Gutzwiller Approach for Elementary Excitations in S=1 Antiferromagnetic Chains
GGutzwiller Approach for Elementary Excitations in S = 1 Antiferromagnetic Chains
Zheng-Xin Liu
Institute for Advanced Study, Tsinghua University, Beijing, 100084, P. R. China
Yi Zhou
Department of Physics, Zhejiang University, Hangzhou, 310027, P.R. China
Tai-Kai Ng
Department of Physics, Hong Kong University of Science and Technology, ClearWater Bay Road, Kowloon, Hong Kong, China
Abstract.
In a previous paper [Phys. Rev. B 85,195144 (2012)], variational MonteCarlo method (based on Gutzwiller projected states) was generalized to S = 1 systems.This method provided very good trial ground states for the gapped phases of S = 1bilinear-biquadratic (BLBQ) Heisenberg chain. In the present paper, we extend theapproach to study the low-lying elementary excitations in S = 1 chains. We calculatethe one-magnon and two-magnon excitation spectra of the BLBQ Heisenberg chainand the results agree very well with recent data in literature. In our approach, thedifference of the excitation spectrum between the Haldane phase and the dimer phase(such as the even/odd size effect) can be understood from their different topology ofcorresponding mean field theory. We especially study the Takhtajan-Babujian criticalpoint. Despite the fact that the ‘elementary excitations’ are spin-1 magnons whichare different from the spin-1/2 spinons in Bethe solution, we show that the excitationspectrum, critical exponent ( η = 0 .
74) and central charge ( c = 1 .
45) calculated fromour theory agree well with Bethe ansatz solution and conformal field theory predictions.PACS numbers: 75.10.Pq, 75.10.Kt, 75.40.Mg, 71.10.Hf
Contents1 Introduction 22 Fermionic mean-field theory and Gutzwiller Projected states for spin S = 1 models 4 a r X i v : . [ c ond - m a t . s t r- e l ] A ug ONTENTS K = 0 . . . . . . . . . . . . . . . . 103.2 Dimer phase: Strong pairing state at K = − K = − A.1 Bogoliubov eigenstates in mean field theory . . . . . . . . . . . . . . . . 17A.2 Projected states in weak pairing phase . . . . . . . . . . . . . . . . . . . 19A.3 Projected states in the strong pairing phase . . . . . . . . . . . . . . . . 23
Appendix B Momentum of projected ground states and excited states 231. Introduction
The Haldane phase[1] reveals important physics in S = 1 spin chains and has beenprofoundly studied in literature. The Haldane phase has a disordered ground stateand a finite excitation gap. Especially, there is spin-1 / Z × Z symmetry breaking[5] and a nonzero string order[6]. These nontrivial properties showsthat the Haldane phase is distinguished from a trivial phase (such as the dimer phaseor the large D phase, where DS z is the single-ion anisotropy term) and was though tobe a topological phase. Recently, it was shown that the Haldane phase is protected bysymmetry, such as Z × Z spin rotation symmetry or time reversal symmetry, and iscalled a symmetry protected topological (SPT) phase[7]. 1-dimensional SPT phases areclassified by projective representations of the symmetry group[8] . New SPT phases asgeneralizations of the Haldane phase are realized in spin chains or ladders[9].Numerous theoretical methods had been applied to study the Haldane phase,such as effective field theory (via nonlinear sigma model plus a topological thetaterm)[1], Bosonization theory[10], Schwinger-Boson mean field theory[11], fermionicmean field theory[12, 13], and various numerical techniques such as density matrixrenormalization (DMRG)[2], exact diagonalization[14], and time-evolution-block-decimation(TEBD)[15]. Recently, variational Monte Carlo (VMC) method wasgeneralized to S = 1 systems[16] and was applied to study the Haldane phase andthe dimer phase of the S = 1 BLBQ Heisenberg chain. Although the energy obtainedis not as accurate as DMRG and TEBD, the advantage of VMC is that we can easilyread out the topological structure of ground states in different phases.The S = 1 BLBQ Heisenberg model[17, 18, 19, 20] H = (cid:88) i [ J S i · S i +1 + K ( S i · S i +1 ) ] , ( J > . (1) ONTENTS S = 1 AKLT model. It has attracted much interest in the quantummagnetism community because of its rich phase diagram. In the antiferromagneticsection (where we can set J = 1), the model contains three phases: the dimerphase at K < −
1, the Haldane phase with − < K < K >
1. In Ref. [16], we revisited this model via VMC method by using Gutzwillerprojected p -wave Bardeen-Cooper-Schrieffer (BCS) wave functions as trial ground statewavefunctions. We found that the optimized projected BCS wavefunctions are veryclose to the true ground states for model (1) in the region K ≤
1. In particular, theoptimized projected BCS state is the exact ground state at the AKLT point K = .Since the pairing symmetry is p -wave, the unprojected BCS states are classified intoweak pairing (topologically non-trivial) and strong pairing (topologically trivial) statesby their different winding numbers[12]. The topology of the BCS state is found to beimportant in distinguishing Haldane and dimer phases: after Gutzwiller projection theweak pairing states become the Haldane phase whereas the strong pairing states becomethe dimer phase. The phase transition between the Haldane phase to the dimer phaseis reflected as a topological phase transition between weak pairing and strong pairingphases.Since Gutzwiller projected BCS wavefunction is a resonating valence bound (RVB)state[21], or a spin liquid state, our VMC approach reveals that the two gapped phasesare two different classes of (fermioinc) RVB states. The topology of the S = 1 BCSmean field state reflects the pairing pattern of the resulting S = 1 RVB state: theprojected weak/strong pairing state is a long/short ranged RVB state. Here long rangemeans that the pairing amplitude a ij [see eq. (9)] between two spins is finite evenif | i − j | → ∞ , while short range means that a ij exponentially decays to zero withincreasing of distance | i − j | . It can be shown straightforwardly that the Haldane phaseis long-ranged fermionic RVB states (this is a new interpretation of the Haldane phase)while the dimer phase is short-ranged fermionic RVB states. The transition point, i.e. the Takhtajan-Babujian(TB) point, between them is a quasi-long-ranged fermionic RVBstate where a ij decays to zero in power law of | i − j | .The success of the Gutzwiller approach in describing the ground states of the BLBQmodel leads us to ask the question that how good this approach is in describing theexcited states. This question is addressed in the present paper. We shall show thatthe one- and two-magnon excitation spectra calculated numerically from the Gutzwillerprojected wavefunctions are consistent with the best available numerical results forthe corresponding excitations in the Haldane phase. Interestingly, the excitations inthe dimer phase have a very different character - there exists only odd/even-magnonexcitations if the length of the chain is odd/even. The TB phase transition point[22]between the Haldane and the dimer phases is studied carefully in this paper wherewe find that the excitation spectrum at the TB point is gapless with the criticalexponent and the central charge agree well with SU (2) Wess-Zumino-Witten fieldtheory predictions[23].This paper is organized as follows. In Section 2, we review the fermionic mean
ONTENTS S = 1 model, and discuss the general properties of the correspondingGutzwiller projected BCS states. The Gutzwiller projected excited states are studiednumerically using Monte Carlo technique and the results are presented in section 3. Ourfindings are summarized in section 4 where some general comments to our approach aregiven.
