice with second and third neighbor couplings, which allows one to tune the classical ground state from flat bands to being governed by shallow dispersive bands for intermediate coupling strengths. Employing the complementary strengths of variational Monte Carlo, pseudo-fermion functional renormalization group, and density-matrix renormalization group, we establish the quantum phase diagram. The U(1) Dirac spin liquid ground state of the nearest-neighbor antiferromagnet remains remarkably robust till intermediate coupling strengths when it transitions into a pinwheel valence bond crystal displaying signatures of half-moons in its structure factor. Our work thus identifies a microscopic setting that realizes one of the proximate orders of the Dirac spin liquid identified in a recent work [Song, Wang, Vishwanath, He, Nat. Commun. 10, 4254 (2019)]. For larger couplings, we obtain a collinear magnetically ordered ground state characterized by star-like patterns.

Classical spin models which admit a completion of squares belong to the distinct genre of "maximally frustrated" Hamiltonians which feature an exponentially large degenerate ground-state manifold [1,2].In two spatial dimensions, a celebrated example is the classical nearest-neighbor Heisenberg antiferromagnet (NNHAF) on the kagome lattice with |S i | = 1 and N the total number of spins.By virtue of the right-hand-side of Eq. (1), any spin configuration which satisfies (S 1 +S 2 +S 3 ) = 0 on each triangle qualifies as a classical ground state.The emergence of such a local constraint leads to the formation of a Coulomb spin liquid [3], with algebraically decaying spin-spin correlations in real space, which gives structure to the exponentially large manifold of degenerate ground states.In Fourier space, these correlations most strikingly manifest themselves in the presence of non-analytic features in the structure factor called pinch points [4,5].Remarkably, this classical Coulomb phase remains stable [6,7] even in the presence of additional couplings along a fine-tuned line when second neighbor (J 2 ) and third neighbor along edges (J 3a ) [see Fig. 1(a)] are concurrently introduced and of equal strength, i.e.J 2 = J 3a (≡ J henceforth).This can be readily understood when diagonalizing the spin exchange Hamiltonian H(k) in momentum space [8][9][10], which reveals that the characteristic flat band of the NNHAF persists [6] up to J/J 1 = 1/5.For J/J 1 > 1/5, a shallow dispersive band starts to cut below the flat band in parts of the Brillouin zone [6], which in turn gives rise to pairs of half-moons, i.e., crescent shaped arcs in the static structure factor [11], with the flat band remaining close-by with a multitude of lowenergy excitations [12].On a deeper level, the formation of half-moons in the static structure factor results from a nonanalyticity in the dispersive-band eigenvectors as a function of momentum and, given the completeness of the eigenvector basis, can be viewed as necessarily arising in order to complement the singularity in the momentum dependence of the flat-band eigenvectors [6,13].With increasing J/J 1 , the radius of the half-moon continuously grows and at J/J 1 = 1, the half-moons from different Brillouin zones coalesce, giving rise to a star pattern in the static structure factor.While in the case of Ising spins, which show a similar sequence of momentum space signatures as a function of J/J 1 , the nature of the half-moons and star phases has a well-understood real-space picture in terms of magnetic clustering of topological charges [14][15][16], for continuous (Heisenberg) spins, the nature of the real-space clustering and its freedom to continuously evolve with J is far more involved and not yet completely understood [6].
Much of the interest in the kagome quantum antiferromagnet as a potential host to highly entangled quantum states owes its origin to the realization that its classical ground state is governed by flat bands -an opportunity for otherwise resid- c p q V + q i E h d E y G 0 L N U E g H a z + a H T v G F V U I c x c q W N H i u / p 3 I i N B 6 I g L b K Y g Z 6 V V v J v 7 n 9 V I T 3 f o Z k 0 l q Q N L F o i j l 2 M R 4 9 j U O m Q J q + M Q S Q h W z t 2 I 6 I o p Q Y 7 N Z 2 s J F H I K S U x u M t x r D O m l f V b 3 r a q 1 Z q 9 T v 8 o i K 6 A y d o 0 v k o R t U R w + o g V q I I k A v 6 B W 9 O c / O u / P h f C 5 a C 0 4 + c 4 q W 4 H z 9 A s W d l Z Q = < / l a t e x i t > 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " Z E K k G F i 1 I c s / V m l u f 0 E 6 w v / 3 T W A = " > A A A B / 3 i c d V D L S s N A F J 3 U V 6 2 v q k s 3 g 0 e u a s Z w 5 A S s w 3 r 8 A 7 S 2 W y g = = < / l a t e x i t > J 2 < l a t e x i t s h a 1 _ b a s e 6 4 = " c f M 4    ual quantum effects to dictate the macroscopic ground state.Thence, tuning the pairwise exchange along the maximally frustrated axis (J 2 = J 3a ≡ J) which, classically, is tuned to have a flat band over an extended region in parameter space, should provide a fertile playground to potentially realize novel states of matter also in the quantum model.For one, the U(1) Dirac spin liquid (DSL) [17][18][19] ground state of the NNHAF [20][21][22][23] is indeed known to be fragile to magnetic order when perturbed by longer-range Heisenberg couplings [24,25] or Dzyaloshinskii-Moriya interactions [26], as expected for algebraic spin liquids, but its fate along the maximally frustrated direction of interest here is unknown.In particular, this parameter axis may afford a higher degree of stability to the U(1) DSL against long-range order, and one may wonder whether the DSL naturally gives way to other exotic quantum phases as one marches along this direction.On a conceptual level, instabilities of the DSL have recently been rigorously classified [19] in field theoretical work.But it remains an open challenge to identify microscopic settings in which these instabilities manifest themselves and what telltale signatures they come along with that might be accessible in experimental studies.
