Breathing kagome XY quantum magnet with four-site ring exchange
TThe breathing kagome XY quantum magnet with four-site ring exchange
Niklas Casper ∗ and Wolfram Brenig † Institute for Theoretical Physics, Technical University Braunschweig, D-38106 Braunschweig, Germany (Dated: October 2, 2020)We study the impact of trimerization (breathing) of the nearest-neighbor (NN) exchange on theplanar XY spin-1 / Z quantum spin liquid by virtue of the ring exchange. Using quantum Monte-Carlo calculations, based on the stochastic series expansion, we present results for the spin stiffness,the quantum phase diagram, the longitudinal static, as well as the transverse dynamic structurefactor. Our results corroborate 3D XY universality also at finite trimerization and suggest a simplecontinuation of the quantum critical point of the uniform case into a line in terms of rescaled exchangeparameters. Moreover, at any trimerization and in the ordered phase the elementary excitationscan be understood very well in terms of linear spin wave theory, while beyond the critical line, inthe spin liquid phase we find signatures of spinon continua. I. INTRODUCTION
Quantum spin liquids (QSL) constitute intriguingforms of magnetic matter, which are attracting great in-terest since several decades by now [1–6]. QSLs are char-acterized by the absence of local magnetic order param-eters, even at zero temperature, they show fractionalizedexcitations, long-ranged topological entanglement, andquantum orders beyond Landau’s paradigm of symme-tries and spontaneous symmetry breaking. Proposals forputative QSLs mostly rest on an underlying gauge struc-ture [7], e.g. U (1) or Z , likely related to the non-localityof the fractional excitations [5].Frustration of spin interactions is a prime ingredientto drive magnetic systems into QSL states. This rendersquantum Monte-Carlo (QMC) calculations widely inap-plicable as an unbiased tool, because of the minus-signproblem [8]. Among the few exceptions, in which a Z QSLs can exactly be shown to exists in D ≥ S z constraints onlocal units and feature ring exchange K ∼ J XY /J z . Thelatter can lead to dynamics dual to quantum dimer mod-els [12, 13] and realizes a similar Z gauge structure [14–20].Consequently the BFG QSLs are topologically orderedwith four-fold ground state degeneracy on the torus in 2Dand represent a deconfined phase [21–23]. The decon-fined non-local elementary excitations are analogous to ∗ [email protected] † [email protected] those of the toric code, i.e. ’electric’ e -particles (spinons,neutral spin-1 / m -particles(vortices, visons) [5, 13, 23, 24]. The e - and m -particlesare relative semions [5, 23], and in the BFG models thespinons are known to have bosonic statistics [11]. Asfor additional fingerprints of a Z QSL, the BFG phasehosts symmetry-protected edge states for open boundaryconditions [25] and displays topological entanglement en-tropy [19, 20].An explicit low-energy Hamiltonian to study the BFGQSL on the kagome lattice is the XY-model with ringexchange on four-site bow-ties [18, 26], as in Fig. 1 H = − (cid:88) (cid:104) ij (cid:105) J ij (cid:0) S + i S − j + h.c. (cid:1) − K (cid:88) (cid:104) ijkl (cid:105)∈ (cid:46)(cid:47) P ijkl , (1)where S ± i are spin-1/2 raising and lowering operators onsites i and J ij ≥ K ≥ P ijkl = S + i S − j S + k S − l + S − i S + j S − k S + l acts on eachbow-tie ( (cid:46)(cid:47) ) as illustrated in Fig. 1(a). This model isamenable to QMC analysis, as it lacks a minus-sign prob-lem. Variants of it, introducing additional interactions,including also the S z components have been considered,focusing either on the spin, or its equivalent hard-coreboson formulation [15–17, 19, 20, 27].With nearest-neighbor exchange J ij = J , Hamilto-nian (1) has been shown to harbor two quantum phasesversus J/K [18]. For K → / J → K/J ) c ≈ . /
2. Approachingthe critical point out of the QSL, the spinons condenseinto the superfluid density (cid:104) b (cid:105) of the hard-core bosons,undergoing a conventional XY transition. The latter fea-tures standard values for the exponent ν (cid:39) . z = 1 for thedynamical critical exponent, consistent with 3D XY uni-versality. However, due to the composite nature of the a r X i v : . [ c ond - m a t . s t r- e l ] S e p hard-core bosons in terms of the spinons, the exponent ofthe equal-time boson-correlation function is a fingerprintof the deconfined quantum critical nature of the transi-tion [28]. In fact, η (cid:63) ≈ . − .
493 has been established[20, 28–33], which is different from standard 3D XY uni-versality, i.e. η ≈ .