2. Fermionic mean-field theory and Gutzwiller Projected states for spin S = 1 models Our theory is based on the fermionic representation for S = 1 systems[12, 16]. Weintroduce three species of fermionic spinons c , c , c − to represent the S = 1 spinoperators as ˆ S a = C † I a C , where a = x, y, z , C = ( c , c , c − ) T and I a is the 3 by 3matrix representation of spin operator. The fermion Hilbert space is identical to thespin Hilbert space when a local particle number constraint c † c + c † c + c †− c − = 1 isimposed on the system.In this fermionic representation, the BLBQ model (1) can be rewritten as H = − (cid:88) (cid:104) i,j (cid:105) [ J ˆ χ † ij ˆ χ ij + ( J − K ) ˆ∆ † ij ˆ∆ ij ] , where ˆ χ ij = (cid:80) m =1 , , − c † mi c mj is the fermion hopping operator and ˆ∆ ij = c i c − j − c i c j + c − i c j is the spin-singlet pairing operator. This Hamiltonian can be decoupledin a mean field theory [12] by introducing short ranged order parameters χ = (cid:104) ˆ χ ij (cid:105) ,∆ = (cid:104) ˆ∆ ij (cid:105) , and the Lagrangian multiplier λ for the particle number constraint. Themean field Hamiltonian is given by H MF = (cid:88) k (cid:34)(cid:88) m χ k c † m,k c m,k − [∆ k ( c † ,k c †− , − k − c † ,k c † , − k ) + h . c . ] (cid:35) = (cid:88) m,k ≥ ε k γ † m,k γ m,k + const , (2)in momentum space where χ k = λ − J χ cos k , ∆ k = − i ( J − K )∆ sin k , and ε k = (cid:112) χ k + | ∆ k | and γ m,k are Bogoliubov eigen-particles [see Eq. (5) for details]. Themean field Hamiltonian describes a p -wave superconductor and may have nontrivialtopology. The topology of the mean field states can be more easily seen in Cartesianbases c x = ( c − − c ) / √ , c y = i ( c − + c ) / √ , c z = c , where the mean field Hamiltonianis rewriten as H MF = (cid:88) m = x,y,z (cid:88) k (cid:20) χ k c † m,k c m,k + ( 12 ∆ k c † m,k c † m, − k + h . c . ) (cid:21) = (cid:88) m,k (cid:16) c † m,k c m, − k (cid:17) H k (cid:32) c m,k c † m, − k (cid:33) , (3)where H k = ( χ k σ z + ∆ k σ y ) = ε k σσσ · nnn k . Since the unit vector nnn k falls in a circle in yz plain, it defines a map from the momentum space k (a circle) to another circle. In ONTENTS N winding = 12 π (cid:90) π − π ˆ x · ( nnn k × ∂ k nnn k ) dk. (4)If ∆ (cid:54) = 0, the topology of the mean-field ground state | G (cid:105) MF is determined by χ k . If | λ | < | J χ | (see Fig. 1), the state has winding number 1 for each species of fermionsand is called a weak pairing state (i.e. a topological superconductor). On the otherhand, if | λ | > | J χ | (see Fig. 2), the state has winding number 0 and is called a strongpairing state (i.e. a trivial superconductor).The mean field Hamiltonian (2) has a global Z symmetry, so the mean field statehas conserved fermion parity. Furthermore, since mean field parameters are fluctuating,the fermions are effectively coupling to a Z gauge field. Particularly, in 1D the onlyeffect of the spacial component of the Z gauge field is the global Z flux, namely, thefermion boundary conditions. We will discuss about the relation between fermion parityand boundary conditions in more detail later. The mean field ground state | G (cid:105) MF is a BCS type wavefunction. After Gutzwillerprojection, the state | ψ (cid:105) = P G | G (cid:105) MF provides a trial ground state for the Hamiltonian (1)(see Appendix A for details). The parameters χ, ∆ , λ are determined by minimizing theenergy of the projected states E trial = (cid:104) ψ | H | ψ (cid:105) / (cid:104) ψ | ψ (cid:105) (the details of the calculations canbe found in Ref. [16]). It was found that the projected weak pairing states corresponds tothe Haldane phase and the projected strong pairing states corresponds to the dimerizedphase.A special property of the Gutzwiller projection for spin-1 systems has to beemphasized here. As mentioned above, S = 1 fermionic mean field states are p -wavesuperconductors, so there are two different topological sectors. The fermion boundaryconditions have different consequences in different topological sectors. The main issueis the fermion parity. Since the total number of fermions in the system is equal to thenumber of lattice sites by construction, therefore the fermion parity of the ground stateis even/odd for chains with even/odd number of sites. Only the mean field states withproper fermion parity can survive after Gutzwiller projection.It was pointed out in Ref. [24, 16] that for a weak pairing state, the fermionparity depends on the boundary condition: it is even/odd under anti-periodic/periodicboundary condition. Roughly speaking, this effect is an analogy of 2D band insulatorswith nonzero winding number (i.e. the Chern number C ), where a 2 π flux causes fermionnumber changing by C owning to Hall effect. In our case the flux is quantized in uniteof π because of pairing. If the winding number is nonzero, then a global π flux (whichswitches the boundary condition) will cause fermion parity change. More precisely,this effect can be easily understood from the dispersion χ k , as shown in Fig. 1. Wefirstly consider the case L =even. Under anti-periodic boundary condition, since ∆ k isalways nonzero, the fermions c m,k , c m, − k are paired into Cooper pairs, so the fermion ONTENTS k vanishes at k = 0 and k = π ,so the fermion modes c m,k =0 and c m,k = π (where m = 1 , , −
1) are unpaired. Since thechemical potential | λ | < | J χ | , the three fermion modes c m,k =0 have negative energyand the other three c m,k = π have positive energy. The three modes c m,k =0 are occupiedin the ground state and therefore the fermion parity is odd. The same results can beobtained for the case L =odd using similar arguments. Therefore when the length of thechain L =even/odd, only the anti-periodic/periodic boundary condition survives afterGutzwiller projection in the weak-pairing phase. As a result, the ground state of aclosed chain in the Haldane phase is unique.In contrast, in the strong pairing phase, the fermion parity is independent on thefermion boundary conditions. We assume L =even first. Under anti-periodic boundarycondition, the fermion parity is obviously even. Under periodic boundary conditions,the unpaired fermion modes with k = 0 and k = π are unoccupied since they havepositive energies, consequently the fermion parity is also even. In other words, meanfield states with both boundary conditions survive after Gutzwiller projection and theyhave the same energy in thermodynamic limit. So the ground state of the dimer phaseis doubly degenerate[16]. If L =odd, the mean field ground state also have even fermionparity under both boundary conditions, but they vanish after Gutzwiller projection.The true ground state of the dimmer model is constructed by Gutzwiller projection ofmean field state with one fermion excitation. In that case the ground state is not aspin-singlet. Since anti-periodic boundary condition is equivalent to a global Z flux throughthe ring formed by the spin chain, we will denote the ground state with anti-periodicboundary condition as | π -flux (cid:105) and denote the one with periodic boundary conditionas | -flux (cid:105) in the following. The subtle boundary condition effect also exists for theexcited states and leads to important distinction between the excitation spectrums inthe Haldane and dimer phases as we shall see in the following.
