Quantum paramagnetism and magnetization plateaus in a kagome-honeycomb Heisenberg antiferromagnet
QQuantum paramagnetism in a kagome-honeycomb Heisenberg antiferromagnet
Meghadeepa Adhikary, Arnaud Ralko, and Brijesh Kumar School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India. Institut N´eel, Universit´e Grenoble Alpes & CNRS, 38042 Grenoble, France. (Dated: February 25, 2021)A spin-1/2 Heisenberg model on honeycomb lattice is investigated by doing triplon analysis andquantum Monte Carlo calculations. This model, inspired by Cu (pymca) (ClO ), has three differentantiferromagnetic exchange interactions ( J A , J B , J C ) on three different sets of nearest-neighbourbonds which form a kagome superlattice. Interestingly, while the model is bipartite and unfrustrated,its quantum phase diagram is found to be dominated by a quantum paramagnetic phase that is bestdescribed as a spin-gapped hexagonal-singlet state. The N´eel antiferromagnetic order survives onlyin a small region around J A = J B = J C . This study provides an understanding of the gappednon-magnetic behaviour observed in Cu (pymca) (ClO ) at low temperatures. I. INTRODUCTION
Models of interacting quantum spins are essential toour understanding of magnetism in real materials. Theycome in different forms, and display a variety of phenom-ena arising from an interplay of competing interactions,quantum fluctuations and lattice geometry [1, 2]. An-tiferromagnetic spin-1/2 Heisenberg model is a problemof fundamental importance to quantum magnetism, andits physics depends sensitively on the underlying lattice.For instance, on honeycomb lattice with uniform nearest-neighbour interactions, it is known to realise N´eel orderin the ground state [3, 4]. But the same spin-1/2 modelon kagome lattice harbours a complex spin liquid groundstate [5–8]. There are materials that realise spin-1/2 hon-eycomb [9–12] or kagome [13, 14] antiferromagnets. Theabsence of magnetic order on kagome lattice is due to itsfrustrated geometry. Such a loss of magnetic order canalso be caused on honeycomb lattice by allowing the ex-change interactions to compete. It can be so done eitherby having further neighbour interactions [15, 16], or atthe very least, by making the nearest-neighbour interac-tions non-uniform. In this paper, we take the latter routeand consider spins on such a non-uniform honeycomb lat-tice whose nearest-neigbhour bonds form kagome super-lattice. We term it as the ‘kagome-honeycomb’ lattice.The motivation for the present study comes from therecent experimental studies of Cu (pymca) (ClO ) [17,18]. This compound is reported to have no magnetic or-der down to 0.6 K, and to exhibit magnetization plateausat 1/3 and 2/3 of the saturation value. The basic modelapplicable to this material is the spin-1/2 Heisenbergmodel on honeycomb lattice with three different nearest-neighbour antiferromagnetic interactions J A , J B and J C ,as shown in Fig. 1. Notably, these three exchange in-teractions form a kagome superlattice on the underlyinghoneycomb. This material thus realizes a Heisenberg an-tiferromagnet on kagome-honeycomb lattice. It can alsobe viewed as a system of hexagons formed by two typesof bonds (say, J B and J C ) and coupled via the third (say, J A ). This is exactly like a model for some spin-1 kagomematerials with a ferromagnetic J A , but antiferromagnetic J B and J C [19–22]. An early example of a frustratedspin-1/2 Heisenberg model with an exact dimer groundstate on kagome-honeycomb lattice occurs in Ref. [23].In this paper, we study the quantum phase diagramof the spin-1/2 Heisenberg antiferromagnet on kagome-honeycomb lattice by doing triplon analysis and unbiasedquantum Monte Carlo (QMC) simulations. The theoryof triplon fluctuations and the observables computed byQMC produce mutually consistent results not only qual-itatively but also quantitatively. Remarkably, in spite ofbeing bipartite and unfrustrated, this model is found torealise in a large part of the phase diagram a quantumparamagnetic phase, while only a small region around J A = J B = J C corresponds to the N´eel antiferromagneticphase. This quantum paramagnetic phase is describedwell as a spin-gapped hexagonal singlet state. This paperis organized as follows: in Sec. II we describe the modeland discuss its key qualitative aspects; in Sec. III we dotriplon analysis of the model, and present the quantumphase diagram obtained from it; in Sec. IV, we presentthe results obtained from QMC simulations. Finally weconclude with a summary and outlook in Sec. V. II. MODEL
The spin-1/2 Heisenberg model on kagome-honeycomblattice is given by the Hamiltonian, ˆ H ABC = ˆ H A + ˆ H B +ˆ H C , whereˆ H A = J A (cid:88) (cid:126)R (cid:16) (cid:126)S (cid:126)R · (cid:126)S (cid:126)R + (cid:126)S (cid:126)R · (cid:126)S (cid:126)R + (cid:126)S (cid:126)R · (cid:126)S (cid:126)R (cid:17) (1a)ˆ H B = J B (cid:88) (cid:126)R (cid:16) (cid:126)S (cid:126)R · (cid:126)S (cid:126)R + (cid:126)S (cid:126)R · (cid:126)S (cid:126)R + (cid:126)S (cid:126)R · (cid:126)S (cid:126)R (cid:17) (1b)ˆ H C = J C (cid:88) (cid:126)R (cid:16) (cid:126)S (cid:126)R · (cid:126)S (cid:126)R + (cid:126)a ) + (cid:126)S (cid:126)R · (cid:126)S (cid:126)R + (cid:126)a ) + (cid:126)S (cid:126)R · (cid:126)S (cid:126)R − (cid:126)a − (cid:126)a ) (cid:17) . (1c)Here, the exchange interactions J A , J B and J C are allantiferromagnetic, and the lattice and the spin labels are a r X i v : . [ c ond - m a t . s t r- e l ] F e b ~ a ~ a A J B J C ⃗ a ⃗ a ⃗ a FIG. 1. The ABC Heisenberg model on kagome-honeycomblattice. The exchange interactions J A , J B , and J C are antifer-romagnetic, and they form a kagome superlattice (indicatedby thin-dotted lines) on the honeycomb structure. The (cid:126)a and (cid:126)a are two primitive vectors of the underlying Bravais lattice. k k Γ KM ( π ,0 )(- π ,0 ) ( π )( - π ) FIG. 2. The first Brillouin zone for the lattice in Fig. 1. Apoint in this zone is the wavevector (cid:126)k = k (cid:126)b + k (cid:126)b , where (cid:126)b are reciprocal to (cid:126)a such that (cid:126)a i · (cid:126)b j = δ ij . The dashedline demarcates the Brillouin zone in the hexagonal form. as shown in Fig. 1. The basic structure is honeycomb, butthe pattern of exchange