Gutzwiller Projected wavefunctions in the fermonic theory of S=1 spin chains
We study in this paper a series of Gutzwiller Projected wavefunctions for spin chains obtained from a fermionic mean-field theory for general spin systems [Phys. Rev. B 81, 224417] applied to the bilinear-biquadratic (-) model. The free-fermion mean field states before the projection are 1D paring states. By comparing the energies and correlation functions of the projected pairing states with those obtained from known results, we show that the optimized Gutzwiller projected wavefunctions are very good trial ground state wavefunctions for the antiferromagnetic bilinear-biquadratic model in the regime (). We find that different topological phases of the free-fermion paring states correspond to different spin phases: the weak pairing (topologically non-trivial) state gives rise to the Haldane phase, whereas the strong pairing (topologically trivial) state gives rise to the dimer phase. In particular the mapping between the Haldane phase and Gutwziller wavefunction is exact at the AKLT point (). The transition point between the two phases determined by the optimized Gutzwiller Projected wavefunction is in good agreement with the known result. The effect of gauge fluctuations above the mean field theory is analyzed.
Slave boson mean field theory is now accepted as a powerful tool in identifying exotic states in strongly correlated electron systems. Affleck8588 , Anderson87 , LeeNagaosaWen06  At half-filling, the slave boson approach reduces to a fermionic representation for the spins where mean-field theories can be built and corresponding trial ground state wavefunctions can be constructed through the Gutzwiller projection technique.Gros89  The approach has generated large variety of wavefunctions used to describe different resonant valence bond (RVB) states of frustrated Heisenberg systems including quantum spin liquid states.Anderson87 , LeeNagaosaWen06 , Gros89 , algebraicSL , Spinliquids 
In a recent paper,LZN  several of us have generalized the fermionic representation to spin systems and have shown that a simple mean-field theory produces results which are in agreement with Haldane conjecture for the one-dimensional Heisenberg model. A natural question is, how about the Gutzwiller projected wavefunctions obtained from these mean-field states? Are they close to the corresponding real ground state wavefunctions? How about more complicated spin models? Here we shall provide a partial answer to these questions by studying the Gutzwiller projected wavefunctions obtained from the mean field states of the bilinear-biquadratic Heisenberg modelnew , Kato97 , BLBQ 
where are spin operators and in one dimension. In some literature, the above Hamiltonian is parametrized as with where is restricted to . We shall use both notations in this paper.
The bilinear-biquadratic Heisenberg model has attracted much interest. At () the Haldane conjecture predicts that the ground state of integer-spin antiferromagnetic Heisenberg Model (AFHM) is disordered with gapped excitations.Haldane83  Later it was shown by Affleck-Kennedy-Lieb-Tasaki (AKLT) that the point () is exactly solvable AKLT-1987  and the resulting state is a translation invariant valence-bond-solid states. Klumper-1991  The 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 trivial trivial product state without a phase transition. People have been trying to use hidden symmetry breaking,Kennedy 88  or equivalently, a nonlocal string order,stringorder89  to characterize the non-trivial order in the Haldane phase. But those characterizations are not satisfactory since the Haldane phase is separated from the trivial product state even when we break all the spin rotation symmetry, in which case there is no hidden symmetry breaking and/or nonlocal string order.GuWen09 , Pollmann09  It turns out that 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.CGW , SPT1D , SPT2D 
The phase diagram of one dimensional bilinear-biquadratic model is given in Fig. 1. The region () is gapped and contains two phases, the the Haldane phase () and the dimer phase (). The question we shall address in this paper is whether the above phase diagram can be (partially) reproduced by using simple Gutzwiller projected wavefunctions obtained from the fermionic mean-field theory. We shall show in the following that the optimized projected mean-field wavefunctions are very close to the true ground states for the 1D antiferromagnetic bilinear-biquadratic model in the regime (). In particular, the optimized projected mean field state is the exact ground state at the AKLT point (). The mean field state is a pairing state of free fermions, which can be a trivial or a non-trivial topological phase, which are classified as weak and strong pairing states by their different winding numbers.LZN  The nature of the topological phase of the mean field state is found to be important in distinguishing between Haldane and dimer phases. We find that after Gutzwiller projection the weak pairing states become the Haldane phase whereas the strong pairing states become the dimer phase. A long-ranged spin-Peierls order emerges in the strong pairing states after Gultzwiller projection although the spin-Peierls correlation is short ranged at mean field level.
