Gutzwiller Approach for Elementary Excitations in $S=1$ Antiferromagnetic Chains

In a previous paper [Phys. Rev. B 85,195144 (2012)], variational Monte Carlo 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 $S=1$ bilinear-biquadratic (BLBQ) Heisenberg chain. In the present paper, we extend the approach to study the low-lying elementary excitations in $S=1$ chains. We calculate the one-magnon and two-magnon excitation spectra of the BLBQ Heisenberg chain and the results agree very well with recent data in literature. In our approach, the difference 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 of corresponding mean field theory. We especially study the Takhtajan-Babujian critical point. Despite the fact that the `elementary excitations' are spin-1 magnons which are different from the spin-1/2 spinons in Bethe solution, we show that the excitation spectrum, critical exponent ($\eta=0.74$) and central charge ($c=1.45$) calculated from our theory agree well with Bethe ansatz solution and conformal field theory predictions.


I. INTRODUCTION
The S = 1 bilinear-biquadratic Heisenberg model 1-3 has attracted much interest in the quantum magnetism community. At K = 0 the Haldane conjecture predicts that the ground state of integer-spin antiferromagnetic Heisenberg model is disordered with gapped excitations. 4 Later it was shown by Affleck-Kennedy-Lieb-Tasaki (AKLT) that the point K/J = 1 3 is exactly solvable 5 and the resulting state is a translation invariant valence-bond-solid states. The S = 1 AKLT state together with all states in the so-called Haldane phase are topologically nontrivial in the sense that they cannot be deformed into the S z = 0 trivial product state without a phase transition. 6 The non-trivial order in the Haldane phase, called symmetry-protected topological order, is described by symmetric local unitary transformation and the projective representation of the symmetry group. 7 More generally, the model (1) contains three phases (see Fig. 1), the dimer phase at K/J < −1, the Haldane phase with −1 < K/J < 1 and a gapless phase at K/J > 1 with antiferro-nematic correlation. In later discussion, we will set J = 1. In Ref. 8, we studied the ground state properties of the S = 1 bilinear-biquadratic Heisenberg model numerically using Gutzwiller projected Bardeen-Cooper-Schrieffer (BCS) type wave functions as trial ground states wavefunctions. We found that the optimized projected BCS wavefunctions are close to the true ground states for the 1D antiferromagnetic bilinear-biquadratic model in the region K ≤ 1. In particular, the optimized projected BCS state is the exact ground state at the AKLT point K = 1 3 . Since the pairing symmetry is p-wave, the unprojected BCS states are classified into weak pairing (topologically non-trivial) and strong pairing (topologically trivial) states by their different winding numbers. 9 The topology of the BCS state is found to be important in distinguishing Haldane and dimer phases: after Gutzwiller projection the weak pairing states become the Haldane phase whereas the strong pairing states become the dimer phase. The phase transition between the Haldane phase to the dimer phase is reflected as a topological phase transition between weak pairing and strong pairing phases. 10 The success of the Gutzwiller approach in describing the ground states of the bilinear-biquadratic model leads us to ask the question whether a similar approach can be used to described the excited states. This question is being addressed in the present paper. We shall show that the one-and two-magnon excitation spectrums calculated numerically from the Gutzwiller projected wavefunctions are consistent with the best available numerical results for the corresponding excitations in the Haldane phase and excitations in the dimer phase have a very different character -there exists only odd/even-magnon excitations if the length of the chain is odd/even. The Takhtajan-Babujian (TB) phase transition point 11 between the Haldane and dimer phases is being studied carefully in this paper where we find that the excitation spectrum at the TB point is gapless with the critical exponent and the central charge agree well with SU (2) 2 Wess-Zumino-Witten field theory predictions. 12 This paper is organized as follows. In Section II, we review the fermionic mean field theory for S = 1 model, and discuss the general properties of the corresponding Gutzwiller projected BCS states. The Gutzwiller projected excited states are studied numerically using Monte Carlo technique and the results are presented in section III. Our findings are summarized in section IV where some general comments to our approach are given.