In BCS superconductors, excitations are formed by adding quasi-particles obtained fromthe Bogoliubov-de Gennes equations to the BCS ground state wavefunction. We mayadd arbitrary number of quasi-particles to form excited states since the system allowsarbitrary fermion number. We shall assume in the following that (low energy) excitationsin the S = 1 spin liquids can be formed by Gutzwiller projecting the excited states ofthe corresponding BCS superconductor. The fixing of fermion parity in spin systemsimposes a constraint on the excited states that can be constructed in this approach.In the fermionic mean field theory of spin-1 / s -wave), the requirement of fixed fermion parity implies that excited states canbe formed only by Gutzwiller Projecting BCS excited states with even number ofquasi-particle excitations. The situation is similar for spin-1 mean field theory (wherethe pairing symmetry is p -wave) in the strong pairing phase. However, the weak ONTENTS
Let us focus on the weak pairing (Haldane) phase. First weconsider a spin excitation formed by simultaneously switching the boundary conditionand adding a Bogoliubov quasi-particle to the system. We shall call the excitationa one-magnon excitation[25]. The one-magnon creation operator with S z = m andmomentum k can be written as γ † m,p ˆ W , where p = k − π and ˆ W is the boundary-twisting operator which switches the periodic boundary condition to anti-periodic andvice versa. Notice that the quasi-particle momentum changes by π after the boundarycondition is switched (see Appendix B for details). We shall see in next section thatafter projection the state P G γ † m,p ˆ W | G (cid:105) MF corresponds to the one-magnon excitationdiscussed in the literature[2, 15]. Notice that the mean-field energy of this excitationhas minimum at p = 0. This explains why the minimal magnon gap opens at k = π . Thetwo-magnon excitation can be obtained by acting the one-magnon creation operator onthe mean field ground state twice before the Gutzwiller projection. Notice the boundarycondition is restored ( ˆ W = I ) for two-magnons.In the following we shall provide more details of the one- and two-magnonexcitations.We first consider the case L =even integer. In this case the ground state is a spin-singlet given by [see Fig.1(a)] | ground (cid:105) = P G | π -flux (cid:105) . A single magnon is a spin-1 excitation represented by [see Fig.1(b)] | (1 , m ); p + π (cid:105) = P G γ † m,p ˆ W | π -flux (cid:105) = P G γ + m,p | (cid:105) , where | (1 , m ); p + π (cid:105) indicates that the one-magnon carries spin quantum numbers( S, m ) = (1 , m ) and lattice momentum p + π . The one-magnon state | (1 , m ); p + π (cid:105) is orthogonal to the ground state | ground (cid:105) because it carries both nonzero spin andmomentum. The energy-momentum dispersion of the one-magnon spectrum will bediscussed in next section.The two-magnon excitations can be constructed similarly and are denoted by | ( S, m ); p, q (cid:105) , where ( S, m ) are the spin quantum numbers and p, q are the momentacarried by the two magnons. Notice that since each magnon carries spin-1, the totalspin of two magnons can be S = 0 , S = 0 , , m = 0 are given by | (0 , p, q (cid:105) = P G ( γ † ,p γ †− ,q + γ †− ,p γ † ,q − γ † ,p γ † ,q ) | π -flux (cid:105) , | (1 , p, q (cid:105) = P G ( γ † ,p γ †− ,q − γ †− ,p γ † ,q ) | π -flux (cid:105) , | (2 , p, q (cid:105) = P G ( γ † ,p γ †− ,q + γ †− ,p γ † ,q + 2 γ † ,p γ † ,q ) | π -flux (cid:105) . We have dropped some unimportant normalization constants in writing down the abovestates. Obviously, the two-magnon states are orthogonal to each other because they
ONTENTS −1 0 1−0.500.51 k/ π χ k / J −1 0 1−0.500.51 k/ π χ k / J −1 0 1−0.500.51 k/ π χ k / J −1 0 1−0.500.51 k/ π χ k / J (a) L=even, a.p.b.c (b) L=even, p.b.c(c) L=odd, a.p.b.c (d) L=odd, p.b.c Figure 1. (Color online) Dispersion of χ k in the weak pairing phase. The red linesshow the chemical potential. The dispersion will open a gap if we turn on the paringterm ∆ k . The asterisks linked by doted lines show the Cooper pair of spinons. Thedots at k = 0 and k = π are marked in red color, meaning that the pairing ∆ k vanishesat these points. The black solid/hollow dots represent occupied/unoccupied unpairedspinons coming from a broken Cooper pair. (a) L =even, anti-periodic boundarycondition(a.p.b.c), no broken Cooper pairs ( ground state ); (b) L =even, periodicboundary condition(p.b.c), one broken Cooper pair ( one-magnon excited state ); (c) L =odd, a.p.b.c, one broken Cooper pair ( one-magnon excited state ); (d) L =odd,p.b.c, no broken Cooper pairs ( ground state ). carry different spin-quantum numbers. It can be also shown that they are orthogonalto the ground state and the one-magnon states[26]. For a given momentum k = p + q ,the total energy E k depends on the momentum distribution ( p, q ) of the two magnonsand the energy-momentum spectrum of the two-magnon states form continuums.The L =odd integer situation can be constructed similarly as for even chains exceptthat | (cid:105) ⇐⇒ | π -flux (cid:105) in writing down the ground and excited state wavefunctions. The fermion parity of the spin chain is independent ofboundary conditions in the strong pairing phase. As a result the ground states aredoubly degenerate and the excitation spectrums are different for chains with even andodd length L ’s.We consider first the case of L =even integer chains. In this case, the ground state ONTENTS −1 0 100.511.52 k/ π χ k / J −1 0 100.511.52 k/ π χ k / J −1 0 100.511.52 k/ π χ k / J −1 0 100.511.52 k/ π χ k / J (d) L=odd, p.b.c(a) L=even, a.p.b.c(c) L=odd, a.p.b.c (b) L=even, p.b.c Figure 2. (Color online) The dispersion of χ k in the strong pairing phase. (a) L =even, a.p.b.c, no broken Cooper pairs( ground state ); (b) L =even, p.b.c, no brokenCooper pair ( another ground state ); (c) L =odd, a.p.b.c, one broken Cooper pair( one-magnon excited state ); (d) L =odd, p.b.c, one broken Cooper pair( degenerate one-magnon excited state ). In contrast to the weak pairing phase (see Fig.1), the spinonsat k = 0 in subfigures (b) and (d) are unoccupied. This