Above results can be understood from the fermionic mean field theory when gauge fluctuations are taken into account. We find that the instantons behave differently in the weak pairing region and strong pairing region. Thus the Haldane phase and the dimer phase can also be distinguished by their different effective gauge theories.
The paper is organized as follows: In section II, we review the fermionic representation for spins and introduce the mean field theory for the bilinear-biquadratic model. The Gutzwiller projected mean-field wavefunctions are introduced in section III as trial ground state wavefunctions for the bilinear-biquadratic model. Using variational Monte-Carlo (VMC) techniqueGros89  we find that the projected mean-field wavefunctions after optimization are very close to the true ground states of the 1D antiferromagnetic bilinear-biquadratic model in the regime (). We show in particular that the optimized projected mean field state is the exact ground state at the AKLT point (). Based on mean-field theory, we construct in section IV the effective low energy theories for the bilinear-biquadratic model. The paper is concluded in section V with some general comments. The mapping between the mean-field zero-energy Majorana end states in the weak-coupling phase and spin-1/2 end states of open spin chains in the Haldane phase is established in the appendix A.
Ii Fermionic representation and mean-field theory for spin bilinear-biquadratic model
The fermionic representation for spins is a generalization of the fermionic representation for spins. In this representation, three fermionic spinon operators are introduced to represent the states at each site, and the spin operator is given as , where is the matrix representation for angular momentum operator with and . As in usual slave particle method, a particle number constraint has to be imposed on each lattice site to ensure a one-to-one mapping between the spin and fermion states.
where is the fermion hopping operator and is the spin singlet pairing operator. A mean field theory for this interacting fermion model can be obtained by introducing the mean field parameters , , and the time averaged Lagrangian multiplier to decouple the model into a fermion bilinear modelLZN ,
It is interesting also to introduce the cartesian coordinate operators, , (these operators annihilate the states, , respectively). In this representation the operators and becomes
and the mean field Hamiltonian reduces to three copies of Kitaev’s Majorana chain model. Kitaev-2001  This representation will be used in our later discussion.
The mean field Hamiltonian can be diagonalized by the standard Bogoliubov-de Gennes (B-dG) transformation. We shall consider periodic/antiperiodic boundary condition here. (The case of open-boundary condition is discussed in appendix A.) In this case the system is translational invariant and become site-independent. The mean-field Hamiltonian is diagonal in momentum space,
where and . ’s are related to ’s by the Bogoliubov transformation,
where , and . The mean field dispersion is gapped when except at the phase transition point .
The ground state of is the vacuum state of the Bogoliubov particles ’s. The parameters and are determined self-consistently in mean-field theory, with is determined by the averaged particle number constraint, i.e.,
where denotes ground state averages. We shall see later that the self-consistently determined mean field parameters are not optimal in constructing Gutzwiller projected wavefunctions. It is more fruitful to treat and as variational parameters in the trial Hamiltonian (3) that generates a trial mean field ground state . The Gutzwiller projected mean-field state will be used as a trial wavefunction for the spin model (1). The optimal mean-field parameters are determined by minimizing the energy of the projected wavefunction .
An important consequence of nontrivial winding number is that topologically protected Majorana zero modes will exist at the boundaries of an open chain in mean-field theory. The mean-field states satisfying Eq. (8) are called weak pairing states. The winding number vanishes if and and these states are called strong pairing states. ReadGreen2000  We shall show later that the weak pairing states become the Haldane phase, while the strong pairing states become the dimer phase after Gutzwiller projection. In later discussion, we will mainly focus on the antiferromagnetic interaction case . To simplify notation we shall set in the following. The value of will be defined again only in exceptional cases.