II. 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. 8,9 We introduce three species of fermionic spinons c 1 , c 0 , c −1 to represent the S = 1 spin operators asŜ a = C † I a C, where a = x, y, z, C = (c 1 , c 0 , c −1 ) T and I a is the 3 by 3 matrix representation of spin operator. The fermion Hilbert space is identical to the spin Hilbert space when a local particle number constraint c † 1 c 1 + c † 0 c 0 + c † −1 c −1 = 1 is imposed on the system. In this fermionic representation, the bilinearbiquadratic model (1) can be rewritten as whereχ ij = m=1,0,−1 c † mi c mj is the fermion hopping operator and∆ ij = c 1i c −1j − c 0i c 0j + c −1i c 1j is the spinsinglet pairing operator. This Hamiltonian can be decoupled in a mean field theory 9 by introducing short ranged order parameters χ = χ ij , ∆ = ∆ ij , and the Lagrangian multiplier λ for the particle number constraint. The mean field Hamiltonian is given by in momentum space where χ k = λ − 2Jχ cos k and ∆ k = −2i(J − K)∆ sin k, and γ m,k are Bogoliubov eigenparticles [see Eq. (A1) for details]. The mean field Hamiltonian describes a p-wave superconductor and may have nontrivial topology. In such paired states with ∆ = 0, the topology of the mean-field ground state |G MF is determined by χ k . The states with |λ| < |2Jχ| (see Fig.  2), called weak pairing states, have winding number 1 for each species of fermions. The states with |λ| > |2Jχ| (see Fig. 3), called strong pairing states, have winding number 0.

A. Gutzwiller Projected Ground states
The mean field ground state |G MF is a BCS type wavefunction. After Gutzwiller projection, the state |ψ = P G |G MF provides a trial ground state for the Hamiltonian (1) (see Appendix A for details). The parameters χ, ∆, λ are determined by minimizing the energy of the projected states E trial = ψ|H|ψ / ψ|ψ (the details of the calculations can be found in Ref. 8). It was found that the projected weak pairing states corresponds to the Haldane phase and the projected strong pairing states corresponds to the dimerized phase.
A special property of the Gutzwiller projection for spin-one systems has to be emphasized here, where three species of fermions are present. The number of fermions in the system is equal to the number of lattice sites by construction and therefore the fermion parity of the ground state is even/odd for chains with even/odd number of sites. It was pointed out in Ref. 8 and 13 that for a close chain in the weak pairing phase, the fermion parity of the mean field state is even/odd under antiperiodic/periodic boundary condition, because the k = 0 states are occupied by three fermions only under the periodic boundary condition.(Since anti-periodic boundary condition is equivalent to a global Z 2 flux through the ring formed by the spin chain, we will denote the ground state with anti-periodic boundary condition as |π-flux and denote the one with periodic boundary condition as |0-flux in the following.) Therefore when the length of the chain L=even/odd, only the anti-periodic/periodic boundary condition survives after Gutzwiller projection in the weak-pairing phase. As a result, the ground state of the Haldane phase is non-degenerate under closed boundary condition. However, in the strong pairing phase, the boundary conditions have no effect on the ground state fermion parity and consequently the dimer phase is doubly degenerate 8 . We note that in the dimer phase the ground state is not a spin-singlet if L =odd.
This subtle boundary condition effect also exists for the excited states and leads to important distinction between the excitation spectrums in the Haldane and dimer phases as we shall see in the following.

B. Gutzwiller Projected excited states
In BCS superconductors, excitations are formed by adding quasi-particles obtained from the Bogoliubov-de Gennes equations to the BCS ground state wavefunction. We may add arbitrary number of quasi-particles to form excited states since the system allows arbitrary fermion number. We shall assume in the following that (low energy) excitations in the S = 1 spin liquids can be formed by Gutzwiller projecting the excited states of the corresponding BCS superconductor. The fixing of fermion parity in spin systems imposes a constraint on the excited states that can be constructed in this approach.
In the fermionic mean field theory of spin-1/2 systems (where the paring symmetry is s-wave), the requirement of fixed fermion parity implies that excited states can be formed only by Gutzwiller Projecting BCS excited states with even number of quasi-particle excitations. The situation is similar for spin-1 mean field theory (where the pairing symmetry is p-wave) in the strong pairing phase. However, the weak pairing phase is more subtle since the fermion parity can be changed by changing the boundary condition of the mean field Hamiltonian. A consequence is that one-magnon 14 excitations are allowed in the Haldane phase.