interactions thereon is kagome.A primitive unit-cell of this so-called kagome-honeycomblattice contains six spins marked here by the integers 1to 6; (cid:126)R denotes the position of a primitive unit-cell. Thevectors (cid:126)a = 3 a ˆ x and (cid:126)a = 3 a ( − ˆ x + √ y ) / J A J A J B J B J C J C (AC-hexagons) ( B C - h e x a go n s ) ( A B - h e x a go n s ) ( C - d i m e r s ) (B-dimers) ( A - d i m e r s ) FIG. 3. Ternary representation of the space of exchange in-teractions such that J A + J B + J C = 1. At the base of thisequilateral triangle, J B = 0, which is the case of independentAC-hexagons. The apex (top corner) with J B = 1 corre-sponds to independent B-dimers. Likewise for the other twosides and corners. The centroid represents the uniform hon-eycomb antiferromagnet. is to study this competition. Throughout this paper, theexchange interactions are taken to have values between 0and 1 in such a way that J A + J B + J C = 1.Since the uniform case, with J A = J B = J C , is knownto realise N´eel order, even when the three exchange inter-actions are unequal, the N´eel order is expected to survivein the vicinity of the point ( J A , J B , J C ) = (1 / , / , / J A , J B , J C ) = (1 , ,
0) , realises independent dimers.The other case in which only one type of bonds have zeroexchange interaction, e.g. ( J A , J B , J C ) = (1 − x, x, x ∈ [0 , J A + J B + J C = 1, the space of interaction parametersis an equilateral triangle shown in Fig. 3. The corners ofthis triangle correspond to independent dimers, and thesides to independent hexagons. The ground state of theABC model is, therefore, bound to exhibit a quantumphase transition from the N´eel antiferromagnetic phasein the interior around the centroid (1 / , / , /
3) to anon-magnetic singlet phase outwards to the three sidesof the ternary diagram. In the following sections, wemake systematic analytical and numerical calculations toobtain the quantum phase diagram of the spin-1/2 ABCmodel on kagome-honeycomb lattice.First we do the triplon analysis with respect to thehexagonal and the dimer singlet states. These are spin-gapped phases, for which the closure of the gap marks aquantum phase transition to the N´eel phase. By follow-ing the triplon gap, and comparing the energies of thesecandidate states, we construct a quantum phase diagram.We then calculate spin stiffness and staggered magneti-zation by doing quantum Monte Carlo simulations. Allthese calculations produce a mutually agreeable phasediagram dominated by a quantum paramagnetic phasebest described as a hexagonal singlet phase.
III. TRIPLON ANALYSIS
The basic framework of triplon analysis is to first iden-tify such building-blocks of the system which in somelimiting case realise singlet ground state locally indepen-dently, and then formulate an effective theory in terms ofthe low-energy triplet excitations of these building-blocksto describe the full system [22, 25–27]. In this spirit, ourABC model can be viewed either as a system of cou-pled dimers, or coupled hexagons. For instance, we canconsider the ABC model (see Fig. 1) as made of the Atype bonds coupled by B and C bonds, or as made of theAB hexagons coupled via C bonds. (The other equivalentchoices can be obtained by permuting A, B, C cyclically.).As mentioned earlier, in the limit of J B = J C = 0, themodel realises an exact dimer singlet (DS) ground stateformed by the direct product of the singlets on the Abonds. When J C = 0, it similarly realises a hexagonalsinglet (HS) ground state exactly. Thus, we have twoways of doing triplon analysis of the ABC model with re-spect to the two natural quantum paramagnetic states,DS or HS. Note that the exact DS state itself is a limitof an exact HS state, e.g. the J B = 0 case of the AB-hexagons is the same as having independent A-dimers.In pictorial terms (see Fig. 3), the corners of the ternarydiagram are the ends of its sides. It requires that weformulate the triplon analysis for the HS case in such amanner that, near the corners of the ternary diagram, itis consistent with the triplon analysis with respect to theDS state. Let us do it now, and see what we learn aboutthe extent of the singlet phases as one moves inwards intothe ternary diagram from its sides and corners. A. Dimer singlet state
Assuming J A to be stronger than J B and J C , we satisfythe Heisenberg interaction on the A-bonds exactly, anddescribe the spin operators in terms of the singlet andtriplet eigenstates thereof. [The same is to be done withrespect to B (or C) bonds, when J B (or J C ) is strongerthan the rest.]. A convenient way to do this is to employbond-operator representation, in which one uses bosonicoperators for the singlet and triplets states of a bond [25,26]. It is simplified by treating the singlet bond-operatoron every A-dimer as a mean singlet amplitude, ¯ s , for the dimer singlet phase. The dynamics of the tripletexcitations (triplons) in the DS phase is described usingthe triplet bond-operators.The ABC model has three A-bonds per unit-cell (redbonds in Fig. 1). We label these bonds as j = I, II, III .Let α = x, y, z denote the three components of a spin.The six spins in a unit-cell at position (cid:126)R in the bond-operator representation (in a basic approximated form)can be written as: S α (cid:126)R ≈ ¯ s t α † (cid:126)R,I + t α(cid:126)R,I ) ≈ − S α (cid:126)R (2a) S α (cid:126)R ≈ ¯ s t α † (cid:126)R,II + t α(cid:126)R,II ) ≈ − S α (cid:126)R (2b) S α (cid:126)R ≈ ¯ s t α † (cid:126)R,III + t α(cid:126)R,III ) ≈ − S α (cid:126)R (2c)where ˆ t α(cid:126)R,j and ˆ t α † (cid:126)R,j are the triplet bond-operators. Thebond-operators are also required to satisfy the constraint,¯ s + (cid:80) α t α † (cid:126)R,j t α(cid:126)R,j = 1, to account for the physical dimen-sion of the Hilbert space on every A-bond.Since the interaction on the A-bonds is treated exactly,we obtain the following expression for that part of theABC model which comes from the A-bonds, i.e. the ˆ H A of Eq. (1a), in terms of the singlet amplitude and thetriplet bond-operators.