038 [34], and therefore is referred toas XY (cid:63) .A natural extension of (1) is to include inhomogene-ity of the NN exchange in terms of trimerization , alsoknown as breathing , in which the spins belonging to up-ward(downward) facing triangles experience different ex-change couplings, J (cid:77) ( (cid:79) ) . Such generalization is moti-vated both by theory and experiment. With respect tothe former, recent analysis of other breathing kagomespin systems, i.e. antiferromagnetic XYZ models indi-cate that trimerization may help to stabilize quantumdisordered phases and QSLs [35–41]. However for BFG-type models such analysis is lacking. With respect toexperiment, several interacting hard-core boson systemson breathing kagome lattices have been realized in recentultracold-atom systems [42–46].In this context, the main purpose of this work is touncover the quantum magnetism of the XY model withring exchange on the breathing kagome lattice versus thetrimerization ratio. Our prime focus will be on the spinstiffness and the static spin structure factor (SSSF) inorder to analyze the shift of the quantum critical point(QCP) versus J (cid:77) /J (cid:79) . In addition we will consider thedynamical spin structure factor (DSSF) in order to shedlight on the elementary excitations.The outline of the paper is as follows. In Section II A,we list several details of the model. Section II B sketchesthe QMC method. In Section II C, the extraction fromQMC of observables relevant to our study is described.Section III is devoted to our results, comprising the spinstiffness in Section III A, the quantum phase diagram inSection III B, the static and dynamic structure factors inSections III C and III D. We conclude in Section IV. Weprovide Appendix A on the single trimer stiffness andAppendix B on the linear spin wave theory (LSWT) forthe breathing XY kagome ferromagnet. II. MODEL AND METHODA. Model
The breathing version of model (1) reads H = −
12 ( (cid:88) (cid:77) (cid:104) ij (cid:105) J (cid:77) S + i S − j + (cid:88) (cid:79) (cid:104) ij (cid:105) J (cid:79) S + i S − j + h.c. ) − K (cid:88) (cid:104) ijkl (cid:105)∈ (cid:46)(cid:47) P ijkl , (2)where J (cid:77) ( (cid:79) ) refer to the solid(dashed) up(down) tri-angles on the kagome lattice, depicted in Fig. 1(a),which is a triangular Bravais lattice with lattice vectors K R R il jkJ M J O − π − π π π − π − π π π G G Γ K K MM (a) (b) FIG. 1. (a) Trimerized kagome lattice, with NN and ringexchange J (cid:77) ( (cid:79) ) and K . (b) First and second Brillouin zonewith high-symmetry path. R , = ( , √ ) , (1 ,
0) for a lattice constant l =1 hereafter,and a three-site basis at r α =0 , , = (0 , , ( , √ ) , ( , a = 1 / N = 3 L . The reciprocal lattice vectors are G , = (0 , π √ ) , (2 π, − π √ ), with G i · R j = 2 πδ ij and theBrillouin zone (BZ) is set by q = n L G + n L G with n , = 0 , , . . . , L −
1, see Fig. 1(b).For the remainder of the text we will employ a numberof dimensionless parameters, i.e. j = J (cid:79) /J (cid:77) , k = K/J (cid:77) and κ = K/ ¯ J . There we define a mean exchange cou-pling ¯ J = ( J (cid:77) + J (cid:79) ) / = ¯ J/J (cid:77) = (1 + j ) /
2. In this work, we will consider theregion of 0 ≤ j ≤ k ≥ B. Quantum Monte-Carlo method
The numerical results of this work are obtained fromQMC calculations, using the stochastic series expansion(SSE) [47–49]. This method is based on an importancesampling of the high temperature series expansion of thepartition function Z = (cid:88) α (cid:88) S M ( − β ) n ( M − n )! M ! (cid:104) α | M (cid:89) a =1 H t a ,p a | α (cid:105) , (3)where β =1 /T is the inverse temperature and M thetruncation order, self-adjusted to the desired pre-cision. As compared to conventional implementa-tions, we enhance the approach by including thering exchange following Ref. [50]. I.e. the en-tries H t a ,p a of the operator string (cid:81) Ma =1 H t a ,p a nowcomprise C -, J -, and K -type operators at plaque-tte p , H C,p = CI ijkl , H J,p = ( S + i S − j + S − i S + j ) I kl /
2, and H K,p = S + i S − j S + k S − l + S − i S + j S − k S + l , including appropri-ate permutations of ijkl . C has to be chosen suchthat all weights of the C -type operators are nonneg-ative. | α (cid:105) = | S z , . . . , S zN (cid:105) refers to the S z basis and S M = [ t , p ][ t , p ] . . . [ t M , p M ] is an index for the oper-ator string.This operator string is sampled, using a Markovian-chain Metropolis scheme, employing three types of up-dates, i.e. (i) diagonal updates which change the numberof C -type operators H C,p in the operator string, (ii) loopupdates which change the type of operators H C,p ↔ H J,p and H J,p ↔ H K,p , and (iii) multibranch cluster updatewhich change the type of operators H C,p ↔ H K,p . Thelatter update refers to the prime new ingredient for ringexchange models [50]. For bipartite lattices the loop up-date comprises an even number of off-diagonal operators( J - and K -type). This ensures positivity of the transitionprobabilities and waives the minus-sign problem. C. Observables
Here we briefly sketch formal details regarding themain physical observables which we evaluate.