In this manuscript, we take an important step in this direction by establishing the quantum counterpart to the classical half-moon phase as a pinwheel valence bond crystal state which the DSL transitions into only for finite coupling strength.We do so by employing complementary numerical quantum many-body approaches to build a detailed picture of the S = 1/2 quantum phase diagram along the max-imally frustrated axis for J/J 1 > 0, resolving the characteristic real-space and Fourier-space signatures of all quantum phases.The numerical approaches include fermionic variational Monte Carlo (VMC) with versatile Gutzwiller projected Jastrow wave functions [27], many-variable variational Monte Carlo (mVMC) with unconstrained optimization of the Bardeen-Cooper-Schrieffer (BCS) pairing function (supplemented with symmetry projectors) [28,29], one-loop pseudofermion functional renormalization group (pf-FRG) [30], and density-matrix renormalization group (DMRG) [31].The resulting quantum phase diagram is shown in Fig. 1, where cumulative and complementary evidence from all employed approaches shows that the ground state remains nonmagnetic over an appreciably wide span of parameter space [see Fig. 1(b)], notably extending far beyond the classical domain (0 J/J 1 0.2) where flat bands are lowest in energy.This nonmagnetic region is composed of two phases: (i) the U(1) Dirac spin liquid (DSL) for 0 J/J 1 0.26 characterized by soft maxima at the pinch points in its spin structure factor χ(k), and (ii) a 12-site unit cell, C 6 symmetric pinwheel valence bond crystal (VBC) for 0.26 J/J 1 0.51, displaying signatures of half-moons in χ(k), see Fig. 1(b).Our analysis indicates the DSL-VBC transition to be first-order as ascertained on finite systems from a sudden change in the spinspin correlation profile and a crossing of the energies.For J/J 1 0.51, the VBC gives way, via a first-order transition, to collinear long-range magnetic order [32,33] with signatures of a star-like pattern in χ(k).
Results.We set the stage, by observing that across our nu- merical approaches we find that the ground state energy is seen to increase with J/J 1 , reflecting an enhanced degree of frustration at variance with conventional expectation that the NNHAF represents the point of maximal frustration, which is relieved upon inclusion of long-range couplings.The presence of a pronounced kink in the evolution of the ground state energy is indicative of a phase transition [see Fig. 2(a)] which we estimate to be at g 1 = 0.27(1) via an analysis of its derivative (from our DMRG calculations).This value is also corroborated by the behavior of the von Neumann entanglement entropy which starts decreasing sharply at g 1 [see Fig. S12] indicating the formation of a less entangled state.
To probe the nature of the ensuing states, let us start by discussing results from our fermionic VMC approach with versatile Gutzwiller-projected wave functions constructed in a manner enabling us to accurately study the competition between nonmagnetic quantum spin liquid (QSL) and VBC phases, together with magnetically ordered states.Such a unified framework has been met with success in its application to a wide range of frustrated spin models [24,25,[34][35][36].Our calculations are performed on 3 × L × L clusters respecting the full symmetry of the kagome lattice.For the S = 1/2 NNHAF, there is emerging consensus towards a U(1) DSL ground state [20,23,37,38], which is known to yield the lowest variational energy [20,37].Upon including a J coupling, we investigate for the potential instability of the U(1) DSL to symmetric Z 2 [39], chiral U(1) [40], chiral Z 2 [41], and lattice nematic Z 2 [42] QSLs.We also probe for possible dimerization tendencies into VBCs with various unit cell sizes up to 36 sites and different symmetries [17,18,[43][44][45].Our analysis finds a remarkable robustness of the U(1) DSL to the above-mentioned potential instabilities over a wide range along the maximally frustrated axis extending up till J/J 1 = 0.26 (1), which we note is beyond the range of J/J 1 for the classical model where the flat band is the lowest in energy [6].