ˆ H A = (cid:88) (cid:126)R (cid:88) j (cid:32) − J A s + J A (cid:88) α t α † (cid:126)R,j t α(cid:126)R,j (cid:33) (3)The triplets on different A-bonds interact and disperse onthe lattice due to ˆ H B and ˆ H C , i.e. Eqs. (1b) and (1c).We use Eqs. (2) to rewrite ˆ H B and ˆ H C in terms of thetriplon operators. The constraint on the bond-operatorsis satisfied on average through a Lagrange multiplier λ by adding the term, λ (cid:80) (cid:126)R,j (cid:16) ¯ s + (cid:80) α t α † (cid:126)R,j t α(cid:126)R,j − (cid:17) , tothe triplon Hamiltonian.We find it convenient to write the triplon Hamiltonianusing canonical “position”and “momentum” operators:ˆ Q α(cid:126)R,j = √ ( t α † (cid:126)R,j + t α(cid:126)R,j ) and ˆ P α(cid:126)R,j = i √ ( t α † (cid:126)R,j − t α(cid:126)R,j ). Theyfollow the relations (cid:104) ˆ Q α(cid:126)R,j , ˆ P α (cid:48) (cid:126)R (cid:48) ,j (cid:48) (cid:105) = iδ j,j (cid:48) δ α,α (cid:48) δ (cid:126)R, (cid:126)R (cid:48) and( ˆ P α(cid:126)R,j ) + ( ˆ Q α(cid:126)R,j ) = 2 t α † (cid:126)R,j t α(cid:126)R,j + 1. Their Fourier trans-formation is defined as: ˆ Q α(cid:126)R,j = √ N uc (cid:80) (cid:126)k ˆ Q α(cid:126)k,j e i(cid:126)k. (cid:126)R andˆ P α(cid:126)R,j = √ N uc (cid:80) (cid:126)k ˆ P α(cid:126)k,j e i(cid:126)k. (cid:126)R , where N uc is the total num-ber of unit-cells, and the wavector (cid:126)k lies in the Brillouinzone drawn in Fig. 2. Moreover, ( ˆ Q α(cid:126)k,j ) † = ˆ Q α − (cid:126)k,j , andlikewise for ˆ P α(cid:126)k,j .We obtain the following effective Hamiltonian for thetriplon dynamics with respect to the DS state.ˆ H tDS = (cid:15)N uc + 12 (cid:88) (cid:126)k,α (cid:110) λ P α † (cid:126)k I P α(cid:126)k + Q α † (cid:126)k V (cid:126)k Q α(cid:126)k (cid:111) (4)Here, λ = λ + J A , (cid:15) = 3¯ s λ + J A − λ − J A ¯ s , and I is the 3 × P α(cid:126)k , Q α(cid:126)k and V (cid:126)k are givenbelow. P α(cid:126)k = P α(cid:126)k,I P α(cid:126)k,II P α(cid:126)k,III , Q α(cid:126)k = Q α(cid:126)k,I Q α(cid:126)k,II Q α(cid:126)k,III (5) V (cid:126)k = λ I − J B ¯ s − J C ¯ s e ik e ik e − ik e − ik e − ik e ik (6)Note that k = (cid:126)k.(cid:126)a , k = (cid:126)k.(cid:126)a and k = k + k . Theeigenvalues of V (cid:126)k are found to be ω (cid:126)k,j = (cid:113) λ ( λ − s ζ (cid:126)k,j ) (7)where ζ (cid:126)k,I = − ( J B + J C ) is (cid:126)k independent, while ζ (cid:126)k,II = 18 (cid:20) J B + J C + (cid:113) J B − J B J C + 9 J C + 8 J B J C f (cid:126)k (cid:21) and ζ (cid:126)k,III = 18 (cid:20) J B + J C − (cid:113) J B − J B J C + 9 J C + 8 J B J C f (cid:126)k (cid:21) depend on (cid:126)k through f (cid:126)k = cos k + cos k + cos k . Know-ing these ω (cid:126)k,j ’s (the triplon dispersions of ˆ H tDS ) gives thefollowing ground state energy per unit-cell. E gDS = (cid:15) + 32 N uc (cid:88) (cid:126)k (cid:88) j ω (cid:126)k,j (8)Minimizing the E gDS with respect to ¯ s and λ leads tothe following equations, λ = J A + λ N uc (cid:88) (cid:126)k,j ζ (cid:126)k,j ω (cid:126)k,j (9a)¯ s = 52 − N uc (cid:88) (cid:126)k,j λ − ¯ s ζ (cid:126)k,j ω (cid:126)k,j (9b)whose self consistent solution determines the dimer sin-glet phase for the ABC model.Before solving these equations for λ and ¯ s , let us alsoformulate a theory of triplon dynamics with respect tothe hexagonal singlet state. Then, we will present anddiscuss their findings together. B. Hexagonal singlet state
When J C = 0, the ABC model is a collection of inde-pendent AB-hexagons (see Figs. 1 and 3). So, when J C - - - - - - J B / J A E n e r g y E i g e n va l u es Singlet ( ) Triplet ( ) Triplet ( ) Singlet ( ) - - - - - - J B / J A E n e r g y E i g e n va l u es Singlet ( ) Triplet ( ) Triplet ( ) Singlet ( ) FIG. 4. Low-energy spectrum of a spin-1/2 hexagon withalternating exchange interactions J A and J B on the nearest-neighbour bonds. The lowest eigenvalue (thick black) corre-sponds to a unique singlet, and the second lowest (dashed red)is a triplet. Then, there are two triplets (dot-dashed blue) andanother unique singlet (thin green). The higher energy part ofthe spectrum is not shown here. The ABC Heisenberg modelof Eq. (1) and Fig. 1 can be considered as a system of suchAB-hexagons coupled via the exchange interaction J C . is non-zero (but somewhat weaker than J A and J B ), itis reasonable to formulate a theory of the ABC model interms of the eigenstates of the AB-hexagons. In doing so,we satisfy two interactions ( J A and J B ) exactly, whichcertainly makes for a better case (than the dimer case ofthe previous section, where only one interaction, J A , wasexactly satisfied).The exact eigenspectrum of the Heisenberg model of asingle AB-hexagon is evaluated in Appendix A, of whichthe lowest few eigenstates are plotted in Fig. 4. Here, theground state is a unique singlet, separated from the firstexcited state (which is a triplet) by a finite energy. Whenthese hexagons are coupled via J C , one would expect theground state of the full model to be a hexagonal singlet(HS) state renormalised by triplet fluctuations, but pro-tected by triplon gap. For sufficiently strong J C , eitherthis triplon gap will close causing a phase transition toan ordered antiferromagnetic phase, or another state maylevel-cross. What one minimally needs to carry out suchan anaylsis is the lowest singlet and triplet eigenstates.But as noted earlier, the triplon analysis based on hexag-onal states is desired to be such that its approach to thedimer limit (for small J B or J A ) is appropriate. Figure 4suggests that we should take into consideration the nexttwo degenerate triplets also, because these two becomedegenerate with the lowest triplet (as for three indepen-dent dimers) when J B tends to zero. Taking three tripletsconsiderably enhances the complexity of the triplon anal-ysis, but it does give us a theory that works very well.These eigenstates are identified by their total spin andtwo other quantum numbers, m and ν corresponding re-spectively to the z -component of the total spin and thethreefold rotation of the hexagon. The ν takes values0 ,
1, ¯1(= −
1) (for the rotation eigenvalues 1 , ω, ω , re-spectively), and m takes values 0 , ± , ± ±
3. Refer toAppendix A for more details. Of the states presentedin Fig. 4, we denote the singlet ground state as | s (cid:105) andits energy as E s ; it belongs to m = 0 , ν = 0 subspace.The triplets are denoted as | t mν (cid:105) with m = 0 , ±