is an important differencebetween the weak pairing phase and the strong pairing phase. wavefunctions are given by [see Fig.2(a),(b)] | ground (cid:105) = P G | π -flux (cid:105) , | ground (cid:105) = P G | (cid:105) ;where | ground (cid:105) carries 0-momentum and | ground (cid:105) carries π -momentum. One-magnonexcitations do not exist in this case since the fermion parity cannot be changed byswitching boundary condition. We can only construct two-magnon excitations.Similar to the ground states, the two-magnon spectra are also doubly degenerate.For simplicity, we only consider excitations above the ground state with π -flux.Employing the same notation as above, we find that the | ( S, m = 0); p, q (cid:105) states aregiven by | (0 , p, q (cid:105) = P G ( γ † ,p γ †− ,q + γ †− ,p γ † ,q − γ † ,p γ † ,q ) | π -flux (cid:105) , | (1 , p, q (cid:105) = P G ( γ † ,p γ †− ,q − γ †− ,p γ † ,q ) | π -flux (cid:105) , | (2 , p, q (cid:105) = P G ( γ † ,p γ †− ,q + γ †− ,p γ † ,q + 2 γ † ,p γ † ,q ) | π -flux (cid:105) . The two magnon excitations form continuum in the energy-momentum spectrumas in the Haldane phase.Another way to understand why one-magnon excitations do not exist for L =even ONTENTS k = 0 have negativeenergy in the weak pairing phase and have positive energy in the strong pairing phase.In the one-magnon excited state of the weak pairing phase (p.b.c), one Bogoliubov quasi-particle is excited whereas the three spinon states at k = 0 are filled. To construct asimilar state in the strong pairing phase, we have to occupy the three spinon states at k = 0 which corresponds to exciting three (gapped) magnons. As a result, a one-magnonexcited state of the Haldane phase becomes a four-magnon excited state in the dimerphase.The L =odd integer chains have a different character. First of all, the “ground”state of the system is not a spin singlet but is a spin-triplet with wavefunctions [seeFig.2(c),(d)] | (1 , m ); p (cid:105) = P G γ † m,p | (cid:105) , | (1 , m ); p + π (cid:105) = P G γ † m,p | π -flux (cid:105) ;with m = 0 , ± p = 0 for | (1 , m ); p (cid:105) and p = π for | (1 , m ); p + π (cid:105) . The energy ofthe system changes continuously and forms a one-magnon excitation spectrum whenwe change p . This can be easily understood, since in the dimer phase, the spins formsinglet pairs (or dimers) at the ground state. When L =odd, not all the spins can formpairs and there must exist odd number of magnons in the system including the groundstate.
3. Numerical results
In this section we discuss our numerical results for various spin excitations weconstructed in the previous section. When L is large, the expectation values of physicalquantities in a Gutzwiller projected state can be calculated with Monte Carlo (MC)method[16, 27]. K = 0We first consider the Heisenberg model ( K = 0). Fig. 3 shows the ground state and theone-magnon excitations for two different chains with chain length L =100 and L =99.We note that the two excitation spectrums almost coincide with each other, showingthat even or odd chain length makes little difference in the Haldane phase. The lowestenergy one-magnon excitation costs energy (0 . ± . J and carries momentum k = π .The maximum of the one-magnon dispersion locates near k = 0 . π . These features agreevery well with the numerical results in Ref. [2, 15] for the one-magnon excitations (wherethe spin gap is 0 . J ).The two-magnon excitations form a continuum spectrum, as shown in the filledarea in Fig.4. The energy cost for the minimal two-magnon excitation is roughly twicethe spin gap. The one-magnon curve merges into the two-magnon continuum below k = 0 . π , suggesting that a single-magnon excitation will decay into two magnons if its ONTENTS k/ π E k L=100 1−magnon excitationsL=99 1−magnon excitationsground state
Figure 3. (Color online) The dispersion of one-magnon excitations for the Heisenbergmodel with length L =100 and L =99. The ground sate energy has been set to 0 andthe energy scale is J = 1. The data for L =100 almost coincide with that of L =99.The averaged one-magnon gap is (0 . ± . J , which opens at k = π . k/ π E k Figure 4. (Color online) Low energy excitations of the Heisenberg model ( L = 100).The red solid circle shows the ground state energy. The blue dotted line decorated withhollow circles shows the one-magnon dispersion. The filled area shows the two-magnon-excitation continuum. The two excited magnons can form total spin S = 0 , , ONTENTS k = 0 . π . This result agrees also with the numerical result forthe two-magnon spectrum in Ref. [2, 15].Depending on the symmetry under exchanging the spin momentum of thetwo magnons, the total spin of two magnons can be either 0,2 (symmetric) or 1(antisymmetric). Fig.4 shows that the two-magnon energy bounds for total spin S = 0 , , k = π . This suggests that there is almost no interaction between the twomagnons except when their total momentum is close to k = π . Near k = π , the S = 1channel is lower in energy then the S = 0 , S = 1 channel while weakly repulsive for the S = 0 , K = − k/ π E k k/ π E k (b)(a) Figure 5. (Color online) The excitations in the strong pairing phase ( K = − L = 99. The one-magnon excitations. Notice that a singlet (dimer) ground statecannot be constructed for odd L . The two solid dark dots show the minimal three-magnon excitation energy at k = 0 and k = π separately; (b) L = 100. The red soliddot shows the ground state energy and filled area is the two-magnon continuum. We shall study spin excitations in the strong pairing phase at K = −