At J/J 1 = 0.26(1), we detect a dimer instability of the DSL towards a VBC ground state in our VMC calculations.This VBC state is found to be characterized by a 2×2 enlarged unit cell with a C 6 invariant pinwheel structure of spin-spin correlations in real space which breaks reflection symmetries [see Fig. 1(b)].The formation of such a VBC state is further corroborated by an enhanced dimer response (see Fig. S4).Interestingly, such a pattern of strong/weak bonds has previously been identified as descending from confinement transitions of Z 2 spin liquids [46] (left panel of Fig. 1 therein), and recently proposed in Ref. [19] [Fig.3(c) therein] as a potential instability of the U(1) DSL resulting from a condensation of a C 6 invariant mass and the associated monopole terms.Our finding of a C 6 symmetric VBC, as opposed to other less symmetric patterns [Fig.2(c) in Ref. [47]], is likely connected to the fact that the imaginary expectation value of the monopole condensation responsible for this reflection symmetry breaking pattern also optimizes the Landau potential [19].It is worth pointing out that our VBC pattern is distinct from the 2 × 2 enlarged VBC patterns previously proposed in Fig. 4 of Ref. [17] and Fig. 5 of Ref. [18] which do not break reflections (though these pattern also minimize the Landau potential as noted in Ref. [19]).While, the DSL to VBC transition is allowed to be continuous, our microscopic calculations find it to be first-order as inferred from a level-crossing of the energies of the two states [see Fig. 2(a)] together with the observation of an abrupt change in the nearest-neighbor spin-spin correlation profile.We show that the energy gain of the VBC w. r. t. the U(1) DSL is non-zero for J/J 1 > 0.26 (1) and remains so on all finite size systems we simulated, indicating size-consistency of the VBC state and its stability in the thermodynamic limit.
Further support for the pinwheel VBC state comes from mVMC calculations at J/J 1 = 0.4, for which we measure the real-space dimer-dimer correlation pattern (see Fig. S6) where the emergence of the C 6 symmetric pinwheel VBC is also manifest.We also construct a symmetry-breaking dimer operator with non-vanishing susceptibility extrapolated to the thermodynamic limit (see Fig. S7).An analysis of the latter suggests a triply-degenerate C 3 -related order parameter, with the three M -points momenta setting the spatial dependence, which signals a VBC behavior with the spontaneous C 3 -symmetry breaking.However, the equal-weight sum of these three basis functions of the dominant irreducible representations results into an effective C 6 symmetric pinwheel pattern as obtained within VMC [see Fig. 1 (b)], which we illustrate in the inset of Fig. S7.The corresponding susceptibility decreases rapidly as J/J 1 → 0, substantiating a transition to a quantum spin liquid phase from the VBC.
To probe the aforementioned VBC order within DMRG, we start by imposing the pinwheel VBC pattern (via small pinning fields) in a trial wavefunction that is then used as initial state for subsequent DMRG calculations performed with the original unperturbed Hamiltonian deep within the three phases of interest, namely, at J/J 1 = 0.2, J/J 1 = 0.4, and J/J 1 = 0.65.This procedure allows us to probe the stability of the initial pinwheel VBC state for these three phases or, alternatively, see its melting into different quantum states.We see that for J/J 1 = 0.4 [see Fig. 3(c)], the removal of the bias hardly affects the initial state thus providing strong support for the pinwheel VBC as true ground state in this regime.This is further corroborated by the fact that at J/J 1 = 0.2 and 0.65, the VBC pattern is progressively washed out [see Fig. 3(b) and see Fig. 3(d)].Together, these results provide a smoking gun signature for the formation of the pinwheel VBC state in the range J ∈ (g 1 , g 2 ) [see Fig. 1(c)].
In Fourier space, the hallmark of the onset of the VBC order, as obtained within pf-FRG, is the splitting of the pinch points (M-points of the extended Brillouin zone), where the maxima of χ(k) are located for the DSL, into two symmetric half-moons resulting in the maxima of the intensity now being located at generic (0, q y ) (and symmetry related) incommensurate points, as captured in an earlier pf-FRG study of the same model [49].Given that the DSL and VBC phases can From pf-FRG, we show for different values of spin-S [48], the evolution with J/J1 of the radius of the halfmoons characterizing the pinwheel VBC.The large-S (classical) result is from Ref. [6]. also be distinguished by comparing χ(k) along two cuts in momentum space, i.e., Γ − K and Γ − M segments, more precisely, we define a "spectral measure" ∆χ k as the difference between the maxima along these two cuts, i.e., The splitting of the pinch point into half-moons correspond to a downturn in the value of ∆χ while the zero crossing of ∆χ indicates that the half-moons become the dominant feature in χ(k).Based on these two signatures, we estimate the onset of VBC from pf-FRG at J/J 1 = 0.30 (2) [see Fig. 2(b) and Fig. S1], in good agreement with the other employed approaches.The evolution of the radius of the half-moon as a function of J/J 1 obtained from pf-FRG is shown in Fig. 4, where for S = 1/2 one observes an appreciable deviation from the reported large-S result [6].For progressively increasing values of S, the known large-S behavior [6] is approached.Within the VMC calculation, the splitting of the pinch point maxima into halfmoons is observed deep inside the VBC phase as shown in Fig. S8.Similarly, deep inside the VBC phase, the χ(k) obtained from mVMC shows maxima at incommensurate (0, k y ) points as shown in Fig. S5.