1. Thelowest energy triplet corresponds to ν = 0 with energydenoted as E t , and the next two triplets correspond to ν = 1 , ¯1 with energy E t . Note that for J B /J A (cid:38) . E t . But unlike the triplets, this second singletmakes no direct matrix elements (of the spin operators)with the singlet ground state. So, its effect on a low-energy theory based on the hexagonal states is negligible;we have checked this. Hence, we consider | s (cid:105) and | t mν (cid:105) only (a total 10 states per hexagon) to formulate a theorywith respect to the hexagonal singlet ground state.Like the bond-operators employed for the dimer case,we now introduce the bosonic operators, ˆ s (cid:126)R and ˆ t mν, (cid:126)R ,corresponding to the hexagonal singlet and triplet statesat position (cid:126)R [22]. Next we replace the singlet operatoron every hexagon by a mean amplitude ¯ s that accountsfor the hexagonal singlet background. Then, we writethe six spins (labelled as l = 1 to 6) on an AB-hexagonin terms of the triplet operators as follows. S zl (cid:126)R ≈ ¯ s (cid:104) C l (ˆ t , (cid:126)R + ˆ t † , (cid:126)R ) + ( C l ˆ t , (cid:126)R + C l ∗ ˆ t , (cid:126)R + h . c) (cid:105) (10a) S + l (cid:126)R ≈ ¯ s (cid:104) C l ¯10 (ˆ t ¯10 , (cid:126)R − ˆ t † , (cid:126)R ) + C l ¯11 (ˆ t ¯11 , (cid:126)R − ˆ t † , (cid:126)R )+ C l ∗ ¯11 (ˆ t ¯1¯1 , (cid:126)R − ˆ t † , (cid:126)R ) (cid:105) (10b)Here, the coefficients C l , C l et cetera are the matrixelements between the singlet and the triplet states. Referto Appendix A for more details on this representation.The constraint in this case is ¯ s + (cid:80) mν t † mν, (cid:126)R t mν, (cid:126)R = 1.The ˆ H A + ˆ H B part of the ABC model in this represen-tation reads as:ˆ H A + ˆ H B ≈ E s ¯ s N uc + (cid:88) (cid:126)R,mν E mν ˆ t † mν, (cid:126)R ˆ t mν, (cid:126)R (11)where E m = E t and E m = E m ¯1 = E t . The in-teraction between the AB-hexagons comes from ˆ H C ,which is now re-expressed using the representation inEqs. (10). Moreover, the constraint is taken into accountby adding the term λ (cid:80) (cid:126)R (¯ s + (cid:80) mν t † mν, (cid:126)R t mν, (cid:126)R −
1) tothe Hamiltonian through a Lagrange multiplier λ . ByFourier transforming the triplon operators as, ˆ t mν, (cid:126)R = √ N uc (cid:80) (cid:126)k e i(cid:126)k · (cid:126)R ˆ t mν,(cid:126)k , we finally get the following triplonHamiltonian for the hexagonal singlet case.ˆ H tHS = (cid:15) N uc + (cid:88) (cid:126)k Ψ † (cid:126)k H (cid:126)k Ψ (cid:126)k (12)Here, (cid:15) = E s ¯ s + λ ¯ s − λ − ( E t + 2 E t ), H (cid:126)k is an18 ×
18 matrix in the Nambu basis given in Appendix B,and Ψ † (cid:126)k is the following row vector of triplon creation and annihilation operators; Ψ (cid:126)k is its Hermitian conjugate.Ψ † (cid:126)k = (cid:16) ˆ t † ,(cid:126)k ˆ t † ,(cid:126)k ˆ t † ¯10 ,(cid:126)k · · · ˆ t , (cid:126) − k ˆ t , (cid:126) − k ˆ t ¯1¯1 , (cid:126) − k (cid:17) (13)We diagonalize ˆ H tHS using Bogloliubov transforma-tion, and obtain nine triplon dispersions, (cid:15) i(cid:126)k , in terms ofwhich the ground state energy can be written as: E gHS = (cid:15) + 1 N uc (cid:88) (cid:126)k (cid:88) i =1 (cid:15) i(cid:126)k (14)The following self-consistent equations for ¯ s and λ areobtained by minimizing E g , i.e. ∂E g ∂ ¯ s = 0 and ∂E g ∂λ = 0. λ = − E s − N uc (cid:88) (cid:126)k (cid:88) i =1 ∂(cid:15) i(cid:126)k ∂ ¯ s (15a)¯ s = 112 − N uc (cid:88) (cid:126)k (cid:88) i =1 ∂(cid:15) i(cid:126)k ∂λ (15b) C. Quantum phase diagram from triplon analysis
We solve Eqs. (9) and Eqs. (15) numerically. It gives usthe triplon dispersions and the ground state energy withrespect to the DS and HS states, respectively. By com-paring their energies, and by following the triplon gap,we obtain a quantum phase diagram presented in Fig. 5.As anticipated, it has in the middle a small region of N´eelantiferromagnetic phase, which is surrounded on all threesides by a quantum paramagnetic phase described prettywell for the most part as an hexagonal singlet phase (withrespect to the AB, BC or AC hexagons in the three tri-angular parts of the ternary diagram).For the concreteness of discussion, let us focus in Fig. 5on the triangular region on right-hand-side, given by0 ≤ J C ≤ / (cid:84) J A ≥ J C (cid:84) J B ≥ J C . It is formedby joining the top corner, right corner and the centroid.(The other two similar regions are related to this one bythe cyclic permutation of J A , J B , J C .). In this regionof the phase diagram, for J C = 0, we have independentAB-hexagons with exact HS ground state having a finiteenergy gap to triplet excitations. We find that for smallnon-zero J C , the triplon excitations of the renormalizedHS state are still gapped, and the mean singlet weightper hexagon, ¯ s , is close to 1. See Fig. 6 for triplon dis-persions in the gapped HS phase. We also find that closerto the corners of the ternary diagram, the results fromthe HS state triplon analysis correctly approach the DScase. See Fig. 7 for the energies of the HS and DS statesfrom triplon analysis as a function of J B for J C = 0 . J C here), the HS state is always lower in energy than the DSstate. Hence, the model clearly realises the HS phase nearthe three sides of the ternary diagram. This behaviourfrom the AB-hexagon side continues upto J C ≈ . J A J A J B J B J C J C (AC-hexagons) ( B C - h e x a go n s ) ( A B - h e x a go n s ) ( C - d i m e r s ) (B-dimers) ( A - d i m e r s ) Hexagonal Singlet Phase H e x a go n a l S i n g l e t P h a s e H e x a go n a l S i n g l e t P h a s e Néel Order
FIG. 5. Quantum phase diagram of the ABC Heisenberg anti-ferromagnet on kagome-honeycomb lattice from triplon anal-ysis. It is dominated by the spin-gapped hexagonal singletphase on the three sides, with a small region of N´eel antifer-omagnetic phase in the middle, and small competing regionsat the interfaces between the hexagonal singlet phases. ▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮▮□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶▶○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ Γ K M Γ J A = J B = J C = ▮ ϵ k □ ϵ k ▲ ϵ k ▶ ϵ k ○ ϵ k ▽ ϵ k ■ ϵ k ◆ ϵ k ϵ k FIG. 6. Triplon dispersions in the gapped HS phase along thesymmetry directions shown in Fig. 2. The triplon energy gapcomes from the Γ point.