Iii Gutzwiller Projected wavefunctions
|comparison||0.2971Sutherland-1975 ||-AKLT-1987 ||-1.4015White93 ||-4Takhatajan-1982 ||-6.7531||-9.5330||-2.7969BB89 ||-7.3518||-4.5939|
|VMC||0.2997111Due to the symmetry, the particle number of should be equal. To this end, we have set .||-||-1.4001||-3.9917||-6.7372||-9.5103||-2.7953222The unit of the energy is , which is normalized to 1.||-7.2901||-4.4946|
The Gutzwiller Projection for systems is in principle the same as Gutzwiller Projection for systems. In the mean field ground state wavefunction, the particle number constraint is satisfied only on average and the purpose of the Gutzwiller projection is to remove all state components with occupancy for some sites i, and thus projecting the wavefunction into the subspace with exactly one fermion per site.
There are however a few important technical difference between spin-1/2 and spin-1 systems. First of all, systems are particle-hole symmetric and the particle number constraint is invariant under particle-hole transformation. As a result, we can always set in the trial wavefunctions. This is not the case for models where the particle number constraint is not invariant under particle-hole transformation. Consequently, in general and should be treated as a parameter determined variationally. Notice that determines the topology of the mean field state and the corresponding Gutzwiller projected states as we shall see in the following.
The second important difference between spin-1/2 and spin-1 systems is that a singlet ground state for systems is composed of configurations with different distribution , where is the number of spins with in a spin configuration and is the number of sites in the spin chain. It is clear that can take any value between and where if is even and if is odd. On the contrary is fixed for spin-1/2 systems. As a result, is always even for spin-1/2 singlet states, but can be even or odd for spin-1 singlet states. For even , is even and the Gutzwiller projected wavefunction is a straightforward projection of the mean-field ground state which is a paired BCS state. For odd , is odd. This means there is one fermion mode () remaining unpaired in the ground state. The projection is similar to the even case except that we have to keep in mind the occupied free fermion mode. The situation for open spin chains is further complicated by the existence of Majorana end modes which is discussed in the appendix A.
The above condition results in a natural choice of boundary condition in constructing the Gutzwiller projected wavefunction. To see this we first note that the values of allowed fermion momentum in periodic and anti-periodic boundary conditions are different. They are given by under periodic (antiperiodic) boundary conditions, where take values . The energy spectrum is doubly degenerate except at the points which exists only under periodic boundary condition (for both even or odd ), and which exists under periodic boundary condition for even and under anti-periodic boundary condition for odd . Notice that at these two points vanishes and they are natural candidates for constructing a singly occupied fermion state. All other momentum ’s are paired at the ground state. The energies at these two points are given by . Notice that and in the weak pairing phase (we choose a gauge where ) whereas both in the strong pairing phase.
We now consider the weak pairing phase. In this case a lowest energy state is formed for chains with odd when the state is occupied, i.e. periodic boundary condition is preferred. On the other hand, for chains with even an anti-periodic boundary condition is preferred so that the state is not available and a paired BCS state is naturally formed.
Next we consider the strong pairing phase. Suppose is even. Under anti-periodic boundary condition, all the fermions are paired in the ground state. Under periodic boundary condition, since , the unpaired fermion modes at and are unoccupied. The remaining fermions are paired. Both of the two boundary conditions are allowed. When is odd, the ground state is not a spin singlet and behaves differently.LZTWN12  In later discussion about the strong pairing phase, we will mainly consider the case where is even.
The above result suggests that the weak-pairing ground state is unique, whereas the strong-pairing ground state is doubly degenerate. To see this we note that for a closed chain, the existence or absence of flux through the ring (which corresponds to the anti-periodic boundary condition or the periodic boundary condition for fermions, respectively) usually result in two degenerate time-reversal invariant fermion states. However, as shown above, for a chain with fixed , we cannot choose boundary condition freely for the Gutzwiller projected state in the weak-coupling phase, whereas both boundary conditions are available in the strong-coupling phase.