3. As wehave pointed out in last section, the L =even and L =odd chains have quite differentproperties. There exist only even/odd-magnon excitations for even/odd L .First we consider even L . Fig. 5(b) shows the two-magnon continuum for L = 100.There is an obvious gap (of order 1 . J ) between the ground state and the two-magnoncontinuum. The two magnons can form states with total spin S =0,1 or 2. The energydifferences between states with different total spin S are small as is clear from the figureexcept at the points k = 0 , π , indicating that the two magnons almost do not interactwith each other except when their total momentum is close to k = 0 or π , similar to theHaldane phase. ONTENTS L . Recall that the singlet ground state does not exist for odd L and the lowest energy states are the states | (1 , m ); p (cid:105) = P G γ † m,p | (cid:105) with p = 0,or | (1 , m ); p + π (cid:105) = P G γ † m,p | π -flux (cid:105) with p = π . The energy dependence of | (1 , m ); p (cid:105) as function of p is shown in Fig. 5(a). We indicate in the figure also the (minimal)3-magnon excitation energies at points k = 0 and k = π . The finite difference in energybetween the one- and three- magnon states indicates that the two-magnon excitationshave a finite gap of order 1 . J . K = − k/ π E k k/ π E k
500 1000012 L (a) (b) Figure 6. (Color online) The one-magnon excitations for the TB point. The resultsfor L =even and L =odd have a little difference. (a) L = 199, the one-magnon gapis closing at k = 0 and k = π ; (b) L = 200, the one-magnon gap closes at k = π ,while the gap at k = 0 is finite. The insect shows that the gap at k = 0 vanishes inpower low L − . . So in thermodynamic limit the one-magnon excitations are gaplessat k = 0 and k = π . Lastly we consider the TB critical point at K = − λ − χ ≈ L =odd and L =even chains. Fig. 6(a) shows that for L = 199, the spinons atmomentum k = 0 and k = π are gapless. Fig. 6(b) shows the data for L = 200,the excitation gap closes at k = π but remains finite at k = 0. However, a finite sizescaling analysis (insert) shows that the gap at k = 0 vanishes in power low of the chainlength L . Thus, we expect that in thermodynamic limit, the one-magnon excitationsare gapless at both k = 0 and k = π .The two-magnon continuum for L = 100 is shown in Fig.7. We expect that theone-magnon dispersion will coincide with the lower energy bound of the two-magnoncontinuum in thermodynamic limit.We now compare our result with the Bethe ansatz solution[22]. In our approach,the elementary excitations are spin-1 magnons whereas the elementary excitations are ONTENTS η and the central charge c from the projected groundstate. The results are shown in Fig.8. The critical exponent is obtained by calculatingthe spin-spin correlation, |(cid:104) S i · S i + x (cid:105)| ∝ [sin( πxL )] − η , k/ π E k Figure 7. (Color online) Excitations at the TB point ( L = 100). The blue dottedline shows the one-magnon dispersion (or the lower bound of four-magnon continuum)and the filled area is the two-magnon continuum of our MC data. The red dash-dottedlines are the boundaries of the two-spinon continuum of the Bethe solution, here wehave enlarged the energy scale 1.1 times to fit our data (the inconsistency of energyscales may be caused by finite size effect or systematic error). −6 −4 −2 0−2−10123 log[sin( π x/L)] − l og [ | S ( ) ⋅ S ( x ) | ] −1 −0.5 00.511.522.5 log[sin( π x/L)]/4 S ( ) ( x ) MC datafitting η =0.74 MC datafitting c=1.45 (a) (b) Figure 8. (Color online) The critical behavior of the TB model ( L = 200). (a) criticalexponent is η = 0 . ± .
01 fitted by |(cid:104) S i · S i + x (cid:105)| ∝ [sin( πxL )] − η ; (b) The central chargefitted by S (2) ( x ) = c log[sin( πxL )] + const is c = 1 . ± . ONTENTS S (2) ( x ) = c Lπ sin( πxL )] + const , where S (2) ( x ) is defined as e − S (2) ( x ) = Tr[ ρ ( x ) ] and ρ ( x ) is the reduced densitymatrix for a x -site subsystem in a L -site chain under periodic boundary condition.Tr[ ρ ( x ) ] can be calculated with MC technique[29, 30]. We note that c = 0 forgapped states such as the Haldane phase and the dimer phase, since the Renyi entropy S (2) ( x ) saturate to a finite constant in large x limit. For the TB model, our results η = 0 . ± . , c = 1 . ± .
02 agree very well with SU (2) Wess-Zumino-Witten fieldtheory predictions η = 0 . , c = 1 . real spin excitation spectrum. We note that the dispersion of thespin-1/2 spinon in the Bethe Ansatz solution is given by ε ( k ) = 2 π sin | k | [22], and theexcitation spectrum is gapless at k = 0 and k = π in the Bethe-Ansatz solution. Theone-magnon dispersion in Fig. 6 is also gapless at k = 0 and k = π , and the shapeis close to a sine function, in agreement with the Bethe solution. A pair of spin-1/2spinons form a spin-singlet continuum and a spin-triplet continuum in the Bethe Ansatzsolution. The two continuums are degenerate in energy. In our approach, the spin-0two-magnon continuum and the spin-1 two-magnon continuum are almost degenerate,and correspond to the two continuums of the Bethe solution mentioned above (also seeFig.7).We note also that a one-magnon excited state can also be viewed as a four-magnonexcitation in our approach (recall that if one approaches the critical point from theHaldane phase, this state is viewed as a one-magnon state; but if one approaches fromthe dimer phase, this state is viewed as a four-magnon state), i.e. the one-magnondispersion curve is nothing but the lower bound of the four-magnon continuum andmay be constructed from the four- or more-spin-1/2-spinon continuum. Furthermore,the spin-2 two-magnon continuum may correspond to part of the four(or more)-spin-1/2-spinon continuum. These observations suggest that the relation between the S = 1magnons in the Gutzwiller projected wavefunction approach and the S = 1 /
4. Conclusion and discussion
Conclusion
To summarize, we have studied in this paper the low energy spin excitationsin the Haldane ( K = 0) and dimer ( K = −
3) phases [including the TB criticalpoint ( K = − ONTENTS Z flux and a spinon in our Gutzwiller projected wavefunction approach. Thecorresponding two-magnon excitation spectrum computed in the Gutzwiller projectedwavefunction also agrees with earlier numerical works and we show evidence that themagnons are weakly scattering with each other (absence of confinement).The excitation spectrum in the dimer phase is computed (to our knowledge, it isthe first time that the energy spectrum of the dimer phase is studied) where we pointout the qualitative differences between L =odd and L =even chains. At the criticalpoint (the TB model), the projected dispersion is gapless at both k = 0 and k = π . Thecritical exponent η = 0 .