1. Spin Stiffness
To obtain the superfluid density, we calculate the spinstiffness (or helicity modulus) ρ S ρ S = ∂ F ( φ ) ∂φ (cid:12)(cid:12)(cid:12)(cid:12) φ =0 , (4)which indicates the presence of long-range order. Here F ( φ ) is the free energy versus a twist of the spins inthe XY-plane, with an angle increasing by φ = Φ /L perbond for any given bond direction. At T = 0, the freeenergy is replaced by the ground state energy E ( φ ). Weimplement the estimator for this quantity following Refs.[51–54], using that the r.h.s. of (4) can be expressed interm of squares of operators S + i S − j , transporting ↑ -spinsalong the ij -bond. I.e. ρ S = 1 dβ d (cid:88) α (cid:104) W α (cid:105) , (5)where d = 2 refers to the dimension and the windingnumber W α is defined by W α = 1 L α (cid:88) b N b,α , (6)where α = x, y is the spatial direction and L α the numberof bonds per spatial direction. The sign of the phase fac-tor is N b,α = ± − spatial direction, which corre-sponds to the operators S + S − or S − S + in the operatorstring.
2. Spin Structure Factors
We evaluate two types of spin correlation functions.First, the longitudinal, i.e zz , static spin structure factor(SSSF) S ( q ) = 1 N (cid:88) i,j e i q · ( r i − r j ) (cid:104) S zi S zj (cid:105) , (7) which is extracted during the simulation by calculatingthe product of the S z -component between all sites andperforming a Fourier transformation.Additionally we evaluate the transverse, i.e. + − , dy-namic spin structure factor (DSSF). In real space and atimaginary time τ this is obtained from the SSE by [47] (cid:10) S + i ( τ ) S − j (0) (cid:11) = (cid:42) M (cid:88) m =0 (cid:18) Mm (cid:19) (cid:18) τβ (cid:19) m (cid:18) − τβ (cid:19) M − m M M − (cid:88) p =0 S + i ( m + p ) S − j ( p ) (cid:43) W , (8)where i, j refer to lattice sites, m + p, p label positionswithin the operator string, i.e. intermediate time slices,and (cid:104) . . . (cid:105) W denotes the Metropolis weight of an operatorstring of length M generated by the SSE [48, 49].From (8) we proceed to momentum space by Fouriertransformation S ( q , τ ) = 1 N (cid:88) i,j e i q · ( r i − r j ) (cid:10) S + i ( τ ) S − j (0) (cid:11) . (9)Since the model comprises three sites per unit cell, S ( q , τ ) is a 3 × q -vector. To further an-alyze this, we first rotate onto the principal axis, i.e. wediagonalize this matrix. Only thereafter, and indepen-dently for each eigenvalue S µ ( q , τ ), µ =0,1,2, we performanalytic continuation to real frequencies ω by S µ ( q , τ ) = (cid:90) ∞ dω S µ ( q , ω ) K ( ω, τ ) , (10)with a kernel K ( ω, τ ) = ( e − τω + e − ( β − τ ) ω ) /π .The inversion (10) is an ill-posed problem, for whichmaximum entropy methods (MEM) have proven tobe well suited. Here we use Bryan’s MEM algo-rithm [55–57]. This method minimizes the functional Q = χ / − ασ , with χ being the covariance of theQMC data with respect to the MEM trial spectrum S ( q , ω ). Overfitting is prevented by an entropy term σ = (cid:80) ω S ( q , ω ) ln[ S ( q , ω ) /m ( ω )]. We have used a flatdefault model m ( ω ), which is iteratively adjusted tomatch the zeroth moment of the trial spectrum. Theoptimal spectrum follows from the average of S ( q , ω ),weighted by a probability distribution P [ α | S ( q , ω )] [55]. III. RESULTS
In this section, we detail our findings for the spin stiff-ness and the quantum phase diagram, as well as thestatic, and dynamic structure factors in order to char-acterize the phases of the model versus ring exchangeand trimerization. − − . . . . . .
12 (a) (b)
T/j ρ S − ρ S t j L . . . . . . L ρ S j κ . . FIG. 2. (a) Spin stiffness ρ S for κ = 0, subtracted bysingle trimer spin stiffness ρ St , for various trimerization ra-tios j = 1 , . , . T ∗ = j/L . Note: x -axisscaled by j . (b) Inset shows spin stiffness versus L = 6 , . . . , T = j/