For J C ≥ .
18, the DS state is found to become lowerin energy than the HS state, but only when either J A or J B is very close to J C . This level-crossing happens acrossthe blue-dotted lines in Fig. 5; also see the inset of Fig. 7.The gapped HS phase with respect to the AB-hexagonsstill holds good for the most part, except very close tothe interface with AC (or BC) hexagonal phase. At theinterface between, say, the AB and AC hexagonal phases,the B and C bonds would naturally compete to partnerwith the A bonds to form the respective HS state. So,when the exchange interactions of comparable values onB and C bonds are strong enough, it is possible that itis favourable for neither of them to partner with A. Thisis what this level-crossing seems to be hinting at. In thepresent analysis, the DS state of A-dimers happens tooffer an alternative for the B and C bonds to be treated E gHS E gDS - - - - - - - J B J C = - - - J C = FIG. 7. Ground state energies of the hexagonal singlet ( E gHS )and the dimer singlet ( E gDS ) phases from triplon analysis for J C = 0 .
1. Inset shows a level-crossing between the DS and theHS states by varying J B for J C = 0 .
25. Such level-crossingsoccur across the blue dotted lines in Fig. 5. ●●●●●●●●●●●●●●●●●●●● ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲▲▲▲▲▲▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼▼▼▼▼▼○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○○○○○○○ ● J B = ■ J B = ◆ J B = ▲ J B = ▼ J B = ○ J B = J C G a p FIG. 8. Triplon gap of the AB-hexagonal singlet state alongfixed J B lines for J C ∈ [0 , J B ]. freely and not bound to A . But it does not exclude thepossibility of an alternate description of this competingcross-over region.At J C = 0 .
23, we for the first time find the HS phaseto become gapless along the J A = J B line. This closingof the triplon gap (at the Γ point in the Brillouin zone)is found to occur in a continuous manner. See Fig. 8 forthe triplon gap in the HS phase. For J C > .
23, we get afinite region of the gapless HS phase in the middle. It isa common knowledge that the gapless triplons describemagnetic order [25, 27]. Hence, what we find here isa quantum phase transition from the gapped hexagonalsinglet phase to the N´eel antiferromagnetic phase. Thethick black line in Fig. 5 is the boundary of this quantumphase transition.Upon increasing the J C further, there comes a stage at J C ∼ .
27, when the gapped HS phase is lost. Now thecompeting region described here as a gapped DS phaseis found to be directly crossed by the N´eel state (e.g.,at J A = 0 .
41 along J B = J C line). This level-crossing(shown by the red dashed lines in Fig. 5) is obtained bycomparing the energy of the DS (and the HS) state withthat of the N´eel state from spin-wave theory; see Ap-pendix C for spin-wave calculation. It ought be pointedout here that, pretty much where the DS state is crossedby the N´eel state, the HS state (although energeticallyslightly ill-favoured here) still exhibits a continuous phasetransition to the Neel phase. These small competing re-gions appear to be more complex. IV. QUANTUM MONTE CARLO SIMULATION
In order to challenge and confirm the quantum phasediagram obtained from triplon analysis, we also employquantum Monte Carlo method to study this problem. Weare able to do so because our ABC Heisenberg model onkagome-honeycomb lattice is bipartite and un-frustrated,and hence amenable to QMC approach. We use the well-known stochastic series expansion (SSE) formulation ofQMC [28, 29], which is exact albeit stochastic. Withinthis framework, the physical quantities such as the stag-gered magnetization, m s , and the spin stiffness, D s , canbe calculated. The latter is defined as D s = N ∂ F ( φ ) ∂φ ,where F is the free energy of the system, N is the totalnumber of spins (sites) of the honeycomb lattice, and φ is the twist angle imposed on the periodic boundary con-dition. This quantity is considered to be a clean markerof the transition from an ordered ( D s (cid:54) = 0) to disorderedphase ( D s = 0). Within the SSE simulations, the spinstiffness is extracted using the winding number fluctua-tions as established in [28]. The former quantity, definedas m s = (cid:104) m s (cid:105) = N (cid:80) i (cid:104) S zi,u − S zi,v (cid:105) , is the order pa-rameter of the N´eel phase. Here, i is summed over thetwo-site unit-cells of the honeycomb lattice, and u and v denote the two sublattices. In the QMC simulationsfor finite size systems, what we calculate is the averagevalue, (cid:104) m s (cid:105) , which in the thermodynamic limit gives thesquare of the N´eel order parameter (i.e., m s ).In Fig. 9, we present the stiffness data from our QMCcalculations for a large lattice of N = 864 sites at a lowtemperature, β = 1 /T = 50. Juxtaposed with the quan-tum phase diagram obtained from triplon analysis, theN´eel phase obtained by spin stiffness exhibits remarkableagreement. The overall shape and extent of the regionwith D s (cid:54) = 0 is not only qualitatively consistent withthe phase boundary from triplon analysis, but it is alsoquantitative. This shows how good the proposed triplondescription is for this model, even by such direct com-parison with a large but finite size data.We improve the phase boundary obtained from QMCby doing a systematic finite size scaling of D s and m s along the J A = J B line. Doing it for the whole phasediagram would be too tedious to extract their thermo-dynamic limit (TL) behaviours. While we consider aninverse temperature of β = 50 for D s , a slightly highertemperature of β = 20 is taken for (cid:104) m s (cid:105) whose approachto TL is found to be slower (and harder) than that of D s . The extrapolated values and error bars are ob-tained by the linear fits of D s and m s with respect to1 / √ N [30]. These TL values of the two quantities, pre- J A J A J B J B J C J C (AC-hexagons) ( B C - h e x a go n s ) ( A B - h e x a go n s ) ( C - d i m e r s ) (B-dimers) ( A - d i m e r s ) Hexagonal Singlet Phase H e x a go n a l S i n g l e t P h a s e H e x a go n a l S i n g l e t P h a s e FIG. 9. The spin-stiffness data (pale yellow circles) fromQMC calculations plotted together with the quantum phasediagram from triplon analysis (Fig. 5). The radii of the circlesindicate the strength of the N´eel order. ● ● ● ●●●●●●●●●●●●●●●● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●● ● ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ���� ���� ���� ���� ���� ���� ���� ����������������������������� � � D s m s Triplon theory
FIG. 10. Thermodynamic limit extrapolations of the spinstiffness, D s , and staggered magnetization, m s , from QMCsimulations are plotted as a function of J C along the J A = J B line. They produce a region of N´eel phase that is consistentwith triplon analysis (blue arrow). sented in Fig. 10 as a function of J C , show an even closeragreement on the boundary of the N´eel phase. When J C goes from 1/3 (centroid) to 0 (AB-hexagon side), theextrapolated values of both D s and m s go continuouslyto zero at J C = 0 . J A = J B line,as J C goes from 1/3 (centroid) to 1 (C-dimer corner),the extrapolated values of D s and m s vanish together at J C = 0 . .