Finally, let us turn to the transition into the star phase.To this end, we show, in Fig. 5, the evolution of the square of the sublattice magnetization m 2 with J/J 1 , as obtained from mVMC, VMC, and DMRG.One observes a sudden change to a finite value of m 2 for J/J 1 > 0.51 (1), indicating the onset of long-range collinear spin order with a 12-site magnetic unit cell (see inset of Fig. 5) [32].While the estimate of the phase boundary from these three approaches shows remarkable agreement, the comparatively smaller values of m 2 inside the ordered phase obtained in DMRG can be ascribed to the quasi one-dimensional character of the cylindrical geometries.The abrupt nature of the jump in the value of m 2 observed in mVMC and VMC, together with the crossing of the energies of the disordered VBC and magnetically ordered states across the transition point (see inset of Fig. 5), lends evidence in favor of a first-order character of the transition.The results from VMC and mVMC are for a 3 × 8 × 8 site cluster [see Fig. S10 for finite-size scaling results of m 2 from VMC to the thermodynamic limit], while those from DMRG are obtained on a YC8-8 cylinder.
Similar conclusions are drawn from VMC via finite-size scaling of m 2 for different values of J/J 1 (see Fig. S10), wherein one observes a jump in the value of m 2 in the thermodynamic limit.The collinear magnetically ordered state displays a starlike pattern of intensity distribution in χ(k) [see Fig. 1(b)] with maxima at the location expected for the octahedral regular magnetic order [33].It is worth noting that for S = 1/2 the phase boundary between the half-moon and star phases considerably shifts to a smaller value of J/J 1 = 0.51(1), compared to the classical boundary at J/J 1 = 1, highlighting significant effects of quantum fluctuations.Discussion.Moving the ground state of the kagome antiferromagnet along the maximally frustrated line is a complicated endeavor -as such it is quite fulfilling to see the remarkable agreement between our complementary numerical approaches yielding a consistent understanding of momentum and real space signatures of the ground state phases and their respective boundaries; a feat that would not have been imaginable only a few years ago.One might hope that the U(1) DSL, half-moon, and star phases will have a window of stability away from the maximally frustrated axis.It would thus be of interest to search and identify materials promising to realize the Dirac spin liquid phase and which lie within this region of stability.The recently studied material YCu 3 (OH) 6 Br 2 [Br x (OH) 1 − x ] [50] wherein signatures of DSL behavior has been presented, could serve as a potential material candidate warranting further investigation.Another interesting candidate material might be the distorted kagome compound Rb 2 Cu 3 SnF 12 [51] where indications for a pinwheel VBC have been reported.One may be able to approach the maximally frustrated line by effectively varying the superexchange couplings by application of hydrostatic or uniaxial pressure to vary the super-exchange bond angles [52].On the theoretical front, given the persistent and enhanced frustration upon inclusion of J, it would be interesting to ascertain the extent of the nonmagnetic phase of the spin S = 1 NNHAF, and decipher the corresponding real-space nature of the half-moon phase.Finally, it would be worth exploring the corresponding quantum phase diagram on the pyrochlore lattice, which similarly at the classical level is host to persistent flat bands, as well as half-moon and star phases [6,15].
During completion of this manuscript, we were made aware of a paper by Lugan et al. studying the same model with a complementary bosonic method.the strength of dimerized bonds while weakening the others, i.e.J 1 → J 1 ± δ with δ = 0.01J 1 [S59-S61].Defining the equal time, real-space spin-spin correlation along such a strengthened dimer bond as S z i S z j + and a completely unperturbed (i.e.δ = 0) reference value S z i S z j 0 , we may define the dimer response as Note that this definition requires the evaluation of two separate FRG runs to compute S z i S z j 0 and S z i S z j + .From pf-FRG, equal-time correlators can be computed as S z i S z j ≡ S z i S z j (t = 0) = dνχ ij (ν).Fig. S4 shows the response obtained for the pinwheel VBC pattern displayed in Fig. 1, where thick red bonds are strengthened and thin bonds are weakened by δ: In the QSL phase, we observe a relatively small value of the dimer response which rises steadily towards the VBC phase up until a distinct maximum at J 2 ≈ 0.45 after which it decreases once more.This is in good agreement with the phase diagram presented in Fig. 1 of the main text.

II. MANY-VARIABLE WAVE FUNCTION (MVMC)
The many-variable variational Monte Carlo (mVMC) method can be successfully used in studies of strongly correlated spin and electronic systems [S62, S63].In particular, the method can be applied to distinguish between quantum spin liquid and valence bond solid phases, such as in the case of the Inspired by the Anderson resonating valence-bond wave function, the mVMC ansatz has the form where single occupation is ensured by the Gutzwiller projector which maps the fermionic Hilbert space to the original Hilbert space of spin operators.The wave-function value σ|φ pair of a specific spin configuration |σ is evaluated using the Slater determinant of the matrix with elements F i, j .Here, σ represents a string of ±1, which, for each lattice site, stands for the respective spin eigenstate in the S z basis.The parameters F i, j are optimised using the stochastic reconfiguration technique [S66], which can be seen as a way of performing imaginary-time evolution in the variational parameters manifold [S27, S67].improve the accuracy of the variational wave functions, we employ quantum-number projections.The point-group symmetry Ĝ is enforced by applying its generators until the symmetry orbit is exhausted where ξ is the desired projection quantum number and |Ψ ξ resulting symmetrized state.The projection onto the total spin S is performed by superposing the SU (2)-rotated wave functions [S29].In this work, for systems with more than 36 sites, we partially impose translational symmetry directly on the variational parameters F i, j .Namely, we introduce translational symmetry modulo 2 × 2 unit cells sublattice structure and enforce the 2 × 2 translations and the point-group symmetries using Eq.(S7).The resulting procedure amounts into 2×2×3 2 ×L 2 variational parameters with L being the number of unit cells in each lattice direction.Such partial translational symmetry imposition is a reasonable compromise between the ability to express complicated wave function and the required time to optimize the wave function.