Weak pairing phase
Let us focus on the weak pairing (Haldane) phase. First we consider a spin excitation formed by simultaneously switching the boundary condition and adding a Bogoliubov quasi-particle to the system. We shall call the excitation a one-magnon excitation 14 . The one-magnon creation operator with S z = m and momentum k can be written as γ † m,pŴ , where p = k − π andŴ is the boundary-twisting operator which switches the periodic boundary condition to antiperiodic and vice versa. Notice that the quasi-particle momentum changes by π after the boundary condition is switched (see Appendix B for details). We shall see in next section that after projection the state P G γ † m,pŴ |G MF corresponds to the onemagnon excitation discussed in the literature 15,16 . Notice that the mean-field energy of this excitation has minimum at p = 0. This explains why the minimal magnon gap opens at k = π. The two-magnon excitation can be obtained by acting the one-magnon creation operator on the mean field ground state twice before the Gutzwiller projection. Notice the boundary condition is restored (Ŵ 2 = I) for two-magnons.
In the following we shall provide more details of the one-and two-magnon excitations.
We first consider the case L=even integer. In this case the ground state is a spin-singlet given by [see Fig.2(a)] |ground = P G |π-flux .
A single magnon is a spin-1 excitation represented by [see Fig.2 where |(1, m); p + π 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 is orthogonal to the ground state |ground because it carries both nonzero spin and momentum. The energymomentum dispersion of the one-magnon spectrum will be discussed in next section. The two-magnon excitations can be constructed similarly and are denoted by |(S, m); p, q , where (S, m) are the spin quantum numbers and p, q are the momenta carried by the two magnons. Notice that since each magnon carries spin-1, the total spin of two magnons can be S = 0, 1 or 2. For example, the states with S = 0, 1, 2 and m = 0 are given by We have dropped some unimportant normalization constants in writing down the above states. Obviously, the two-magnon states are orthogonal to each other because they carry different spin-quantum numbers. It can be also shown that they are orthogonal to the ground state and the one-magnon states. 17 For a given momentum k = p + q, the total energy E k depends on the momentum distribution (p, q) of the two magnons and the energy-momentum spectrum of the two-magnon states form continuums.
The L=odd integer situation can be constructed similarly as for even chains except that |0-flux ⇐⇒ |π-flux in writing down the ground and excited state wavefunctions.

Strong pairing phase
The fermion parity of the spin chain is independent of boundary conditions in the strong pairing phase. As a result the ground states are doubly degenerate and the excitation spectrums are different for chains with even and odd length L's.
We consider first the case of L=even integer chains. In this case, the ground state wavefunctions are given by [see Fig.3(a),(b)] where |ground 1 carries 0-momentum and |ground 2 carries π-momentum. One-magnon excitations do not exist in this case since the fermion parity cannot be changed by switching 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 states are given by The two magnon excitations form continuum in the energy-momentum spectrum as in the Haldane phase. Another way to understand why one-magnon excitations do not exist for L=even chains in the strong pairing phase is to compare the corresponding mean-field spectra in Fig.2(b) and Fig.3(b). We note that the three spinon modes at k = 0 have negative energy 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 a similar 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-magnon excited state of the Haldane phase becomes a four-magnon excited state in the dimer phase.
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 [see Fig.3 m); p 1 and p = π for |(1, m); p + π 2 . The energy of the system changes continuously and forms a one-magnon excitation spectrum when we change p. This can be easily understood, since in the dimer phase, the spins form singlet pairs (or dimers) at the ground state. When L=odd, not all the spins can form pairs and there must exist odd number of magnons in the system including the ground state.