10, for various trimerizationratios j = 1 , . κ = 0 ,
3. Errorbars smallerthan marker size.
A. Spin Stiffness
We use the stiffness to extract the quantum criticalpoint. To this end we rely on the fact that scaling the-ory [58] has been proven to apply for j = 1 [18] and weanticipate that chosing j (cid:54) = 1 should not change this.This implies that the stiffness fulfills the scaling rela-tion ρ S = L − z F (( κ C − κ ) L /ν , β/L z ) with a universalscaling function F . In turn, using z = 1 for the dy-namical critical exponent, and fixing the temperaturesuch that L z /β = c is constant, i.e. T = T (cid:63) = c/L ,the function ρ S L will collapse onto a single curve ver-sus ( κ C − κ ) L /ν for all L . This allows to determine theQCP if c is chosen such, that T (cid:63) is low enough for ρ S torepresent the zero temperature limit. Furthermore, since ρ S ( κ = const ., L ) = const . for L (cid:29) T →
0, thisimplies that F ( x, const) ∝ x ν .Prior to analyzing ρ S L in this way, we therefore needto fix c . For this, Fig. 2(a) depicts the spin stiffness ver-sus temperature for various trimerization at κ = 0 whereLRO is certain at T = 0 for j = 1. To focus on the col-lective properties, we correct for the finite spin stiffnesspresent already on isolated trimers ρ St with j = 0, i.e. weconsider ρ S − ρ St . Several points can be read off from thisfigure. First, there is a clear low- T crossover regime to astate with a finite stiffness, which decreases with increas-ing trimerization. Choosing T (cid:63) well below this regime issufficient for the scaling analysis to describe zero temper-ature behavior. Second, the crossover regimes for differ-ent j can be made to coincide, if T is scaled by j . Third,the figure shows that the crossover regime is rather in-sensitive with respect to L for the systems sizes we haveinvestigated. Summarizing these points, c (cid:39) j is a suffi-cient choice and we use T (cid:63) = j/L for the remainder of − −
200 000 . . κ − κ C ) L /ν ρ S / L − d − z j = 1 L − −
200 0 00 . . . . . κ − κ C ) L /ν j = 0 . FIG. 3. Scaling behavior at T = T ∗ of ρ S /L − d − z versus( κ − κ C ) L /ν at (a) j = 1 and (b) j = 0 .
5. For j = 1 ( j = 0 . κ C ≈ .
04 ( κ C ≈ . ∝ ( κ C − κ ) ν . this work.We also note from Fig. 2(a), that the low- T saturationvalue of ρ S − ρ St depends little on system size. Suffi-ciently close to criticality, this is to be expected fromscaling. I.e., for any temperature on the low- T satura-tion plateau of ρ S ( t ) , the universal function F ( x, y ) is inits asymptotic regime F ( x, const) ∝ x ν . At fixed κ ( C ) therefore, ρ S ∝ L − z ( L /ν ) ν = const versus L . For ρ St the latter is satisfied trivially. The inset Fig. 2(b) demon-strates, that the independence of system size of ρ S at low T remains valid over a wide range of j - and κ -values, ir-respective of criticality, rendering the usage of the scalingfunction rather robust.Setting T = T (cid:63) , we now extract the QCP by opti-mizing the collapse of all of our results for ρ S L z ver-sus ( κ C − κ ) L /ν for L = 6 . . .
12 at fixed j , for various j = 0 . . . . (cid:63) universality es-tablished, i.e. z = 1 and ν = 0 . κ C asthe single fit parameter.Results for this optimization procedures are depictedin Fig. 3 for two different j . Similar behavior is obtainedfor all other j considered. The collapse is clearly evi-dent and leads to a critical ring exchange κ C ≈ . .
97 at j = 1 and 0 .
5, respectively. The value for κ C ( j = 1) agrees with previous findings [18]. The fig-ure displays an additional function ∝ ( κ − κ C ) ν , whichcan be superimposed to fit the QMC results very wellin the ordered phase. Apart from the collapse itself, thisprovides further support for the validity of the scaling hy-pothesis also at finite trimerization. In the LRO phase at j (cid:54) = 1, and significantly off from criticality, non-universalcorrections in terms of an intermediate maximum in ρ S L superfluid Z -QSL(a)0 2 4 6 8 10 12 1400 . k j max k ρ S k C superfluid Z -QSL(b)0 2 4 6 8 10 12 1400 . κ j max κ ρ S κ C FIG. 4. Quantum phase diagram versus (a) ring exchange k , (b) dimensionless ring exchange κ and trimerization ratio j . Green symbols, critical coupling (a) k C , (b) κ C for SFL-QSL QCP. Orange squares, location of ρ S -maximum. If notvisible, errrobars smaller than marker size. Note: opposite y -axis directions in (a) and (b). appear at finite κ . We observe this maximum at all j (cid:54) = 1investigated. I.e. the stiffness provides no evidence forphases other than SFL and QSL at j (cid:54) = 1. B. Quantum Phase Diagram