In the following we shall report our numerical results for the antiferromagnetic bilinear-biquadratic Heisenberg model in the parameter range (). As in case, the optimization of energy of the projected wavefunctions is carried out using a variational Monte Carlo method (VMC). We set the length of the chain to be , and has taken MC steps in our numerical work. We note that because of the difference between and systems as discussed above, the VMC for spin-1/2 Gutzwiller projection procedure has to be modified for models. We shall not go into these technical details in this paper. We shall first report in section III.1 our overall numerical results and phase diagram which are in good agreements with known results as long as the ground state is a spin-singlet. Our Gutzwiller projected wavefunction provides a good description of the system even around the critical point () between the Haldane and dimer phases. In section III.2 we illustrate analytically that a projected BCS state becomes the exact ground state of the model at the AKLT point ().
iii.1 Overall results and phase diagram
Our numerical results of Gutzwiller projection for the bilinear-biquadratic Heisenberg model is summarized in Table 1. The ground state energy computed from the (optimized) Gutzwiller projected wavefunction is compared with the exact result or results obtained from the ‘infinite time-evolving block decimation’ algorithm.Vidal  Here the unit of energy is set as , except at the point , () where the energy is measured by .
From Table 1, we see that agreement in energy is better than in the range (). note25  The optimized parameters given in the table suggests that the system is in the weak pairing phase when () and is the strong pairing phase otherwise. The critical point at () will be studied more carefully in the following. To understand the nature of the weak- and strong- pairing phases we first study the spin-spin or dimer-dimer correlation functions at and .
At the self-consistent mean field solution gives . The energy of corresponding Gutzwiller projected wavefunction is per site, which is already quite close to the known ground state energy White93 , Ng95  with a difference . The correlation length determined from a fitting to the spin-spin correlation function is roughly 3.25 units of lattice constant, which is smaller than the value given in literature.Nomura89 , White93 , Ng95 
Our result can be further improved by optimizing the parameters . The optimal parameters we obtain are (here we normalize because the wavefunction before the projection is only dependent on and ). The energy of the projected state is which is further improved by . The spin-spin correlation function is plotted in Fig. 2, The correlation matches very well with , indicating the rotational invariance of the projected wavefunction. The correlation length determined from the optimized wavefunction is 5.92 lattice constants, which is very close to the accepted value .
A trademark for the Haldane phase is the existence of spin-1/2 end states. Indeed, end Majorana fermion states are observed to exist in the weak-pairing phase of the fermionic mean-field theory.LZN  The question is whether these Majorana end states become spin-1/2 end states after Gutzwiller projection. This question is discussed in appendix A where we show how the Majorana fermion end states turn into spin-1/2 end states after Gutzwiller projection.
Next we consider . In this case, the mean field solution has , and the energy of the projected mean-field state is . The energy can be lowered by optimizing the mean field parameters. The optimal parameters found from the VMC are with energy . The Gutzwiller projected wavefunction is obviously translationally invariant and does not explicitly break the translation symmetry. To see that the state described a dimer state, we compute the spin-Peierls correlation function. The result is shown in Fig. 3. We note that the spin-Peierls correlation is clearly short-ranged in the Heisenberg model() but is long-ranged in the model. The “weak-pairing/Haldane” and “strong pairing/ dimer” mapping is in agreement with the ground state degeneracy we deduced in last section. It is remarkable that in the strong pairing phase the spin-Peierls correlation becomes long-ranged only after the Gultzwiller projection and is short-ranged before projection.
Now we examine the phase transition point between the Haldane phase and the dimer phase. Fig. 4 shows the spin Peierls correlation at distance as a function of for the projected optimal trial wavefunctions near the phase transition point. Within numerical error our results show that and the spin-Peierls correlation vanishes at the point . The spontaneous breaking of translation symmetry indicates that the transition is of second order, consistent with exact solution at the Takhatajan-Babujian (TB) point .Takhatajan-1982  We would like to point out that the transition point determined by the self-consistent mean field theory is , which is far away from the exact result.
iii.2 AKLT point: the projected wavefunction as exact ground state
The validity of the Gutzwiller projected wavefunction approach for spinchains is further supported by an exact result at the AKLT point AKLT-1987  (), where we find that the AKLT state can be exactly represented as a Gutzwiller projected BCS wavefunction with parameters .
