Fermionic theory for quantum antiferromagnets with spin S�12

Zheng-Xin Liu,1 Yi Zhou,2,* and Tai-Kai Ng1

1Department of Physics, Hong Kong University of Science and Technology,Clear Water Bay Road, Kowloon, Hong Kong2Department of Physics, Zhejiang University, Hangzhou 310027, People’s Republic of China

�Received 8 June 2010; revised manuscript received 16 September 2010; published 13 October 2010�

The fermion representation for S= 12 spins is generalized to spins with arbitrary magnitudes. The symmetry

properties of the representation is analyzed where we find that the particle-hole symmetry in the spinon Hilbertspace of S= 1

2 fermion representation is absent for S�12 . As a result, different path-integral representations and

mean-field theories can be formulated for spin models. In particular, we construct a Lagrangian with restoredparticle-hole symmetry, and apply the corresponding mean-field theory to one-dimensional �1D� S=1 and S=3 /2 antiferromagnetic �AFM� Heisenberg models, with results that agree with Haldane’s conjecture. For aS=1 open chain, we show that Majorana fermion edge states exist in our mean-field theory. The generalizationto spins with arbitrary magnitude S is discussed. Our approach can be applied to higher dimensional spinsystems. As an example, we study the geometrically frustrated S=1 AFM on triangular lattice. Two spin liquidswith different pairing symmetries are discussed: the gapped px+ ipy-wave spin liquid and the gapless f-wavespin liquid. We compare our mean-field result with the experiment on NiGa2S4, which remains disordered atlow temperature and was proposed to be in a spin-liquid state. Our fermionic mean-field theory provide aframework to study S�

12 spin liquids with fermionic spinon excitations.

DOI: 10.1103/PhysRevB.82.144422 PACS number�s�: 75.10.Kt, 75.10.Jm, 71.10.Hf

I. INTRODUCTION

Quantum magnetism is one of the oldest central problemsin condensed matter and many body physics. Few exact re-sults are known except at one dimension �1D� or when long-range spin order is established at low temperature. In thelater case, Landau’s symmetry-breaking paradigm is appli-cable, and the long-range spin order determines the low-energy physics of the system. The low-energy spin excita-tions are spin waves or magnons.1 Systematic corrections tospin-wave theory can be obtained through semiclassical ap-proach �1 /S expansion2� and/or nonlinear � model.3 How-ever, it was suggested that for some antiferromagnetic mate-rials, strong quantum fluctuations due to small spinmagnitude and low dimensionality combined with geometricfrustration may lead to quantum coherent, spin disorderedground states.4,5 In such cases, a quantum fluid approach6 isprobably a better starting point to describe the low-energyphysics instead of usual spin-wave or semiclassical ap-proach.

There exist at present two common quantum fluid ap-proaches based on different “particle” representations ofspins. In the “bosonic” approach6,7 spins are represented byspin up and down Schwinger bosons with the constraint thattotal boson number at each site is 2S. A spin-disordered stateis obtained in a mean-field theory as long as Bose condensa-tion does not occur. The low-lying excitations coming fromthis theory are bosonic. The other possibility is to representspins by fermions. Up to now, the fermionic approach ismainly restricted to S= 1

2 spin systems.6,8 Because of thePauli exclusion principle, we cannot put more than two spinup or down fermions on a site to form S�

12 objects as in

bosonic approach. One way out is to introduce more speciesof fermions. It was proposed in Ref. 9, that �2S+1� speciesof boson/fermions can be used to construct the spin-S swap-ping operators. Later, it is shown that the fermionic represen-

tation can be used to describe a S=1 valence bond solidstate.10

In this paper, we systematically study the fermionic rep-resentation for arbitrary spins. To construct a spin operatorwith spin magnitude S, we follow Ref. 9 and introduce dif-ferent fermion operators for different Sz states. We find thatfor half-odd-integer spins, the spin operator thus constructedis invariant under a SU�2� transformations,8 i.e., the symme-try group is SU�2� group whereas for integer spins the sym-

metry group is U�1��Z2. Fermionic path-integral formula-tions for spin systems with arbitrary spin S can then beformulated and the corresponding mean-field theories can bestudied. We shall see that one of the mean-field theories ap-plied to 1D antiferromagnetic �AFM� Heisenberg model pro-duces mean-field excitation spectrums which agrees withHaldane’s conjecture.3 Moreover, there exist zero-energyMajorana edge modes for open integer spin chains. We alsoapply our approach to two-dimensional �2D� antiferromag-netic Heisenberg models on triangular lattices where we ob-tain two translational and rotational invariant spin-liquid so-lutions, one �with px+ ipy-wave paring symmetry� is gappedand the other one �with f-wave pairing symmetry� has nodesat the spinon Fermi level. Comparing with experiments onNiGa2S4, we argue that the f-wave pairing spin liquid is aplausible ground state.