41 on thered segment from theory. It is thus evident that the HSstate triplon analysis provides a very good theory of thismodel to describe the thermodynamic properties, evenif the tiny competing regions (not identified by our SSEcalculations) leave room for some improvements.
V. CONCLUSION
The quantum phase diagram of an antiferromagneticspin-1/2 Heisenberg model on kagome-honeycomb lat-tice is obtained by a combined study based on triplonanalysis and QMC simulations. The findings from thetwo approaches are mutually consistent both qualita-tively and quantitatively. Interestingly, while the modelis unfrustrated and bipartite, its phase diagram is dom-inated by a quantum paramagnetic phase that is bestdescribed as hexagonal singlet state. The N´eel antifer-romagnetic order appears only in a small region aroundthe uniform honeycomb case. The exchange interactionsin the material Cu (pymca) (ClO ), which motivatedthe present study, are reported to be ( J A , J B , J C ) ≈ (0 . , . , .
10) [17]. It is a point safely inside the hexag-onal singlet phase; see Fig. 9. Hence, in view of our find-ings, this material realizes spin-gapped hexagonal singletstate at low enough temperatures. The reported low-temperature specific heat and magnetic susceptibility ofthis material are indeed suggestive of a non-magneticgapped phase [17]. There is a scope for better under-standing this material in the light of the theory presentedin this paper.
ACKNOWLEDGMENTS
M.A. acknowledges DST (India) for INSPIRE fel-lowship, and thanks Pratyay Ghosh for helpful discus-sions. B.K. acknowledges SERB (India) research grantfor project No. CRG/2019/003251. We also acknowl-edge the DST-FIST-funded HPC facility at the School ofPhysical Sciences, JNU for computations.
Appendix A: Heisenberg problem and triplonrepresentation on a single AB-hexagon
The Hamiltonian of a single spin-1/2 AB hexagon canbe written as:ˆ h AB = J A (cid:16) (cid:126)S · (cid:126)S + (cid:126)S · (cid:126)S + (cid:126)S · (cid:126)S (cid:17) + J B (cid:16) (cid:126)S · (cid:126)S + (cid:126)S · (cid:126)S + (cid:126)S · (cid:126)S (cid:17) . (A1)The total spin, S total , and its z -component, S ztotal , aretwo conserved quantities of this Hamiltonian. Let thequantum number corresponding to S ztotal be m , in termsof which the Hilbert space of six spin-1/2’s, {| ↑(cid:105) , | ↓(cid:105)} ⊗ , TABLE I. ν Basis states for m = 0 |↑↓↑↓↑↓(cid:105) , |↓↑↓↑↓↑(cid:105) ,0 √ ( |↑↑↑↓↓↓(cid:105) + |↑↓↓↓↑↑(cid:105) + |↓↓↑↑↑↓(cid:105) ), √ ( |↑↑↓↑↓↓(cid:105) + |↓↑↓↓↑↑(cid:105) + |↓↓↑↑↓↑(cid:105) ), √ ( |↓↓↑↓↑↑(cid:105) + |↑↓↑↑↓↓(cid:105) + |↑↑↓↓↑↓(cid:105) ), √ ( |↑↓↓↑↓↑(cid:105) + |↓↑↓↑↑↓(cid:105) + |↓↑↑↓↓↑(cid:105) ), √ ( |↓↓↓↑↑↑(cid:105) + |↓↑↑↑↓↓(cid:105) + |↑↑↓↓↓↑(cid:105) ), √ ( |↓↑↑↓↑↓(cid:105) + |↑↓↑↓↓↑(cid:105) + |↑↓↓↑↑↓(cid:105) ) √ ( ω |↑↑↑↓↓↓(cid:105) + ω |↑↓↓↓↑↑(cid:105) + |↓↓↑↑↑↓(cid:105) ),1 √ ( ω |↑↑↓↑↓↓(cid:105) + ω |↓↑↓↓↑↑(cid:105) + |↓↓↑↑↓↑(cid:105) ), √ ( ω |↓↓↑↓↑↑(cid:105) + ω |↑↓↑↑↓↓(cid:105) + |↑↑↓↓↑↓(cid:105) ), √ ( ω |↑↓↓↑↓↑(cid:105) + ω |↓↑↓↑↑↓(cid:105) + |↓↑↑↓↓↑(cid:105) ), √ ( ω |↓↓↓↑↑↑(cid:105) + ω |↓↑↑↑↓↓(cid:105) + |↑↑↓↓↓↑(cid:105) ), √ ( ω |↓↑↑↓↑↓(cid:105) + ω |↑↓↑↓↓↑(cid:105) + |↑↓↓↑↑↓(cid:105) ) √ ( ω |↑↑↑↓↓↓(cid:105) + ω |↑↓↓↓↑↑(cid:105) + |↓↓↑↑↑↓(cid:105) ),¯1 √ ( ω |↑↑↓↑↓↓(cid:105) + ω |↓↑↓↓↑↑(cid:105) + |↓↓↑↑↓↑(cid:105) ), √ ( ω |↓↓↑↓↑↑(cid:105) + ω |↑↓↑↑↓↓(cid:105) + |↑↑↓↓↑↓(cid:105) ), √ ( ω |↑↓↓↑↓↑(cid:105) + ω |↓↑↓↑↑↓(cid:105) + |↓↑↑↓↓↑(cid:105) ), √ ( ω |↓↓↓↑↑↑(cid:105) + ω |↓↑↑↑↓↓(cid:105) + |↑↑↓↓↓↑(cid:105) ), √ ( ω |↓↑↑↓↑↓(cid:105) + ω |↑↓↑↓↓↑(cid:105) + |↑↓↓↑↑↓(cid:105) ) can be sectorized into seven parts for m = 0 , ± , ± , ± |↑(cid:105) and |↓(cid:105) are the eigenstates of an individual S z operator, with eigenvalues and − respectively. ThisHamiltonian also has a threefold rotational symmetry, R π . Furthermore, (cid:104) S ztotal , R π (cid:105) = 0. Hence, the ba-sis states in each fixed m sector can be further groupedinto smaller sectors using the rotational quantum num-ber, ν = 0 , , ¯1 corresponding respectively to the three-fold rotation eigenvalues 1 , ω, ω . Here, ¯1 stands for − h AB in matrix form in each of these ( m, ν )subspaces separately, and find the complete eigenspec-trum for different values of J B /J A varying from 0 to 1.The ground state of ˆ h AB is a nondegenerate unique sin-glet state (i.e., S tot = 0) in the 8-dimensional ( m, ν ) =(0 ,