III. NUMERICAL RESULTS
In this section we discuss our numerical results for various spin excitations we constructed in the previous section. When L is large, the expectation values of physical quantities in a Gutzwiller projected state can be calculated with Monte Carlo (MC) method. We first consider the Heisenberg model (K = 0). Fig. 4 shows the ground state and the one-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, showing that even or odd The two-magnon excitations form a continuum spectrum, as shown in the filled area in Fig.5. The energy cost for the minimal two-magnon excitation is roughly twice the spin gap. The one-magnon curve merges into the two-magnon continuum below k = 0.4π, suggesting that a single-magnon excitation will decay into two magnons if its momentum is less than k = 0.4π. This result agrees also with the numerical result for the twomagnon spectrum. 15,16 Depending on the symmetry under exchanging the spin momentum of the two magnons, the total spin of two magnons can be either 0,2 (symmetric) or 1 (antisymmetric). Fig.5 shows that the two-magnon energy bounds for total spin S = 0, 1, 2 excitations are almost the same, with small deviations appearing only near momentum k = π. This suggests that there is almost no interaction between the two magnons except when their total momentum is close to k = π. Near k = π, the S = 1 channel is lower in energy then the S = 0, 2 channels. Comparing with the energy sum of two one-magnon states, we find that the interaction between the two magnons is attractive for the S = 1 channel while weakly repulsive for the S = 0, 2 channels, qualitatively consistent with results obtained in Ref. 15, where the repulsive interaction in S = 0, 2 channels is stronger than the attractive interaction in S = 1 channel. We shall study spin excitations in the strong pairing phase at K = −3. As we have pointed out in last section, the L =even and L =odd chains have quite different properties. There exist only even/odd-magnon excitations for even/odd L.
First we consider even L. Fig. 6(b) shows the twomagnon continuum for L = 100. There is an obvious gap (of order 1.1J) between the ground state and the twomagnon continuum. The two magnons can form states with total spin S =0,1 or 2. The energy differences between states with different total spin S are small as is clear from the figure except at the points k = 0, π, indicating that the two magnons almost do not interact with each other except when their total momentum is close to k = 0 or π, similar to the Haldane phase.
Next we consider odd L. Recall that the singlet ground state does not exist for odd L and the lowest energy states are the states |(1, m); p 1 = P G γ † m,p |0-flux with p = 0, or |(1, m); p + π 2 = P G γ † m,p |π-flux with p = π. The energy dependence of |(1, m); p 1 as function of p is shown in Fig. 6(a). We indicate in the figure also the (minimal) 3-magnon excitation energies at points k = 0 and k = π. The finite difference in energy between the oneand three-magnon states indicates that the two-magnon excitations have a finite gap of order 1.5J. Lastly we consider the TB critical point at K = −1 which can be solved exactly with Bethe ansatz. 11 At this point, the optimal variational parameters satisfies λ − 2χ ≈ 0 and the mean field excitation spectrum is gapless. 8 The one-magnon spectrum after Gutzwiller projection is plotted in Fig. 7. There is a slight difference in energy between L =odd and L =even chains. Fig. 7(a) shows that for L = 199, the spinons at momentum k = 0 and k = π are gapless. Fig. 7(b) shows the data for L = 200, the excitation gap closes at k = π but remains finite at k = 0. However, a finite size scaling analysis (insert) shows that the gap at k = 0 vanishes in power low of the chain length L. Thus, we expect that in thermodynamic limit, the one-magnon excitations are gapless at both k = 0 and k = π. The two-magnon continuum for L = 100 is shown in Fig.8. We expect that the one-magnon dispersion will coincide with the lower energy bound of the two-magnon continuum in thermodynamics limit. We now compare our result with the Bethe ansatz solution. 11 In our approach, the elementary excitations are spin-1 magnons whereas the elementary excitations are pairs of spin-1/2 spinons in the Bethe ansatz solution. Therefore, the two approaches do not seem to give the same result at first glance. The correctness of our approach can be verified by checking the critical behavior of the projected state. We numerically calculate the critical exponent η and the central charge c from the projected ground state. The results are shown in Fig.9. The critical exponent is obtained by calculating the spin-spin correlation, We note that c = 0 for gapped 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 = 0.74±0.01, c = 1.45±0.02 agree very well with SU (2) 2 Wess-Zumino-Witten field theory predictions η = 0.75, c = 1.5, 12,22 suggesting that our spectrum is correct in at least the continuum limit. The agreement of our result with WZW field theory predictions suggests that although the elementary excitations in our approach differ from those in the Bethe ansatz solution, there is a one-to-one mapping between the two approaches in the construction of the real spin excitation spectrum. We note that the dispersion of the spin-1/2 spinon in the Bethe Ansatz solution is given by ε(k) = 2π sin |k|, 11 and the excitation spectrum is gapless at k = 0 and k = π in the Bethe-Ansatz solution. The one-magnon dispersion in Fig. 7 is also gapless at k = 0 and k = π, and the shape is close to a sine function, in agreement with the Bethe solution. A pair of spin-1/2 spinons form a spin-singlet continuum and a spin-triplet continuum in the Bethe Ansatz solution. The two continuums are degenerate in energy. In our approach, the spin-0 two-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 see Fig.8).
We note also that a one-magnon excited state can also be viewed as a four-magnon excitation in our approach (recall that if one approaches the critical point from the Haldane phase, this state is viewed as a one-magnon state; but if one approaches from the dimer phase, this state is viewed as a four-magnon state), i.e. the onemagnon dispersion curve is nothing but the lower bound of the four-magnon continuum and may 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 = 1 magnons in the Gutzwiller projected wavefunction approach and the S = 1/2 spinons in Bethe Ansatz solution at the TB critical point is highly nonlinear.