74 and the central charge c = 1 .
45 we obtained agree very wellwith literature[15, 23, 31]. k/ π E k k/ π E k k/ π E k (a) (c)(b) Figure 9. (Color online) The mean field dispersion of (a) the weak pairing phase,(b) the critical point, (c) the strong pairing phase. Comparing with Figs. 3,6,5(a),one finds that the one-magnon energy dispersions are dramatically changed after theGutzwiller projection.
We note that the one-magnon dispersions in Figs. 3, 6, 5(a) are qualitativelydifferent from the corresponding mean field dispersions before Guzwiller projection (seeFig.9). In the weak paring phase, the minimal mean field gap opens at k = 0, butafter projection, the minimal one-magnon gap opens at k = π . In the strong pairingphase, the mean field dispersion is asymmetric by reflection along k = 0 . π , while afterprojection the one-magnon curve becomes more symmetric. Especially, the mean fielddispersion is gapless only at k = 0 at the TB point, but the magnons are gapless at both k = 0 and k = π after Gutzwiller projection. These features indicates that only themean-field states after Gutzwiller projection correctly describe the physical propertiesof the spin system (1). Discussion
The existence of one-magnon excitation in the spin-one BLBQHeisenberg spin chain reflects a fundamental difference between integer and half-odd-integer spin systems: for integer spin systems, it is possible to form a spin-singlet statefor a system with both even and odd number of sites whereas for half-odd-integer spinsystems, singlet state exists only in systems with even number of sites. For a spin chain
ONTENTS L , the one-magnon excitation in the Haldane phase can be understood withthe single-mode approximation: the ground state (which is a L -site singlet state) isreconstructed into a ( L − AppendicesA. Details for Gutzwiller projected states
A.1. Bogoliubov eigenstates in mean field theory
In momentum space, the mean field Hamiltonian (2) can be diagnolized into Bogoliubovparticles: H = (cid:88) m,k ≥ ε k γ † m,k γ m,k , m = 1 , , − , (5) γ ,k = u k c ,k + v ∗ k c † , − k ,γ † , − k = u k c † , − k − v k c ,k , ONTENTS γ ,k = u k c ,k − v ∗ k c †− , − k ,γ †− , − k = u k c †− , − k + v k c ,k , where ε k = (cid:112) χ k + ∆ k , u k = cos θ k , v k = i sin θ k and tan θ k = i ∆ k χ k .In the following, we will provide some eigen states of above Hamiltonian.(A) Ground state ( E = 0), | G (cid:105) MF = exp { (cid:88) m,n,k a k Γ mn c † m,k c † n, − k }| vac (cid:105) = (cid:89) k (1 + a k Γ mn c † m,k c † n, − k ) | vac (cid:105) = (cid:89) k (1 + a k c † ,k c †− , − k ) (cid:89) q> (1 − a q c † ,q c † , − q ) | vac (cid:105) (6)where a k = v ∗ k u k and Γ mn is the CG coefficient: Γ , − = Γ − , = − Γ , = 1 and othersequal to zero.(B) Excited states by breaking the pair c † ,p c † , − p :1), one-spinon excitation ( E = ε p , two-fold degenerate ) | p (cid:105) MF = γ † ,p | G (cid:105) MF = c † ,p | G (cid:105) MF , | − p (cid:105) MF = γ † , − p | G (cid:105) MF = c † , − p | G (cid:105) MF , E = 2 ε p ) | , p, − p (cid:105) MF = (1 + a − p c † ,p c † , − p ) | G (cid:105) (cid:48) MF . where | m ; p (cid:105) MF means the excited spinon carry spin momentum S z = m and latticemomentum p , and | G (cid:48) (cid:105) MF = (cid:89) k (1 + a k c † ,k c †− , − k ) (cid:89) q> ,q (cid:54) = p (1 − a q c † ,q c † , − q ) | vac (cid:105) . (C) Excited states by breaking the pair c † ,p c †− , − p :1), one-spinon excitation ( E = ε p , two-fold degenerate ) | p (cid:105) MF = γ † ,p | G (cid:105) MF = c † ,p | G (cid:105) MF , | − − p (cid:105) MF = γ †− , − p | G (cid:105) MF = c †− , − p | G (cid:105) MF , E = 2 ε p ) | , − p, − p (cid:105) MF = (1 − a − p c † ,p c †− , − p ) | G (cid:105) ” MF . (7)where | G (cid:48)(cid:48) (cid:105) MF = (cid:89) k (cid:54) = p (1 + a k c † ,k c †− , − k ) (cid:89) q> (1 − a q c † ,q c † , − q ) | vac (cid:105) . Similarly, we can obtain more excited states by breaking more BCS pairs. However,when performing Gutzwiller projection, there will be a subtle problem in the weakpairing phase owning to the dependence of fermion parity on boundary conditions.