0) subspace. See Table I for the basis states with m = 0. Let us denote this state as | s (cid:105) and call the corre-sponding ground state energy as E s .The first excited state of ˆ h AB is a triplet (i.e., S tot = 1).The three eigenstates forming this triplet come from the ν = 0 sectors of the m = 0 , ± m = 1, see Table II. Next in thespectrum we find two more triplets. Of these, one setof triplet comes from ν = 1 and m = 0 , , ¯1; the secondtriplet is formed in the subspaces given by ν = ¯1 and m = 0 , , ¯1. Let the 9 eigenstates in these 3 triplets bedenoted as | t mν (cid:105) . The energy corresponding to | t m (cid:105) isdenoted as E t and is shown by red line in 4. This energylevel remains the second lowest all along J B /J A = 0 → | t m (cid:105) and | t m ¯1 (cid:105) are degenerate, and have theenergy E t shown by blue line in Fig. 4. The E t is the TABLE II. ν Basis states for m = 1 √ ( |↑↑↑↓↑↓(cid:105) + |↑↓↑↓↑↑(cid:105) + |↑↓↑↑↑↓(cid:105) ),0 √ ( |↑↑↑↓↓↑(cid:105) + |↑↓↓↑↑↑(cid:105) + |↓↑↑↑↑↓(cid:105) ), √ ( |↑↑↓↑↑↓(cid:105) + |↓↑↑↓↑↑(cid:105) + |↑↓↑↑↓↑(cid:105) ), √ ( |↑↑↓↑↓↑(cid:105) + |↓↑↓↑↑↑(cid:105) + |↓↑↑↑↓↑(cid:105) ), √ ( |↑↑↑↑↓↓(cid:105) + |↑↑↓↓↑↑(cid:105) + |↓↓↑↑↑↑(cid:105) ) √ ( ω |↑↑↑↓↑↓(cid:105) + ω |↑↓↑↓↑↑(cid:105) + |↑↓↑↑↑↓(cid:105) ),1 √ ( ω |↑↑↑↓↓↑(cid:105) + ω |↑↓↓↑↑↑(cid:105) + |↓↑↑↑↑↓(cid:105) ), √ ( ω |↑↑↓↑↑↓(cid:105) + ω |↓↑↑↓↑↑(cid:105) + |↑↓↑↑↓↑(cid:105) ), √ ( ω |↑↑↓↑↓↑(cid:105) + ω |↓↑↓↑↑↑(cid:105) + |↓↑↑↑↓↑(cid:105) ), √ ( ω |↑↑↑↑↓↓(cid:105) + ω |↑↑↓↓↑↑(cid:105) + |↓↓↑↑↑↑(cid:105) ) √ ( ω |↑↑↑↓↑↓(cid:105) + ω |↑↓↑↓↑↑(cid:105) + |↑↓↑↑↑↓(cid:105) ),¯1 √ ( ω |↑↑↑↓↓↑(cid:105) + ω |↑↓↓↑↑↑(cid:105) + |↓↑↑↑↑↓(cid:105) ), √ ( ω |↑↑↓↑↑↓(cid:105) + ω |↓↑↑↓↑↑(cid:105) + |↑↓↑↑↓↑(cid:105) ), √ ( ω |↑↑↓↑↓↑(cid:105) + ω |↓↑↓↑↑↑(cid:105) + |↓↑↑↑↓↑(cid:105) ), √ ( ω |↑↑↑↑↓↓(cid:105) + ω |↑↑↓↓↑↑(cid:105) + |↓↓↑↑↑↑(cid:105) ) third lowest upto J B /J A = 0 . m, ν ) = (0 ,
0) subspace, shown by greenline in Fig. 4.Next we derive a representation of the six spinsof the hexagon in terms of the singlet ground stateand the 3 triplets, i.e. a total of 10 eigenstates: {| s (cid:105) , | t (cid:105) , | t (cid:105) , | t ¯10 (cid:105) , | t (cid:105) , | t (cid:105) , | t ¯11 (cid:105) , | t (cid:105) , | t (cid:105) , | t ¯1¯1 (cid:105)} . Weignore the singlet excited state mentioned above, becauseit doesn’t form a matrix element with the singlet groundstate. We also ignore all the other higher energy eigen-states, because we want to develop a description that isessentially minimal.For the ten low-energy eigenstates identified above, weintroduce ten bosonic operators as follows. | s (cid:105) = ˆ s † | (cid:105)| t mν (cid:105) = ˆ t † mν | (cid:105) (A2)Here, the creation of a boson by applying ˆ s † on the vac-uum | (cid:105) corresponds to having the singlet ground state | s (cid:105) on the hexagon; likewise for ˆ t † mν . Since the auxiliarybosonic Fock space is infinite dimensional, the bosons are required to satisfy the constraint, ˆ s † ˆ s + (cid:80) m,ν ˆ t † mν ˆ t mν = 1,to conform to the dimension of the spin Hilbert space.We can write the six spins of a hexagon in terms ofthese 10 eigenstates. This is a reasonable approxima-tion to formulate an effective low-energy theory. Weevaluate the matrix elements of every component of thesix spins ( l = 1 , (cid:104) s | S zl | t mν (cid:105)| s (cid:105)(cid:104) t mν | corresponds to (cid:104) s | S zl | t mν (cid:105) ˆ s † ˆ t mν . In a physically motivated simplificationof this representation, we treat ˆ s and ˆ s † in mean-field ap-proximation by the mean singlet amplitude ¯ s . This ¯ s ismeant to describe the mean-field hexagonal singlet (HS)state on the full lattice. Finally we keep only those termswhich are directly coupled to ¯ s , i.e. the terms which makethe HS state quantum fluctuate directly through tripletexcitations. With these simplifications, we get the fol-lowing triplon representation of the spins on a hexagon. S zl ≈ ¯ s (cid:104) C l (ˆ t + ˆ t † ) + (cid:0) C l ˆ t + C l ∗ ˆ t + h . c . (cid:1)(cid:105) (A3) S + l ≈ ¯ s (cid:104) C l ¯10 (ˆ t ¯10 − ˆ t † ) + C l ¯11 (ˆ t ¯11 − ˆ t † ) + C l ∗ ¯11 (ˆ t ¯1¯1 − ˆ t † ) (cid:105) (A4)where C l = (cid:104) s | S zl | t (cid:105) , C l = (cid:104) s | S zl | t (cid:105) , C l ¯10 = (cid:104) s | S + l | t ¯10 (cid:105) , and C l ¯11 = (cid:104) s | S + l | t ¯11 (cid:105) are the matrix ele-ments in terms of which the other matrix elements canbe expressed as C lmν = C l ∗ m ¯ ν and C lmν = C l ¯ mν . Moreover,the coefficients corresponding to the third and fifth spinsare related to that of the first spin as: C mν = ω ν C mν and C mν = ω ν C mν . Similarly, the coefficients correspondingto the fourth and sixth spins are related to that of thesecond spin as: C mν = ω ν C mν and C mν = ω ν C mν . Appendix B: Hamiltonian matrix and Bogoliubovdiagonalization for the HS state triplon dynamics