Magnetic properties of variational wave functions can be assessed by computing the structure factor χ(k) as the equaltime momentum-resolved spin-spin correlation function where r i indicates the position of the lattice site i including sublattice displacement.In Fig. S5, we present the spin structure factor at J/J 1 = 0.4 as computed by mVMC.In a non-magnetic phase, the properties of the wave function are assessed by measuring the dimer-dimer correlation function χ D b, b = Db Db − Db Db for all pairs of bonds in the system, 0 b, b < N bonds , where Db = Ŝi • Ŝj , with i, j being sites at ends of the bond b.In Fig. S6, we show the dimer-dimer correlations between the base bond (located in a distant unit cell) and other bonds.To carry out a quantitative assessment of the VBC character of the ground state, we need to define suitable scalar order parameters to perform an infinite-volume extrapolation of the dimer order.Thus, we regard χ Following this procedure, we obtain leading eigenvalues and eigenstates of the χ D b, b matrix at J/J 1 = 0.2, 0.3, 0.4 on finite-size lattices with L = 4, 6, 8 and 10.In Fig. S7, we show an equal-weight superposition of the three degenerate leading eigenstates at the M -points.Other eigenvalues are an order of magnitude smaller, and are thus not shown.We extrapolate the corresponding susceptibility to the thermodynamic limit and obtain, for J/J 1 = 0.4, nonvanishing susceptibility extrapolation of 7.6(3) × 10 −3 .This signals presence of symmetry breaking through establishment of a dimer order at J/J 1 = 0.4.Similar extrapolations at J/J 1 = 0.2, 0.25, 0.3 and 0.35 yield 3.8(2) × 10 −3 , 4.1(2)×10 −3 , 5.1(3)×10 −3 and 6.9(2)×10 −3 , respectively.By fitting the susceptibility dependence on J/J 1 with a hyperbolic tangent ansatz, we estimate the inflection point to be at J/J 1 = 0.32 (3).This provides an estimate of the transition point from the QSL to the VBC phase.We emphasize that the dimer-dimer susceptibility within mVMC remains finite in the QSL phase.This is related to the fact that the mVMC   wave function cannot efficiently express the ground state of the U (1) DSL phase, unlike the dimerized VBC case.This is confirmed by the comparison of the mVMC variational energy to the one of the DMRG approach in the two phases.

III. VARIATIONAL MONTE CARLO (VMC)
The variational Monte Carlo (VMC) [S27] approach employed in this work shares several common aspects with the mVMC method introduced in the previous section.Both techniques rely on the Abrikosov fermion representation of spin operators, introduced in Eq. (S1).Within this fermionic formulation, suitable variational states for the Heisenberg model are obtained by projecting a fermionic wave function to the spin Hilbert space.The projection, which enforces the single fermionic occupation of each lattice site, is achieved by means of a Gutzwiller-projector P ∞ G [see Eq. (S6)], and can be performed exactly by an appropriate Monte Carlo sampling [S27].The variational Ansätze of VMC, discussed in this section, differ from those of the mVMC approach in the choice of the fermionic states to be projected.
Within the VMC approach, the variational state is obtained by projecting a Slater determinant, |Φ 0 , which is the ground state of an auxiliary quadratic Hamiltonian The parameters t ij (hoppings) and h µ i (fictitious magnetic field) of H 0 are optimized in order to minimize the variational energy of the projected state [S69].The complete expression for the variational wave function is where, in addition to the projected Slater determinant, we have included the long-range spin-spin Jastrow factor [S27] The pseudopotential parameters v i,j are assumed to be translationally invariant, and numerically optimized along with the fermionic parameters t ij and h µ i .The optimization of the variational wave function is achieved through the stochastic reconfiguration method [S27, S66, S67]

A. Spin liquid to pinwheel VBC transition in VMC
For small values of the ratio J/J 1 , the optimal variational wave function for the model is the U(1) DSL [S20, S37].Upon increasing J/J 1 , the system undergoes a phase transition to the pinwheel VBC at J/J 1 = 0.26(1) (see Fig. 2 of the main text).The variational Ansatz for the VBC is obtained by considering a 2 × 2 enlarged unit cell of 12 sites, which can accommodate the pinwheel pattern depicted in Fig. 1 (b) of the main text.The variational parameters of the VBC Ansatz are the inequivalent hoppings within the enlarged unit cell, from first-to third-neighbors (the latter ones being limited to the J 3a -bonds).The number of independent hopping parameters is reduced from 72 to 12 by applying the C 6 rotational symmetry of the kagome lattice.Finally, an underlying sign structure for the hoppings is imposed, to reproduce the flux pattern of the U(1) DSL (similarly to the approach used in Ref. [S43]).For this reason, the VBC wave function can be regarded as an instability of the DSL state.We find that the energy of the VBC state is lower than the one of the DSL for J/J 1 > 0.26, signalling the transition to the pinwheel VBC phase.Deep inside the VBC phase, the static structure factor χ(k) displays signatures of half-moons, as shown in Fig. S8.