Using all κ C ( j ) obtained by the data collapse, we are ina position to extend the quantum phase diagram (QPD)derived in Ref. [18] from the line j = 1 onto the ( κ, j )-plane. This is shown in Fig. 4(b). Remarkably, if ex-pressed in terms of κ = 2 K/ ( J (cid:77) + J (cid:79) ) = 2( K/J (cid:77) ) / (1 + j )the transition resides at a fixed location versus j withinthe error of the data collapse. Speaking differently, andequally remarkable, removing the implicit j -dependencefrom the x -axis, as in Fig. 4(a), the QPD reveals an in-crease in the tendency to form the QSL as the trimer-ization increases. This can be understood by the de-crease with j of the boson kinetic energy which drivesthe SFL phase. In fact, performing linear spin-wave the-ory (LSWT) exactly at K = 0, see Appendix B, eachspin is connected to four neighboring ones, two of themby J (cid:77) and two of them by J (cid:79) . Therefore the leadingorder-1 /S contribution to the energy is proportional to¯ , see Eqns. (B4, B5).It is tempting to speculate about the QPD as j → j = 0 directly. Second, strictly at j = κ = 0,there can be no SFL LRO phase, since the system is alattice of disconnected trimers. Finally, it is likely, thatthe QSL remains existent for κ > κ C , also at j = 0. For κ < κ C , one scenario could be that the superfluid den-sity vanishes as j →
0, consistent with Fig. 2, such that ρ S ≤ ρ St on the line j = 0 for all κ < κ C . However, thisrenders the nature of the state for κ < κ C at j = 0 un-clear. Another scenario, suggested by the maximum in (a) − π π . κ = 0 (b) . < κ C (c) . > κ C (d) − π π (e) (f)(g) − π π − π π (h) − π π (i) − π π . . j FIG. 5. SSSF S ( q ) for various trimerization ratios j = 1 , . , . κ = 0, 8( < κ C ),and 13( > κ C ). QMC at L = 12, T = T ∗ . ρ S could be, that weak LRO, driven by ring exchange, re-mains present even at j = 0, but with a non-monotonousbehavior versus κ < κ C . This remains an open issue. C. Static Spin Structure Factor
The SSSF S ( q ) is equivalent to the static density-density correlation function within the hard-core bosonpicture. As such it has been used to check if the tran-sition from the SFL into the anticipated Z -QSL is ac-companied by a density wave, i.e. by any peaks scaling ∝ N . This has been excluded in [18] at j = 1, provingthat the non-SFL phase does not break discrete latticesymmetries and thereby further corroborating its QSLnature.In view of the QPD in Fig. 4, the short-range nature ofthe zz -spin-correlations will remain true also for j < κ . This is shown in Fig. 5,which depictes contours of S ( q ) at T = T ∗ and for L = 12at κ = 0, 8( < κ C ), and 13( > κ C ).Indeed, the changes in this figure along the vertical j -direction in each column are small only. The evolutionof the contours along the κ -direction within the j = 0row are consistent with that in [18]. For κ (cid:28)
1, themaximum intensity in S ( q ) resides at the K -points, dueto the NN-correlations from j . As κ increases, the nextNN-correlations produce maximum intensity at the M -points and minimum intensity at the Γ - and K -points. (a) j = 10123 ω / J (cid:77) . . S ( q , ω ) (b) j = 10 0 .
05 0 . S ( q , ω ) (c) j = 10 0 .
05 0 . S ( q , ω ) (d) j = 0 . ω / J (cid:77) (e) j = 0 . (f) j = 0 . (g) j = 0 . M (cid:48) K (cid:48) Γ0123 ω / J (cid:77) (h) j = 0 . M (cid:48) K (cid:48) Γ (i) j = 0 . M (cid:48) K (cid:48) Γ FIG. 6. DSSF in SFL phase at κ = 0 for three trimeriza-tion ratios j = 1 , . , .
25 (rows), along high-symmetry pathFig. 1(b) in first and second BZ. Columns refer to the threeeigenmodes Eq. (10). QMC at L = 12, T = T ∗ . Solid redline: LSWT dispersions Eqns. (B4, B5). D. Dynamic Spin Structure Factor
As to be expected from the QPD, the elementary exci-tations fall into one of two classes. I.e. for κ<κ C they aremagnons of a planar FM, with a broken residual U (1)-symmetry and for κ>κ C they represent the deconfinedspinons and visons of the Z -QSL. While the DSSF canbe used directly to map out the magnon spectrum, it can-not do so for the fractionalized one-particle excitations.However, since the DSSF comprises a two-particle corre-lation function in terms of the latter, it can access thetwo-spinon spectrum, which is a continuum at each fixedtotal momentum. Using QMC at j = 1, magnons (two-spinon continua) have been verified in the FM (QSL)phase directly for Hamiltonian (2) [27], and for a closelyrelated model of the BFG class [26]. Here we considerthe DSSF for j ≤
1. Primarily, we focus on the magnonexcitations at κ = 0, since the dynamics in the QSLphase is driven by the ring exchange, where effects of thetrimerization are not expected to be significant.Fig. 6 details the evolution with j of the spectra ofthe three diagonal modes S µ ( q , ω ) of the DSSF at κ = 0.The dispersions are shown along the high symmetry pathof the extended BZ depicted in Fig. 1(b). Each rows ofthe figure display the typical signatures of a FM with athree-site unit cell. I.e. there is one ”acoustic” Goldstonemagnon and two ”optical” magnon branches. Properly,the dominant spectral weight occurs in the Goldstonemode at the ordering vector, i.e. the Γ -point. The effectof the trimerization can be seen clearly along the columnsof Fig. 6, i.e. as j is decreased, implying a reductionof the magnon kinectic energy, the bandwidth for bothtypes of magnons shrinks accordingly, with however the (a) κ = 0 , × . ω / J (cid:77) (b) κ = 0 , × . (c) κ = 8 , × . ω / J (cid:77) (d) κ = 8 , × . (e) κ = 13 , × . M (cid:48) K (cid:48) ω / J (cid:77) (f) κ = 13 , × . M (cid:48) K (cid:48) Γ j = 1 j = 0 .