The H (cid:126)k in Eq. (12) is an 18 ×
18 matrix in the Nambubasis. We can write it as, H (cid:126)k = M (cid:126)k W (cid:126)k W † (cid:126)k M ∗− (cid:126)k , where M (cid:126)k and W (cid:126)k are two 9 × M (cid:126)k = D (cid:126)k A (cid:126)k A − (cid:126)k ∗ D (cid:126)k A (cid:126)k A − (cid:126)k ∗
00 0 D (cid:126)k A (cid:126)k A − (cid:126)k ∗ A (cid:126)k ∗ D (cid:126)k A (cid:126)k A (cid:126)k ∗ D (cid:126)k A (cid:126)k
00 0 A (cid:126)k ∗ D (cid:126)k A (cid:126)k A − (cid:126)k A (cid:126)k ∗ D (cid:126)k A − (cid:126)k A (cid:126)k ∗ D (cid:126)k
00 0 A − (cid:126)k A (cid:126)k ∗ D (cid:126)k (B1a) W (cid:126)k = B (cid:126)k A − (cid:126)k ∗ A (cid:126)k B (cid:126)k − A − (cid:126)k ∗ − A (cid:126)k B − (cid:126)k − A − (cid:126)k ∗ − A (cid:126)k A (cid:126)k ∗ B (cid:126)k ∗ B (cid:126)k − A (cid:126)k ∗ − A (cid:126)k B (cid:126)k − A (cid:126)k ∗ − A − (cid:126)k B (cid:126)k A − (cid:126)k B − (cid:126)k B − A − (cid:126)k B − (cid:126)k − A (cid:126)k ∗ − A − (cid:126)k B − (cid:126)k − A − (cid:126)k ∗ (B1b)The elements of these matrices are given as follows: D (cid:126)k = λ + E t J C ¯ s C C f (cid:126)k D (cid:126)k = λ + E t J C ¯ s Re ( ω C C ∗ γ (cid:126)k ) D (cid:126)k = λ + E t J C ¯ s Re ( ω C C ∗ ¯11 γ (cid:126)k ) (B2a) A (cid:126)k = J C ¯ s C C γ (cid:126)k + ω C C γ − (cid:126)k ) A (cid:126)k = J C ¯ s C C γ (cid:126)k + ω C C γ − (cid:126)k ) A (cid:126)k = J C ¯ s ( C C f (cid:126)k ) ∗ A (cid:126)k = J C ¯ s C C f (cid:126)k ) ∗ (B2b) B (cid:126)k = J C ¯ s C C f (cid:126)k B (cid:126)k = J C ¯ s C C f (cid:126)k B (cid:126)k = − J C ¯ s C C f (cid:126)k B (cid:126)k = J C ¯ s Re ( ω C C ∗ γ (cid:126)k ) B (cid:126)k = − J C ¯ s Re ( ω C C ∗ ¯11 γ (cid:126)k ) (B2c) where f (cid:126)k = cos k + cos k + cos k f (cid:126)k = cos k + ω cos k + ω cos k γ (cid:126)k = e − ik + e − ik + e ik γ (cid:126)k = ωe ik + ω e ik + e − ik (B3)for k , k , k defined in the main text [see below Eq. (6)].To diagonalize the triplon Hamiltonian H tHS ofEq. (12), as per the prescription due to Bogoliubov, wefirst multiply H (cid:126)k with the matrixΛ = (cid:18) I − I (cid:19) (B4)from the left hand side; here I is a 9 × H (cid:126)k . Its eigenvaluescome in pairs, i.e. for every positive eigenvalue thereoccurs a negative eigenvalue with same magnitude. Ofthese, the positive eigenvalues are the triplon dispersions (cid:15) i(cid:126)k in Eq. (14). Appendix C: Spin-wave analysis of the ABC model
Consider the perfect N´eel antiferromagnetic state onthe kagome-honeycomb lattice. In a unit-cell (say, AB-hexagon) at position (cid:126)R , the odd-numbered spins, as-sumed to be aligned in the + z direction, can be written1in the Holstein-Primakoff representation as S z , (cid:126)R = S − ˆ a † , (cid:126)R ˆ a , (cid:126)R , S +1 , (cid:126)R ≈ √ S ˆ a , (cid:126)R (C1a)and likewise for (cid:126)S , (cid:126)R and (cid:126)S , (cid:126)R . Correspondingly, theeven-numbered spins are pointed along − z direction.Hence, in the Holstein-Primakoff representation, S z , (cid:126)R = − S + ˆ a † , (cid:126)R ˆ a , (cid:126)R , S +2 , (cid:126)R ≈ √ S ˆ a † , (cid:126)R (C1b)and likewise for (cid:126)S , (cid:126)R and (cid:126)S , (cid:126)R . We apply this to theABC model [Eq. (1)], together with the Fourier trans-formation, ˆ a l, (cid:126)R = √ N uc (cid:80) (cid:126)k e i(cid:126)k · (cid:126)R ˆ a l,(cid:126)k for l = 1 to 6. Wefinally get the following spin-wave Hamiltonian:ˆ H SW = − S ( S + 1)( J A + J B + J C ) N uc + S (cid:88) (cid:126)k Φ † (cid:126)k h (cid:126)k Φ (cid:126)k (C2)where Φ † (cid:126)k = (cid:16) ˆ a † ,(cid:126)k ˆ a † ,(cid:126)k · · · ˆ a † ,(cid:126)k ˆ a , − (cid:126)k ˆ a , − (cid:126)k · · · ˆ a , − (cid:126)k (cid:17) is a Nambu row vector, and h (cid:126)k = (cid:18) A (cid:126)k B (cid:126)k B (cid:126)k A (cid:126)k (cid:19) is a 12 ×
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