B. Pinwheel VBC to collinear magnetic order transition in VMC
At J/J 1 = 0.51(1), a phase transition from the pinwheel VBC to the magnetic phase with collinear order is observed.The auxiliary Hamiltonian H 0 for the magnetic state features FIG.S10.Finite-size scalings of the square of the sublattice magnetization m 2 the collinear spin order showing its behavior at the VBC-magnetic order transition.We employed 3 × L × L clusters with L = 6, 8, 10, 12.The values of J/J1 are reported in the box in lower-left corner.The inset shows the finite-size scaling of m 2 within the VBC regime, as a function of 1/L 2 .For J/J1 > 0.51 (i.e., within the collinear magnetic phase), m 2 extrapolates to a nonzero value for L → ∞. same hopping parametrization of the VBC Ansatz, with the addition of a fictitious magnetic field h µ i which reprothe collinear order sketched in the inset of Fig. 5.The fictitious collinear field h µ i is taken along the S x direction.the presence of the Jastrow factor, which is a function of S z operators, introduces transverse spin fluctuations on of the ordered fermionic state.Although the variational parametrization allows for a continuous transition between VBC and the collinear ordered states, the transition turns to be of the first order.Indeed, we detect the presence of two energy minima when optimizing the variational energy, an absolute minimum and a metastable state with higher energy.One of the minima corresponds to the VBC state, i.e., it is characterized by a vanishingly small magnetic field h µ i the thermodynamic limit and a dimer pattern like the one of Fig. 1 (b); the other minumum, instead, corresponds the magnetically ordered phase.The relative positions of the two minima swap at J/J 1 = 0.51 (1), and magnetic sets in for larger values of J/J 1 .In the collinear phase, the static structure factor shows the presence of Bragg peaks at the ordering vectors, as shown in Fig. S9 for J/J 1 = 0.6.The first-order nature of the VBC-collinear transition is confirmed by the sudden jump of the sublattice magnetization, shown in Fig. S5 of the main text for a 3 × 8 × 8 lattice.A finite-size scaling analysis of m 2 the presence of an abrupt change at the phase boundary also in the thermodynamic limit (see Fig.In our DMRG calculations, we find g 1 is occurring at J/J 1 = 0.27(1) which is signalled in (i) a discontinuity in the derivative of ground state energy with respect to J as plotted in Fig. 2 < l a t e x i t s h a 1 _ b a s e 6 4 = " B d H i 2 k a M u a h a q w c a x 6 J o H G Z d K n 4 = " > A A A B / n i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e 5 K f B y D X j x G N A 9 I l j A 7 O 5 sM m Z 1 Z Z m a F E A L + g F f 9 A 2 / i 1 V / x B / w O J 8 k e T G J B Q 1 H V T X d X k H C m j e t + O 7 m V 1 b X 1 j f x m Y W t 7 Z 3 e v u H / Q 0 D J V h N a J 5 F K 1 A q w p Z 4 L W D T O c t h J F c R x w 2 g w G t x O / + U S V Z l I 8 m m F C / R j 3 B I s Y w c Z K D 9 7 Z R b d Y c s v u F G i Z e B k p Q Y Z a t / j T C S V J Y y o M 4 V j r t u c m x h 9 h Z R j h d F z o p J o m m A x w j 7 Y t F T i m 2 h 9 N T x 2 j E 6 u E K J L K l j B o q v 6 d G O F Y6 2 E c 2 M 4 Y m 7 5 e 9 C b i f 1 4 7 N d G 1 P 2 I i S Q 0 V Z L Y o S j k y E k 3 + R i F T l B g + t A Q T x e y t i P S x w s T Y d O a 2 8 F i G V I m x D c Z b j G G Z N M 7 L 3 m W 5 c l 8 p V W + y i P J w B M d w C h 5 c Q R X u o A Z 1 I N C D F 3 i F N + f Z e X c + n M 9 Z a 8 7 J Z g 5 h D s 7 X L 6 + P l g w = < / l a t e x i t > 1/5 < l a t e x i t s h a 1 _ b a s e 6 4 = " o N H M y Z u V w R g R 6 I K G M U d B z S 7 J k M s = " > A A A B / H i c b V D L S g N B E J y N r x h f U Y 9 e B o P g K e x K U I 9 B L x 4 T M A 9 I l j A 7 2 5 s M m Z l d Z m a F s M Q f 8 K p / 4 E 2 8 + i / + g N / h J N m D S S x o K K q 6 6 e 4 K E s 6 0 c