IV. CONCLUSION AND DISCUSSION
Conclusion To summarize, we have studied in this paper the low energy spin excitations in the Haldane (K = 0) and dimer (K = −3) phases [including the TB critical point (K = −1)] for the one-dimensional bilinear-biquadratic Heisenberg model using a Gutzwiller Projected wavefunction approach.
We find that the so-called one-magnon excitation observed previously in other numerical methods in the Haldane phase can be explained as a composite object of global Z 2 flux and a spinon in our Gutzwiller projected wavefunction approach. The corresponding two-magnon excitation spectrum computed in the Gutzwiller projected wavefunction also agrees with earlier numerical works and we show evidence that the magnons are weakly scattering with each other (absence of confinement).
The excitation spectrum in the dimer phase is computed (to our knowledge, it is the first time that the energy spectrum of the dimer phase is studied) where we point out the qualitative differences between L =odd and L =even chains. At the critical point (the TB model), the projected dispersion is gapless at both k = 0 and k = π. The critical exponent η = 0.74 and the central charge c = 1.45 we obtained agree very well with literature. 12,16,22 We note that the one-magnon dispersions in Figs. 4, 7, 6(a) are qualitatively different from the corresponding mean field dispersions before Guzwiller projection (see Fig.10). In the weak paring phase, the minimal mean field gap opens at k = 0, but after projection, the minimal one-magnon gap opens at k = π. In the strong pairing phase, the mean field dispersion is asymmetric by reflection along k = 0.5π, while after projection the one-magnon curve becomes more symmetric. Especially, the mean field dispersion 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 the mean-field states after Gutzwiller projection correctly describe the physical properties of the spin system (1).
Discussion The existence of one-magnon excitation in the spin-one bilinear-biquadratic Heisenberg 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 state for a system with both even and odd number of sites whereas for half-oddinteger spin systems, singlet state exists only in systems with even number of sites. For a spin chain with length L, the one-magnon excitation in the Haldane phase can be understood with the single-mode approximation: the ground state (which is a L-site singlet state) is reconstructed into a (L−1)-site singlet plus a singlet spin. The single spin is propagating and forms as a magnon. It is obvious that it is not possible to form similar excitations above a singlet ground state of half-odd-integer spin systems. The Gutzwiller projected wavefunction approach captures this important difference between integer and half-integer spin systems nicely.
The distinction between topologically trivial and nontrivial phases in the Gutzwiller wavefunction approach for the 1D S = 1 bilinear-biquadratic Heisenberg model suggests the interesting possibility that similar distinction may be found in higher dimensional spin systems. In particular, fractional spin excitations which obey fermonic or non-Abelian statistics 21,23 may exist in 2D topologically ordered phases.
We thank Hong-Hao Tu for very helpful discussions about the TB point. We also thank T. Senthil In momentum space, the mean field Hamiltonian (2) can be diagnolized into Bogoliubov particles: where ε k = χ 2 k + ∆ 2 k , u k = cos θ k 2 , v k = i sin θ k 2 and tan θ k = i∆ k χ k . In the following, we will provide some eigen states of above Hamiltonian.
(C) Excited states by breaking the pair c † 1,p c † −1,−p : 1), one-spinon excitation (E = ε p , two-fold degenerate) 2), two-spinon excitation (E = 2ε p ) 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 weak pairing phase owning to the dependence of fermion parity on boundary conditions.