50 0 . FIG. 7. Cumulated eigenmodes of DSSF for three ring ex-change values, in SFL phase at κ = 0 , κ = 13 (third row) for two trimerization ra-tios j = 1 , .
5, along high-symmetry path Fig. 1(b) in first andsecond BZ. QMC at L = 6 and T = T ∗ . Intensities rescaledfor visibility. Solid red line in first row: LSWT dispersionsEqns. (B4, B5). Goldstone nature of the acoustic mode remaining intact.To obtain a qualitative account of the magnon energiesto leading O (1 /S ) we have performed LSWT. See Ap-pendix B. The LSWT dispersions are included in Fig. 6.Their locations agree remarkably well with the QMC re-garding the Goldstone mode. For the two optical modes,comparison of the QMC spectra and the LSWT disper-sions suggest, that diagonalizing S ( q , τ ), as described af-ter Eq. (9) entails a mixing of the two optical modes.Irrespective of the qualitative agreement between QMCand LSWT, the figure also clearly demonstrates, that athigher energies the QMC spectra tend to broaden, hint-ing at the relevance of magnon-magnon interactions.Fig. 7 depicts the evolution of the DSSF with κ cross-ing over from the SFL into the QSL phase, along two lineswith j = 1 and 0.5. For simplicity, each spectrum refersto a trace over the three diagonal modes. The values of κ are chosen to be deep in the SFL phase, κ = 0, as well ascloser to the QCP, both, in the SFL and QSL phase, i.e.at κ = 8 and 13, respectively. Several points should benoted. First, in the SFL phase at κ = 0 and 8, the spec-tra corroborate a clear Goldstone mode behavior, with apronounced spectral weight at the Γ -point. Second, inthe QSL phase, the spectral weight may show remnantsof this at j = 1. However clearly, for j = 0 .
5, the DSSF inthe QSL regime is rather featureless versus momentum.Third, introducing ring exchange, the spectrum is shiftedto higher energies and is broadened (note the different y -scales of the panels). Fourth, Fig. 7 displays an openingof a gap at the Γ -point as κ increases. It is temptingto associate this with the onset of the two-spinon gap for κ > κ C . To substantiate this, larger systems are requiredto prove a vanishing gap for κ < κ C . This is beyond ouranalysis. IV. CONCLUSION
To summarize, using extensive quantum Monte-Carlocalculations, we have investigated the role of trimeriza-tion in the frustrated planar XY spin-1 / Z QSL states, the scaling behavior andthe universality class, as well as the static structure fac-tor of the model. Regarding the impact of trimerizationon the excitations, we find that linear spin waves remaina reasonably valid description in the LRO phase, whilespinon continua in the QSL phase may even be inten-sified in the non-uniform case. Open questions remain,regarding a non-monotonous behavior of the spin stiff-ness with ring exchange at strong trimerization, whichmay signal new types of ground states. Regarding actualrealizations in optical lattices or local-moment systems,our study shows that, in terms of the boson exchangeparameters, trimerization enlarges the QSL regime.
ACKNOWLEDGMENTS
Work of N.C. and W.B. has been supported in partby the State of Lower Saxony through QUANOMET(project NP-2). Work of W.B. has been supported in partby the DFG through project A02 of SFB 1143 (project-id 247310070). W.B. also acknowledges kind hospitalityof the PSM, Dresden. This research was supported inpart by the National Science Foundation under GrantNo. NSF PHY-1748958.