ONTENTS A.2. Projected states in weak pairing phase
Now we consider the mean field low energy excited states in the weak pairing phase,and their Gutzwiller projection. We will treat L =even and L =odd separately.The following property of pfaffian is useful in case that not all fermions are paired.Assuming A is an n -dimensional skew symmetric matrix, then we have,Pf A = n (cid:88) i =1 ( − i a i Pf A (cid:48) = n (cid:88) i =1 ( − i a ni Pf A (cid:48)(cid:48) , (8)where A (cid:48) ( A (cid:48)(cid:48) ) mean A with the first( n th) and the i th row and column are removed. L =even .1), Ground stateThe fermion parity is even under anti-periodic boundary condition (we note it as | π -flux (cid:105) ) and odd under periodic boundary condition ( | (cid:105) ). Only the former survivesafter Gutzwiller projection and will be the ground state: | G (cid:105) MF = | π -flux (cid:105) = exp { (cid:88) k a k c † ,k c †− , − k } exp {− (cid:88) q> a q c † ,q c † , − q }| vac (cid:105) = (cid:89) { i,j } (1 + a ij c † ,i c †− ,j ) (cid:89) { r,s } (1 − a rs c † ,r c † ,s ) | vac (cid:105) . where a − k = − a k and a ij = 1 L (cid:88) k a k sin[ k ( i − j )] . (9)Projected mean field ground state is the approximate ground state: | ground (cid:105) = P G | G (cid:105) MF (10)= (cid:88) α sgn( i , ..., i n , j , ..., j n , r , ..., r n )Pf A ( α )Pf B ( α ) | α (cid:105) where i , ..., i n ( j , ..., j n , r , ..., r n ) are the positions of the 1(-1,0)-component spins inconfiguration α . Obviously, 2 n + n = L . The sign sgn( i , ..., i n , j , ..., j n , r , ..., r n ) =( − P , where P is the the permutation number by permuting i , ..., i n , j , ..., j n , r , ..., r n into the standard order 1 , , ..., L . The matrices A ( α ) and B ( α ) are defined as A ( α ) = a i j ... a i j n ... . . . ... a i n j ... a i n j n a j i ... a j i n ... . . . ... a j n i ... a j n i n , (11) ONTENTS B ( α ) = − a r r ... − a r r n − a r r ... − a r r n ... ... . . . ... − a r n r − a r n r ... . (12)2), 1-magnon excited statesA 1-spinon excited mean field state is given as | (1 , p (cid:105) MF = γ † ,p ˆ W | π -flux (cid:105) = γ † ,p | (cid:105) = (cid:89) k (cid:54) =0 (1 + a k c † ,k c †− , − k ) c † , c †− , (cid:89) q> (1 − a q c † ,q c † , − q ) c † ,p c † , | vac (cid:105) , where | ( S, m ); p (cid:105) MF means that the spinon carries spin-quantum number (
S, m ) andmomentum p , and ˆ W is the boundary-condition twisting operator that switches periodicboundary condition to anti-periodic boundary condition and vice versa (namely, ˆ W addsa global Z flux through the whole system).Gutzwiller projected mean field 1-spinon excited states are approximate 1-magnonexcited states: | (1 , p + π (cid:105) = P G γ † ,p ˆ W | π -flux (cid:105) = (cid:88) α sgn( i , ..., i n , j , ..., j n , r , ..., r n )Pf A ( α )Pf B p ( α ) | α (cid:105) , (13)where A ( α ) = A ( α ) − − − ... − ... − ... − − ... − , (14) B p ( α ) = e ipr e ipr B ( α ) ... ... e ipr n − e ipr − e ipr ... − e ipr n − − ... − . Notice that momentum of the magnon is equal to the sum of the spion and the extra Z flux, which gives p + π (for details, see Section B).3), 2-magnon excited statesNow we consider two-spinon (or two-magnon) excited states. We can either excitetwo c spinons or one c spinon plus one c − spinon. But these states do not respectthe symmetry of the spin Hamiltonian since they do not carry correct spin quantum ONTENTS S =0,1,2) of the two magnons (the S =0,2 statesare symmetric under exchanging the spin quantum numbers of the two spinons, whilethe S = 1 states are anti-symmetric under exchanging the spin quantum numbers ofthe two spinons), an excited eigenstate is a superposition the two states listed above.Owning to the degeneracy, we only consider the ( S, | (0 , p, q (cid:105) MF = ( γ † ,p γ †− ,q + γ †− ,p γ † ,q − γ † ,p γ † ,q ) | G (cid:105) MF , | (2 , p, q (cid:105) MF = ( γ † ,p γ †− ,q + γ †− ,p γ † ,q + 2 γ † ,p γ † ,q ) | G (cid:105) MF , | (1 , p, q (cid:105) MF = ( γ † ,p γ †− ,q − γ †− ,p γ † ,q ) | G (cid:105) MF . (15)Above we have assumed that p + q (cid:54) = 0. If p + q = 0, then the corresponding 2-spinonexcited state should be constructed as mentioned in appendix A.1.The corresponding Gutzwiller projected states are listed below: | (0 , p, q (cid:105) = P g | (0 , p, q (cid:105) MF= (cid:88) α sgn( i , ..., i n , j , ..., j n , r , ..., r n ) , × (cid:2) Pf A spq ( α )Pf B ( α ) − Pf A ( α )Pf B pq ( α ) (cid:3) | α (cid:105)| (2 , p, q (cid:105) = P g | (2 , p, q (cid:105) MF= (cid:88) α sgn( i , ..., i n , j , ..., j n , r , ..., r n ) × (cid:2) Pf A spq ( α )Pf B ( α ) + 2Pf A ( α )Pf B pq ( α ) (cid:3) | α (cid:105) , | (1 , p, q (cid:105) = P g | (1 , p, q (cid:105) MF= (cid:88) α sgn( i , ..., i n , j , ..., j n , r , ..., r n ) × Pf A apq ( α )Pf B ( α ) | α (cid:105) . (16)where A ( α ) and B ( α ) are given in (11) and (12) respectively, and A spq ( α ) = e ipi e iqi ... ... A ( α ) e ipi n e iqi n e ipj e iqj ... ... e ipj n e iqj n − e ipi ... − e ipi n − e ipj ... − e ipj n − e iqi ... − e iqi n − e iqj ... − e iqj n ,A apq ( α ) = ONTENTS e ipi e iqi ... ... A ( α ) e ipi n e iqi n − e ipj e iqj ... ... − e ipj n e iqj n − e ipi ... − e ipi n e ipj ... e ipj n − e iqi ... − e iqi n − e iqj ... − e iqj n ,B pq ( α ) = e ipr e iqr e ipr e iqr B ( α ) ... ... e ipr n e iqr n − e ipr − e ipr ... − e ipr n − e iqr − e iqr ... − e iqr n .L =odd .1), Ground state | G (cid:105) MF = | - flux (cid:105) = exp { (cid:88) k (cid:54) =0 a k c † ,k c †− , − k } exp { (cid:88) q> a q c † ,q c † , − q } c † , c †− , c † , | vac (cid:105) = (cid:89) { i,j } (1 + a ij c † ,i c †− ,j ) (cid:89) { rs } (1 − a rs c † ,r c † ,s ) × ( (cid:88) i (cid:48) c † ,i (cid:48) )( (cid:88) j (cid:48) c †− ,j (cid:48) )( (cid:88) r (cid:48) c † ,r (cid:48) ) | vac (cid:105) , (17) where a ij is defined in (9).The projected mean field ground state is given by P G | G (cid:105) MF = (cid:88) α sgn( i , ..., i n , j , ..., j n , r , ..., r n ) × Pf A ( α )Pf B ( α ) | α (cid:105) (18) where the matrices A is given in (14) and B is defined as ( n =odd) B ( α ) = − a r r ... − a r r n − a r r ... − a r r n ... ... . . . ... ... − a r n r − a r n r ... − − ... − . | (1 , p (cid:105) MF = γ † ,p ˆ W | - flux (cid:105) = γ † ,p | π - flux (cid:105) = (cid:89) k (1 + a k c † ,k c †− , − k ) p (cid:89) q> (1 − a q c † ,q c † , − q ) c † ,p | vac (cid:105) , (19) ONTENTS After projection, the excited state is given by | (1 , p + π (cid:105) = P G γ † ,p ˆ W | - flux (cid:105) = (cid:88) α sgn( i , ..., i n , j , ..., j n , k , ..., k n )Pf A ( α )Pf B p ( α ) | α (cid:105) where the matrices A ( α ) is given in (11) and B p ( α ) is defined as B p ( α ) = − a r r ... − a r r n e ipr − a r r ... − a r r n e ipr ... ... . . . ... ... − a r n r − a r n r ... e ipr n − e ipr − e ipr ... − e ipr n . L = even, we have | (0 , p, q (cid:105) = P g | (0 , p, q (cid:105) MF = (cid:88) α sgn × (cid:2) Pf A s ,pq ( α )Pf B ( α ) − Pf A ( α )Pf B ,pq ( α ) (cid:3) | α (cid:105) , | (2 , p, q (cid:105) = P g | (2 , p, q (cid:105) MF = (cid:88) α sgn × (cid:2) Pf A s ,pq ( α )Pf B ( α ) + 2Pf A ( α )Pf B ,pq ( α ) (cid:3) | α (cid:105) , | (1 , p, q (cid:105) = P g | (1 , p, q (cid:105) MF = (cid:88) α sgn × Pf A a ,pq ( α )Pf B ( α ) | α (cid:105) , where sgn = sgn( i , ..., i n , j , ..., j n , r , ..., r n ) , and the matrices A and B are definedsimilar to previous cases and will not be repeated here.A.3. Projected states in the strong pairing phase For L =even, single-spinon excitations (or generally odd number of spinon excitations)do not exist. And the method to obtain projected two-spinon excited states are similarto the weak pairing phase. When L =odd, even-spinon excitations (including ‘0-spinonexcitation’ state) are not allowed, and only odd-spinon excitations exist. The methodto obtain projected 1-spinon excited states are similar to the weak pairing phase, exceptthat both P G γ † ,p | π -flux (cid:105) and P G γ † ,p | (cid:105) are allowed here. B. Momentum of projected ground states and excited states
Firstly, let us consider the Heisenberg model in the case L =even. The groundstate is the projected mean field ground state with anti-periodic boundary condition | ground (cid:105) = P G | π -flux (cid:105) , which yields a i,j + L = − a ij . (20)where a ij is defined in (9), which is antisymmetric a ij = − a ji and translational invariant a ij = a ( i − j ). (20) is equivalent to a ( r − L ) = − a ( r ) = a ( − r ). ONTENTS | α (cid:105) = | m m ...m L (cid:105) has a weight[see eq. (10)] f ( α ) = sgn( α ) × Pf A ( α )Pf B ( α ) . The weight of translated configuration T | α (cid:105) = | m m ...m L m (cid:105) is f ( T α ) = ( − L − sgn( α ) × Pf A ( T α )Pf B ( T α ) , where the phase factor ( − L − is owning to moving a fermion from site 1 to site L , and A ( T α ) , B ( T α ) can be obtained from A ( α ) , B ( α ) by the following replacement (assuming i, j (cid:54) = L ): a ij → a i +1 ,j +1 = a ij ,a iL → a i +1 , = − a iL ,a Lj → a ,j +1 = − a Lj , Thus, A ( T α ) , B ( T α ) just defer from A ( α ) , B ( α ) by multiplying a minus sign to thecollum a iL and the row a Lj . As a result, we have Pf A ( T α )Pf B ( T α ) = − Pf A ( α )Pf B ( α )and f ( T α ) = ( − L sgn( α ) × Pf A ( α )Pf B ( α ) = f ( α ) . This proves that the projected state has zero lattice momentum.Now we look at the ‘one-magnon’ excited states. They exist at periodic boundarycondition, namely, a i,j + L = a ij , or equivalently a ( r − L ) = a ( r ). Suppose the excitedspinon carry momentum p . Assuming p (cid:54) = 0, then we have f ( α ) = sgn( α ) × Pf A ( α )Pf B p ( α ) . and f ( T α ) = ( − L − sgn( α ) × Pf A ( T α )Pf B p ( T α ) .A ( T α ) , B p ( T α ) can be obtained from A ( α ) , B p ( α ) by the following replacement(assuming i, j (cid:54) = L ): a ij → a i +1 ,j +1 = a ij ,a iL → a i +1 , = a iL ,a Lj → a ,j +1 = a Lj ,e ipr → e ip ( r +1) . From (13) and (14), we have Pf A ( T α ) = Pf A ( α ) , Pf B p ( T α ) = e ip Pf B p ( α ).Consequently, f ( T α ) = ( − L − sgn( α ) × e ip Pf A ( α )Pf B p ( α )= e i ( p + π ) f ( α ) . This shows that the total momentum of the wavefunction is π + p . The lowestenergy spinon carry momentum p = 0, so the lowest-energy ‘one-magnon’ state carrymomentum π + p = π (in other words, the minimal spin gap opens at momentum k = π ). ONTENTS L =odd, the ground state carryzero momentum, and the lowest-energy ‘one-magnon’ state carry momentum k = π ± πL .In thermodynamic limit L → ∞ , the minimal one-magnon gap opens at momentum k = π .Now we go to the strong pairing phase. When L =even, there are two degenerateground states. Above we have shown that the state P G | π -flux (cid:105) carries zero momentum.It is easy to show that the other ground state P G | (cid:105) carries π momentum. Bothstates are translationally invariant. However, since they are degenerate, a superpositionof these two states is also a ground state of the spin Hamiltonian. The resultant statedo not have certain momentum, and is no longer invariant under translation. This isthe reason why the ground states have nonzero spin-Peierls correlation. References [1] F. D. M. Haldane, Physics Letters A 93, 464 (1983); Phys. Rev. Lett. 50, 1153 (1983).[2] S. R. White and D. A. Huse, Phys. Rev. B 48, 3844 (1993); S. Qin, T. K. Ng and Z.-B. Su, hys.Rev. 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