It is convenient to adopt the cartesian coordinate in the discussion. Firstly we consider the closed boundary condition. Since the MF Hamiltonian in (3),(II) contains three identical copies for the flavors , we may concentrate on a single flavor and define , where and are Majorana operators satisfying where and . Setting , and adopting close boundary condition ( for periodic boundary condition and for antiperiodic boundary condition), the mean-field Hamiltonian can be mapped into Kitaev’s Majorana chain, Kitaev-2001 
it is easy to see that the fermion parity of the ground state of the mean-field Hamiltonian is given by
and is even(odd) under anti-periodic (periodic) boundary condition. The total fermi parity is the product of the three flavors . Notice that the periodic boundary condition is ruled out by the particle number constraint for even . This is a general property of the BCS Hamiltonian in weak-coupling phase as we have discussed before.
Since all terms in (9) are commuting, the ground state of one term in the Hamiltonian (9) provides a supporting Hilbert space for the reduced density matrix constructed for the whole ground state. The ground state of for two neighboring sites and is two-fold degenerate,
Since there are three flavors , the ground state of the two sites is a product state , and are 8-fold degenerate. It is easy to see by direct computation that the Gutzwiller projection kills half of these states, and the surviving four states are
The first one is a spin singlet, and the remaining three form a () triplet. The absence of spin-2 states for every two neighboring sites is a fingerprint of the spin-1 AKLT state. AKLT-1987  Thus, we prove that the Gutzwiller projected trial state with is equivalent to the spin-1 AKLT state.
We note that the above proof can be extended straightforwardly to the symmetric AKLT models. HHTu08  When is odd, the Majorana fermions at the edge form the irreducible spinor representation of group. So, after Gutzwiller projection, the ground state of -copies of Kitaev’s Majorana chain model exactly describes the ground state of the -AKLT model. When is even, the ground state is dimerized since the spinor representation of is reducible. The equivalence between the ground state of the -AKLT model and the projected ground states of -copies of Kitaev’s Majorana chain remains valid. note1 , HHTu11  In that case, the Majorana fermions at the edge form a direct sum of two versions of irreducible spinor representations.
Iv effective low energy theory
The success of the Gutzwiller projected wavefunction in describing the ground state properties of the Haldane and dimer phases suggests that the low energy properties of these phases may be well described by effective field theories constructed from the corresponding mean-field states. We provide two approaches in this section. The first one is a gauge field description by integrating out the fermions, and the other one is an effective (Majorana) fermionic field theory.
The mean-field theory provides a correct description of the low energy properties of the Haldane and dimer phases when gauge fluctuations or instantons are taken into account. Since the mean field state is a fermion paired state with finite gap, the resulting low energy effective theory is gauge theory. Usually, the gauge theory in (1+1)D is always confined since the instantons (ie the vortices in (1+1)D discrete space-time) have a finite action. However, the low energy effective gauge theory obtained from our models has different dynamical properties.
In the weak pairing phase, a instanton gives rise to a fermionic zero mode.LZTWN12  As a result, the action of separating two instantons is proportional to , where is the time-distance between the instantons and is the excitation gap.LZTWN12  This means that the instantons are confined and consequently the gauge theory is deconfined (this situation still holds if the Hamiltonian contains a dimerized interaction). The vortex changes the fermion parity of the mean field ground state. Owing to the particle number constraint, a permitted instanton operator should be a composition of a vortex and a spinon operator. Thus, an instanton carries -momentum and spin-1. An example of such instanton operator is given as
Above instanton operator creates a magnon with momentum .