d 1 v p 7 C x u b W 9 U 9 w t 7 e 0 f H B 6 V j 0 / a O k 4 V h R a N e a y 6 A d H A m Y S W Y Y Z D N 1 F A R M C h E 4 z v Z 3 7 n C Z R m s X w 0 k w R 8 Q Y a S R Y w S Y 6 W m N y h X 3 K o 7 B 1 4 n X k 4 q K E d j U P 7 p h z F N B U h D O d G 6 5 7 m J 8 T O i D K M e c 6 m O / v 4 f + k W z b t m l m 9 r Z Z a V 6 u I 8 u A M n I N L Y I M 6 a I E b 0 A Y d g M E E P I I n 8 G w 8 G C / G q / G 2 b M 0 Z q 5 l T s A b j / Q s b y 5 b n < / l a t e x i t > g 2 < l a t e x i t s h a 1 _ b a s e 6 4 = " z z 8 O I U g i n 0 e 6 B K p h t k f Z p d / s l t 8 5 I 0 6 e k e u t + I c c g p v I B b v Y E 7 c e s J v I D n s P N Z m M S C B 0 X V e 9 S j w l h w g 7 7 / 7 S w t r 6 y u r W c 2 s p t b 2 z u 7 7 t 5 + x a h E M y g z J Z S u h d S A 4 B L K y 9 l 3 o p P 5 L C o O t + O y u r a + s b m 6 m t 9 P b O 7 t 5 + 5 u C w a s J Y c 6 j w U I a 6 7 j M D X s m b 8 + y 8 O x / O 5 3 w 1 4 y x u T s g S n K 9 f L Y S d e A = = < / l a t e x i t > octahedral < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 Z P Y i 4 n s / 3 I s K G r q e 0 s a l Z 0 g 3 9 s = " > A A A B / 3 i c d V D L S s N A F J 3 4 r P V V d e l m s A i u Q t K m b b o r u h F X F e w D 2 l A m k 2 k 7 d D I J M x O h h C 7 8 A b f 6 B + 7 E r Z / i D / g d T t o K V v T A h c M 5 9 3 L v P X 7 M q F S W 9 W G s r W 9 s b m 3 n d v K 7 e / s H h 4 W j 4 7 a M E o F J C 0 c s E l 0 f S c I o J y 1 F F S P d W B A U + o x 0 /

J 1 <
l a t e x i t s h a 1 _ b a s e 6 4 = " B U 5 D K 8 9 C u w x v W W n G 8 d C T y r r r S w I = " > A A A B / 3 i c d V D L S s N A F J 3 4 r P V V d e l m s A l e k Q e 5 I k 7 Q I I z F 5 I a / k z X g 2 3 o 0 P 4 3 O + W j D y m 2 O y A O P r F 3 a M m m Q = < / l a t e x i t > (c) < l a t e x i t s h a 1 _ b a s e 6 4 = " C M a 2 M C 5 D o s T I B u P N x 7 B I 6 q W i d 1 Y 8 v S 8 V y l d p R F l w A A 7 B M f D A O S i D W 1 A F N Y D B I 3 g B r + D N e X b e n Q / n c 9 q a c d K Z f T A D 5 + s X x k O X M A = = < / l a t e x i t > NN-AFM < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 j y x D e E 8 4 7 Q v / 2 d H O z c V A I T L Z f k = " > A A A B 7 n i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e 5 K U I 9 B L 5 J T B P O A Z A m z k 9 l k y O z s M t M r h J C P 8 O J B E a 9 + j z f / x k m y B 0 0 s a C i q u u n u C h I p D L r u t 5 N b W 9 / Y 3 M p v F 3 Z 2 9 / Y 0 L 8 v e V b n y U C l V b 7 M 4 8 n A C p 3 A O H l x D F e 6 h D g 1 g M I J n e I U 3 J 3 F e n H f n Y 9 G a c 7 K Z Y / g D 5 / M H e C G O X A = = < / l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " / 6 / L 6 p L e d A Y e 9 3 a c v g m D H d z V 5 Q Q FIG. 1.The kagome skymap.(a) Illustration of first-(J1), second-(J2), and third-neighbor interactions along edges (J3a) of the kagome lattice for the considered model.(b) The S = 1/2 quantum phase diagram with (top panel) representative real-space spin-spin correlation profiles, with red (blue) bonds denoting antiferromagnetic (ferromagnetic) correlations, and (lower panel) spin structure factors of the different phases evaluated at J/J1 = 0.1 (DSL), J/J1 = 0.4 (pinwheel VBC), and J/J1 = 0.9 (collinear order) from pf-FRG.(c) estimates of the phase boundaries (g1 and g2) obtained from the various approaches employed in this work.While we see agreement, within error bars, for g1 for all approaches, the pf-FRG result for g2 (marked by an asterisk), shows a notable deviation whose origin we discuss in Appendix I B.

FIG. 2 .
FIG. 2. Transition into half-moon phase.(a) From VMC, the evolution with J/J1 of the energy per site of the DSL and VBC states (3 × 12 × 12 lattice).DMRG energies are also shown for comparison.(b) From pf-FRG, the variation of the spectral measure ∆χ k (see text below) with J/J1 evaluated at the lowest simulated RG cutoff Λ/Z = 0.01 where Z = J 2 1 + 2J 2 .