Appendix A: Spin Stiffness of XY Trimer At j = k = 0, our model simplifies to disconnectedXY trimers on a triangular lattice. In turn, even in thislimit the model exhibits a finite extensive spin stiffness,resulting from the trimers. Since this does not imply acollective LRO state, we analyze the stiffness obtainedfrom QMC by subtracting the bare trimer stiffness. Thelatter can be obtained analytically by introducing a twist φ along the x -direction of the trimer. This affects onlythe S z = ± / |↑↓↓(cid:105) and |↓↑↑(cid:105) . For the S z = 1 / H S z =1 / ( φ ) = − J (cid:77) |↑↑↓(cid:105) |↑↓↑(cid:105) |↓↑↑(cid:105) (cid:32) (cid:33) e i φ e − i φ S z = − /
2. Moreover H trimer ( φ ) |↑↑↑ ( ↓↓↓ ) (cid:105) = 0 |↑↑↑ ( ↓↓↓ ) (cid:105) . The stiffness is obtained fromthe free energy by ˜ ρ St ( T ) = ∂ F ( φ ) /∂φ | φ =0 . Straight-forward algebra yields˜ ρ St ( T ) = (cid:34) e T + 3) T + 3)2( e T − T − (cid:35) − . (A2)Fig. 8 depicts ˜ ρ St ( T ) /
4, where the factor of 4 refers to L α in Eq. (6) for trimers on a non-interacting kagomelattice with j = 0, where L α = 2 because of the J (cid:77) and(vanishing) J (cid:79) bonds per spatial direction. The figurealso proves that QMC data for ρ St ( T ) for a single trimerfrom Eqns. (5, 6) agrees with Eq. (A2). Appendix B: Linear Spin Wave Theory
In the SFL phase and for κ = 0, the QMC spectra canbe contrasted against magnon excitations obtained fromlinear spin wave theory (LSWT). For the latter we usethe conventional Holstein-Primakoff representation witha quantization axis along x and expanded up to leadingorder 1 /S , i.e. S xm = S − a † m a m S ym ≈ √ S (cid:0) a m − a † m (cid:1) (B1)with Boson operators a ( † ) m . Inserting this into Eq. (2) andafter Fourier transformation we get H (2) = S (cid:88) q Ψ † q (cid:18) z ¯ J · − Γ( q ) Γ( q )Γ( q ) z ¯ J · − Γ( q ) (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) M ( q ) Ψ q , (B2)where z = z (cid:77) + z (cid:79) = 2 + 2 = 4 is the coordination num-ber, Ψ q = ( a † , q a † , q a † , q a , q a , q a , q ) is a spinor of cre-ation and annihilation operators, with 0 ,
1, and 2 refer-ring to kagome basis of the triangular lattice, and Γ( q )encodes the hopping matrix elementsΓ( q ) = 12 J (cid:77) e i q · r + J (cid:79) e − i q · r J (cid:77) e i q · r + J (cid:79) e − i q · r J (cid:77) e − i q · r + J (cid:79) e i q · r J (cid:77) e i q · ( r − r ) + J (cid:79) e − i q · ( r − r ) J (cid:77) e − i q · r + J (cid:79) e i q · r J (cid:77) e − i q · ( r − r ) + J (cid:79) e i q · ( r − r ) . (B3)The magnon dispersions result from the parauni-tary secular equation det | S Λ · M ( q ) − ω | = 0, whereΛ = (cid:0) − (cid:1) . We find ω , /J (cid:77) = ¯ (cid:104) ∓ (cid:112) (3 j − + 8[1 + j γ ( q )] (cid:105) (B4) ω /J (cid:77) = 32 (1 + j ) = 6¯ (B5)with γ ( q ) = cos(2 q · r )+cos(2 q · r )+cos(2 q · ( r − r )). InEq. (B4), ω corresponds to the minus sign on the righthand side and the three branches ω and ω , compriseone acoustic and two optical modes. One of the latteris completely flat, i.e. at ω the magnons are localizedfrom local interference effects, which are a reoccurringtheme for NN-hopping models on the kagome lattice. Atthe K -point, the gap between the 0- and 1-mode satisfies ∆ ( K , j ) ≡ ω ( K , j ) − ω ( K , j ) = √ (1 − √ j ). I.e. for j = 1, the optical and acoustic mode display a touchingpoint at this momentum. − − . . . T ρ S ( T ) ˜ ρ St ( T ) / FIG. 8. Comparison of spin stiffness versus T for singleXY trimer ˜ ρ St ( T ) / , 153160 (1973).[2] P. Fazekas and P. W. Anderson, Phil. Mag. , 423(1974).[3] G. Misguich and C. Lhuillier, in Frustrated Spin Systems(World Scientific, 2005), pp. 229-306.[4] L. Balents, Nature , 199-208 (2010).