In the strong pairing phase, due to the absence of fermion zero modes, the vortices in (1+1)D space-time have a finite action and consequently have a finite density. However, since the instanton carry crystal momentum, there will be an extra phase factor associated with it. When sum over the contribution of instantons at all spacial positions, the phase factors will cause cancelation. Consequently, the effect of instantons is suppressed, and the gauge theory is still deconfined.Wenbook  On the other hand, if the Hamiltonian have a translation symmetry breaking term (such as a dimerized interaction), then the action of the instantons will not be canceled and will confine the gauge filed. (This is a remarkable difference between the strong pairing phase and the weak pairing phase.) Since a instanton carries momentum and zero spin, we can give an example of such an operator
The finite action of the instantons indicates that the ground state is degenerate and has a finite spin-Peierls order.
At the transition point, the mean field dispersion is gapless at due to , and the low energy excitations are consist of three species of Majorana fermions with energy . Notice that three Majorana fermions form a spin-1/2 object. Lecheminant-2002  Consequently, after Gutzwiller projection, the elementary excitations carry spin-1/2. Notice that the number of excited Majorana fermions must be even in a physical state, so the spin-1/2 excitation must appear in pairs. This physical picture arising from mean-field theory agrees well (in long wavelength limit) with result coming from the Bethe ansatz solution of the TB model. Takhatajan-1982 
Since the effective gauge fields are deconfined in both the Haldane and the dimer phases of model (1), it will be a good approximation to ignore the gauge field and consider the fermion theory only. Tsvelik proposed an effective Majorana field theory to describe the low-energy physics close to the TB point Tsvelik-1990 
where and are right and left moving Majorana fermions. Marginal terms (four-fermion interactions) are neglected. This theory describes the Haldane phase for and the dimer phase for . Thus, the quantum criticality at the TB point belongs to Ising universality class. Our mean field Hamiltonian (5) in Majorana fermion representation is the same as (11) in long wave length limit (strictly speaking, one can only compare the mean field theory with the effective field theory after renormalization). The fermion mass and the velocity are related to the mean field parameters up to renormalization factors
Notice that the effective theory (11) is only valid when the spin Hamiltonian is translationally invariant. Otherwise, the gauge field is confined in the dimer phase, and (11) will not describe the low energy behaviors near the transition point correctly.
V discussion and conclusion
We give a few comments about the regimes () and () in the following.
In the region () the pairing term in (2) becomes irrelevant and consequently in our mean-field theory. In this case, the (trial) mean field ground state has a -filled fermi sea, whose fermi points are located at . Physically, for (), the marginally irrelevant instability of the model (1) is the antiferro-nematic order. Kato97 , Nematic  Notice that there should be no true long-ranged antiferro-nematic order in the ground state in one dimension. Nevertheless antiferro-nematic correlation is not included in the mean field ansatz we propose here and we do not expect the corresponding Gutzwiller projected wavefunctions will describe the ground state well in this regime. One need to introduce new mean field parameters which is beyond the scope of the present paper.
The situation may be different at the special point () which corresponds to the integrable SU(3) ULS model. Bethe ansatz solution at this special point indicates that excitations above the SU(3) singlet ground state are gapless Sutherland-1975  and are described by a SU(3) Wess-Zumino-Novikov-Witten (WZNW) model with marginally irrelevant perturbations. Affleck-1986  The physics of the SU(3) ULS model can be obtained from the Gutzwiller projected wavefunction. The pairing term in (3) vanishes at () and the MF Hamiltonian becomes a free fermion model. Moreover since the fermion bands are filled with fermi points at . In this case the variational wavefunction is an SU(3) singlet and is the exact ground state of the SU(3) Haldane-Shastry model with inverse-square interactions. Kawakami-1992  Since the effective field theory of the SU(3) Haldane-Shastry chain is just an unperturbed version of the SU(3) WZNW model, Kawakami-1992  which shares similar low-energy physics with the ULS model, we expect that the Gutzwiller projected fermi sea state is a good variational wavefunction for the () antiferromagnetic bilinear biquadratic model (see also Tab. 1).