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 ndimensional skew symmetric matrix, then we have, where A (A ) mean A with the first(nth) and the ith row and column are removed. L=even.
Notice that momentum of the magnon is equal to the sum of the spion and the extra Z 2 flux, which gives p + π (for details, see Section B).

3), 2-magnon excited states
Now we consider two-spinon (or two-magnon) excited states. We can either excite two c 0 spinons or one c 1 spinon plus one c −1 spinon. But these states do not respect the symmetry of the spin Hamiltonian since they do not carry correct spin quantum numbers. According to the total spin (S=0,1,2) of the two magnons (the S=0,2 states are symmetric under exchanging the spin quantum numbers of the two spinons, while the S = 1 states are anti-symmetric under exchanging the spin quantum numbers of the two spinons), an excited eigenstate is a superposition the two states listed above. Owning to the degeneracy, we only consider the (S, 0)-component of the excited states: Above we have assumed that p + q = 0. If p + q = 0, then the corresponding 2-spinon excited state should be constructed as mentioned in appendix A 1.

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 similar to the weak pairing phase. When L =odd, evenspinon excitations (including '0-spinon excitation' state) are not allowed, and only odd-spinon excitations exist. The method to obtain projected 1-spinon excited states are similar to the weak pairing phase, except that both P G γ † 0,p |π-flux and P G γ † 0,p |0-flux are allowed here. Firstly, let us consider the Heisenberg model in the case L =even. The ground state is the projected mean field ground state with anti-periodic boundary condition |ground = P G |π-flux , which yields where a ij is defined in (A5), which is antisymmetric a ij = −a ji and translational invariant a ij = a(i − j). where the phase factor (−1) L−1 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 = L): a ij → a i+1,j+1 = a ij , a iL → a i+1,1 = −a iL , a Lj → a 1,j+1 = −a Lj , Thus, A(T α), B(T α) just defer from A(α), B(α) by multiplying a minus sign to the collum a iL and the row a Lj . As a result, we have PfA(T α)PfB(T α) = −PfA(α)PfB(α) and f (T α) = (−1) L sgn(α) × PfA(α)PfB(α) = f (α).
This proves that the projected state has zero lattice momentum.
This shows that the total momentum of the wavefunction is π + p. The lowest energy spinon carry momentum p = 0, so the lowest-energy 'one-magnon' state carry momentum π + p = π (in other words, the minimal spin gap opens at momentum k = π).
Repeating above argument, we can show that when L =odd, the ground state carry zero momentum, and the lowest-energy 'one-magnon' state carry momentum k = π ± π L . 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 degenerate ground states. Above we have shown that the state P G |π-flux carries zero momentum. It is easy to show that the other ground state P G |0-flux carries π momentum. Both states are translationally invariant. However, since they are degenerate, a superposition of these two states is also a ground state of the spin Hamiltonian. The resultant state do not have certain momentum, and is no longer invariant under translation. This is the reason why the ground states have nonzero spin-Peierls correlation.