[5] L. Savary and L. Balents, Rep. Prog. Phys. , 016502(2017).[6] J. Knolle and R. Moessner, Annu. Rev. Condens. MatterPhys. , 451 (2019).[7] X.-G. Wen, Phys. Rev. B , 165113 (2002).[8] P. Henelius and A. W. Sandvik, Phys. Rev. B , 1102(2000).[9] A. Kitaev, Ann. Phys. (NY) , 2 (2006).[10] J. Nasu, M. Udagawa, and Y. Motome, Phys. Rev. Lett. , 197205 (2014).[11] L. Balents, M. P. A. Fisher, and S. M. Girvin, Phys. Rev.B , 224412 (2002).[12] D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. ,2376 (1988).[13] R. Moessner and S. Sondhi, Phys. Rev. Lett. , 1881(2001).[14] R. Moessner, S. L. Sondhi, and E. Fradkin, Phys. Rev.B , 024504 (2001).[15] D. N. Sheng and L. Balents, Phys. Rev. Lett. , 146805(2005).[16] S. V. Isakov, Y. B. Kim, and A. Paramekanti, Phys. Rev.Lett. , 207204 (2006).[17] S. V. Isakov, A. Paramekanti, and Y. B. Kim, Phys. Rev.B , 224431 (2007).[18] L. Dang, S. Inglis, and R. G. Melko, Phys. Rev. B ,132409 (2011).[19] S. V. Isakov, M. B. Hastings, and R. G. Melko, NaturePhys. , 772 (2011).[20] S. V. Isakov, R. G. Melko, and M. B. Hastings, Science , 193 (2012).[21] F. J. Wegner, Journal of Mathematical Physics , 2259(1971).[22] X. G. Wen, Phys. Rev. B , 2664 (1991).[23] A. Yu. Kitaev, Annals of Phys. , 2 (2003).[24] S. Sachdev, Rep. Prog. Phys. , 014001 (2018). [25] Y.-C. Wang, C. Fang, M. Cheng, Y. Qi, and Z. Y. Meng,arXiv:1701.01552.[26] J. Becker and S. Wessel, Phys. Rev. Lett. , 077202(2018).[27] J. Becker and S. Wessel, Phys. Rev. B , 241113(2019).[28] A. V. Chubukov, T. Senthil, and S. Sachdev, Phys. Rev.Lett. , 2089 (1994).[29] A. V. Chubukov, S. Sachdev, and T. Senthil, NuclearPhysics B , 601 (1994).[30] T. Senthil and O. Motrunich, Phys. Rev. B , 205104(2002).[31] S. V. Isakov, T. Senthil, and Y. B. Kim, Phys. Rev. B , 174417 (2005).[32] H. G. Ballesteros, L. A. Fern´andez, V. Mart´ın-Mayor,and A. Mu˜noz Sudupe, Physics Letters B , 125(1996).[33] P. Calabrese, A. Pelissetto, and E. Vicari, Phys. Rev. E , 046115 (2002).[34] M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi,and E. Vicari, Phys. Rev. B , 214503 (2001).[35] F. Mila, Phys. Rev. Lett. , 2356 (1998).[36] M. Mambrini and F. Mila, Eur. Phys. J. B , 651 (2000).[37] M. E. Zhitomirsky, Phys. Rev. B , 214413 (2005).[38] R. Schaffer, Y. Huh, K. Hwang, and Y. B. Kim, Phys.Rev. B , 054410 (2017).[39] C. Repellin, Y.-C. He, and F. Pollmann, Phys. Rev. B , 205124 (2017).[40] Y. Iqbal, D. Poilblanc, R. Thomale, and F. Becca, Phys.Rev. B , 115127 (2018).[41] M. Iqbal, D. Poilblanc, and N. Schuch, arXiv:1912.08284.[42] L. Santos, M. A. Baranov, J. I. Cirac, H.-U. Everts,H. Fehrmann, and M. Lewenstein, Phys. Rev. Lett. ,030601 (2004).[43] B. Damski, H. Fehrmann, H.-U. Everts, M. Baranov,L. Santos, and M. Lewenstein, Phys. Rev. A , 053612(2005).[44] P. Windpassinger and K. Sengstock, Rep. Prog. Phys. , 086401 (2013).[45] G.-B. Jo, J. Guzman, C. K. Thomas, P. Hosur, A. Vish-wanath, and D. M. Stamper-Kurn, Phys. Rev. Lett. , ,011601 (2020).[47] A. W. Sandvik, J. Phys. A , 3667 (1992).[48] A. W. Sandvik, Phys. Rev. B , R14157 (1999).[49] O. F. Sylju˚asen and A. W. Sandvik, Phys. Rev. E ,046701 (2002).[50] R. G. Melko and A. W. Sandvik, Phys. Rev. E , 026702(2005).[51] A. Cuccoli, T. Roscilde, V. Tognetti, R. Vaia, and P. Ver-rucchi, Phys. Rev. B , 104414 (2003). [52] A. W. Sandvik, Phys. Rev. B , 11678 (1997).[53] E. L. Pollock and D. M. Ceperley, Phys. Rev. B , 8343(1987).[54] K. Harada and N. Kawashima, J. Phys. Soc. Jpn. ,2768 (1998).[55] J. Skilling and R. K. Bryan, Mon. Not. R. Astron. Soc. , 111 (1984).[56] R. Bryan, Eur. Biophys. J. , 165174 (1990).[57] M. Jarrell and J. Gubernatis, Phys. Rep. , 133 (1996).[58] M. P. A. Fisher, P. B. Weichman, G. Grinstein, andD. S. Fisher, Phys. Rev. B40