Lastly we consider the () regime. There are two phases. The ferromagnetic phase is located at (). This phase is again beyond our present mean field theory which is designed to describe spin-singlet states. The remaining part () also belongs to the dimer phase and is described by the projected BCS state in strong-pairing regime. The difference between this dimer phase and the dimer phase at () is that the hopping term in (2) becomes irrelevant and vanishes when (), see Tab. 1 for three examples.
To conclude, we introduce in this paper Gutzwiller projection for the mean field states obtained from a fermionic mean-field theory for systems. The method is applied to study the one-dimensional bilinear-biquadratic Heisenberg model. We find that the topology of the mean field state determines the character of the Gutzwiller projected wavefunction. The projected weak pairing states belong to the Haldane phase while the projected strong pairing states belong to the dimer phase. This result is consistent with the gauge theory above the (trial) mean field ground state. Our theory agrees well with the Majorana effective field theory at the TB critical point and the method can be generalized to higher dimensions or other spin models.
We thank Tao Li, Ying Ran, Su-Peng Kou, Yong-Shi Wu, Zheng-Yu Weng, Hui Zhai, Xie Chen and Meng Cheng for helpful discussion. We especially thank Liang Fu and Yang Qi for very helpful discussion about the dimer phase. This work is supported by HKRGC through grant no. CRF09/HKUST3. YZ is supported by National Basic Research Program of China (973 Program, No.2011CBA00103), NSFC (No.11074218) and the Fundamental Research Funds for the Central Universities in China.
Appendix A End states in open Heisenberg chains
It was shown in Ref. LZN, that the mean field ground state in the fermionic mean-field theory for Heisenberg model is topologically nontrivial. As a result, each of the three flavors of fermions have one Majorana zero mode located at each end of an open chain with wavefunction exponentially decay away from the chain end. (We adopt the cartesian coordinate here). It was shown in Ref. Lecheminant-2002, that three Majorana fermions together may represent a spin-1/2 object via , where and is a Majorana fermion operator corresponding to the flavor (see also section IIIB). This suggests that the mean field Majorana zero modes may form spin-1/2 states at each end, in agreement with known results.Ng9394 , White93 , Ng95  In this appendix we show how the Majorana zero modes become the spin-1/2 edge states after Gutzwiller projection. First we demonstrate numerically the existence of Majorana end modes in mean-field theory for open spin chain.
The existence of majorana fermion edge mode in mean field theory can be seen from the mean-field dispersion for open chains (Fig. 5), where the existence of two zero-energy modes (for each favor) are clear. These two Majorana modes, noted as and can be combined into a zero-energy complex fermion mode which can either be occupied or unoccupied. Correspondingly the fermion number of the flavor fermion can be either even or odd (fermion parity) in the ground state.Kitaev-2001  We note that strictly speaking, under open boundary condition, the mean field parameters , and vary from site to site and should be determined self-consistently.Ng9394  We assume for simplicity that they take uniform values determined by closed spin chain.
For the three flavors of fermions, there are a total of six Majorana zero modes, resulting in 8-fold degenerate mean field ground states. Half of these states have odd total number of fermions, and half have even. For chains with fixed length , only half of them survives for the same reason as discussed in section III.
The remaining four states have different fermion parity distributions . One of them is (even, even, even). In this case, the three flavors have equal weight in the projected state, corresponding to a spin-singlet. The other three are (even, odd, odd), (odd, even, odd) or (odd, odd, even). These three states form a spin-triplet because they can be transformed from one to another by a global spin rotation. These four states remains degenerate (in the limit) after Gutzwiller projection, suggesting that the majorana fermion states become stable end spin states after projection.
To confirm the above picture we calculate the correlation function , where and measures the total spin accommodated from sites to and from to , respectively. Note that and . We first compute in the singlet state (even, even, even). The result is shown in Fig. 6 where we find that the correlation function approaches when , indicating that the magnitude of effective end spins ( and is exactly .
The same analysis can also be applied to the triplet states. We consider in Fig. 7 the correlation in the (odd, odd, even) state. It is clear that when , confirming the existence of spin-1/2 effective end spin in the weak-pairing phase.
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