FIG. 4 .
FIG. 4.Half-moon radii.From pf-FRG, we show for different values of spin-S[48], the evolution with J/J1 of the radius of the halfmoons characterizing the pinwheel VBC.The large-S (classical) result is from Ref.[6].

FIG. 5 .
FIG.5.Transition into the star phase.The behavior of the square of the sublattice magnetization m 2 with J/J1 near the transition from the pinwheel VBC into collinear magnetic order [illustrated in the inset, with blue and red spins pointing in opposite directions].The results from VMC and mVMC are for a 3 × 8 × 8 site cluster [see Fig.S10for finite-size scaling results of m 2 from VMC to the thermodynamic limit], while those from DMRG are obtained on a YC8-8 cylinder.
FIG.S4.pf-FRG dimer response for the pinwheel VBC pattern in Fig.1of the main text as a function of J/J1.The dimer response is calculated via Eq.(S4).The phase boundaries shown as grey dotted lines are obtained as described in the sections above.

J 1 -J 2
Heisenberg model on the square lattice [S64, S65].In this work, we employ the mVMC implementation from Ref. [S28, S29].The construction of the variational states relies on the Abrikosov fermion representation of spin degrees of freedom, as given in Eq. (S1).
FIG. S5.Static (equal-time) spin structure factor at J/J1 = 0.4 as computed by mVMC.The color plot shows the isotropic structure factor χ(k) [Eq.(S8)] in the kx − ky plane.The results have been obtained on a 3 × 8 × 8 finite cluster.The white hexagon with solid (dashed) lines delimits the first (extended) Brillouin zone.The green box highlights the "half-moon" feature of the correlation pattern.
D b, b as a matrix in the bond indices and we diagonalize it; the resulting set of eigenvalues/eigenvectors pairs (λ, A λ b ) is used to define the operators Ôλ = b A λ b Db , each of them corresponding to a certain momentum and irreducible representation of the lattice point group.The tendency to establish a finite expectation value of one of these operators, and thus spontaneously break the corresponding lattice symmetry, is measured by the susceptibility χ Ôλ = Ô † λ Ôλ − Ô † λ Ôλ = λ extrapolated to the thermodynamic limit [S68].
FIG. S7.Extrapolation of the maximum eigenvalue of the χ D ij dimer-dimer correlation matrix at J/J1 = 0.4 as the function of inverse lattice volume.The leading eigenvalue is triply-degenerate, while the other eigenvalues are an order of magnitude smaller.The inset shows an equal-weight superposition of the three basis functions of the dominant irreducible representation, all three connected by 2π/3-rotations.
FIG. S8.Static (equal-time) spin factor of the VBC phase as computed by VMC [Eq.(S8)].Results for J/J1 = 0.5 are shown (on a 3 × 24 × 24 lattice).In (a) we plot χ(k) as a function of |k|, along the kx = 0 cut in momentum space shown in Fig. S1 (orange vertical line).The inset highlights the dip in the profile of χ(k) (with error bars) at the pinch-point positions, which signals the appearance of half-moons.The color plots in (b) and (c) show the value of χ(k) in the kx − ky plane.The results of (b) are obtained with the bare VBC wave function, which breaks the reflection symmetry of the kagome lattice.In (c) we show symmetrized results for the structure factor, where the reflection symmetry is imposed a posteriori.The white hexagon with solid (dashed) lines delimits the first (extended) Brillouin zone.
FIG. S9.Static (equal-time) spin structure factor of the collinear ordered phase at J/J1 = 0.6 as computed by VMC.The color plot shows the isotropic structure factor χ(k) in the kx − ky plane.The results have been obtained on a 3 × 12 × 12 finite cluster.The hexagon with solid (dashed) lines delimits the first (extended) Brillouin zone.

A
. Spin liquid to pinwheel VBC transition in DMRG FIG. S12.DMRG results for the spin liquid to pinwheel VBC transition.Top: The ground state energy as a function of J/J1 and its derivative (middle panel).Bottom: The von Neumann entanglement entropy SN calculated across the central bond using the matrix product ground state obtained via DMRG.The kink in the ground state energy, leading to a step-function behavior in its derivative, and the sharply kinked, non-monotonous behavior of SN are all indicative of a first-order phase transition.Its location is estimated at J/J1 ≈ 0.27(1) indicated by the dashed line, consistent with results from VMC calculations, see Fig.2(a).
FIG. S6.Real-space correlations pattern Da Db − Da Db measured within mVMC on the 3 × 8 × 8 kagome for J/J1 0.4.Here Da is the dimer operator placed on the "base" bond (in a distant unit cell) and Db is the dimer operator on other bonds.Red (blue) color in the figure represents positive (negative) values of the correlator, while its absolute magnitude (multiplied by 100) is marked near each bond.The correlations were measured on a non-symmetrized mVMC wave function for which the pinwheel dimer pattern is more pronounced.