It should be noted that rather different fermionic/bosonicspinon approaches have also been applied to SU�N� spinmodels where the dynamic symmetry is generalized fromSU�2� to SU�N�, and the Hamiltonian is composed of SU�N�generators.6,11,12 In these approaches, N species of particlesare introduced to construct the SU�N�-spin operator andHamiltonians.12 The major difference between our approachand the SU�N� approaches is that �2S+1� species of fermionsare introduced to form a SU�2� spin �or irreducible represen-tation of SU�2� group� in our approach. Unlike the SU�N�models,12 our approach is not limited to bipartite lattices.

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The paper is organized as follows. In Sec. II, we introducethe general fermionic spinon representation for arbitrary spinS and discuss the symmetry properties associated with therepresentation. In Sec. III, we introduce the path-integral for-malism where different ways of imposing particle numberconstraints are presented. A particle-hole symmetric way ofimposing constraint is discussed. In Secs. IV A and IV B, wefocus on 1D Heisenberg models for spin-1 and spin-3/2, re-spectively, and study two versions of the mean-field theories.The cases for general spin S are discussed in Sec. IV C. InSec. V we study 2D AFM models on triangular lattice. Ourpaper is summarized and concluded in Sec. VI.

II. GENERAL FERMIONIC SPIN REPRESENTATION

We begin with the fermionic representation for spins. Forspin S= 1

2 , two species of fermions c↑ and c↓ are introducedto construct the three spin operators Sx,y,z. To generalize thisfermionic representation to arbitrary spin S, we introduce2S+1 species of fermionic operators cm satisfying anticom-mutation relations,

�cm,cn†� = �mn, �1�

where m ,n=S ,S−1, . . . ,−S. The spin operator can be ex-pressed in terms of cm and cn

†’s,

S = C†IC ,

where C= �cS ,cS−1 , . . . ,c−S�T and Ia �a=x ,y ,z� is a �2S+1�� �2S+1� matrix whose matrix elements are given by

Imna = �S,m�Sa�S,n .

It is easy to see that the operators Sa satisfy the SU�2� angu-

lar momentum algebra, �Sa , Sb�= i�abcSc. Under a rotationaloperation, C is a spin-S “spinor” transforming as Cm

→DmnS Cn and S is a vector transforming as Sa→RabSb, here

DS is the 2S+1-dimensional irreducible representation ofSU�2� group generated by I and R is the adjoint representa-tion.

As in the S=1 /2 case, a constraint that there is only onefermion per site is needed to project the fermionic systeminto the proper Hilbert space representing spins, i.e.,

�Ni − Nf��phy = 0, �2�

where i is the site index and Nf =1 �the particle representa-tion, one fermion per site�. Alternatively, it is straightforwardto show that the constraint Nf =2S �the hole representation, asingle hole per site� represents a spin equally. The Nf =1representation can be mapped to the Nf =2S representationvia a particle-hole transformation. For S= 1

2 , the particle pic-ture and the hole picture are identical, reflecting an intrinsicparticle-hole symmetry of the underlying Hilbert spacewhich is absent for S�1.

Following Affleck et al.,8 we introduce another spinor C= �c−S

† ,−c−S+1† ,c−S+2

† , . . . , �−1�2ScS†�T, whose components can

be written as Cm= �−1�S−mc−m† , where the index m runs from

S to −S as in C. To examine whether C is really a spinor, we

construct a spin singlet state for two spins at site i and j,Ci

†Cj2S+1

�vac= 12S+1

�m=−SS �−1�S−m�S ,mi�S ,−m j. Therefore Ci

†Cj

is a scalar operator as Ci†Cj, meaning that C and C must

behave identically under spin rotation. Consequently, the

spin operators can also be written in terms of C,

S = C†IC .

Combining C and C into a �2S+1��2 matrix �= �C , C�,8 wecan reexpress the spin operators as

S =1

2Tr��†I�� �3�

and the constraint can be expressed as

Tr���z�†� = 2S + 1 − 2Nf = � �2S − 1� , �4�

where the + sign implies Nf =1 and − sign implies Nf =2S.We are interested in two kinds of operations �or groups�

acting on �. The first kind belongs to a �2S+1�-dimensionalirreducible representation of SU�2� group, acting on the leftof �. Suppose G is an element of this irreducible represen-tation, then G†IaG=RabIb �R belongs to the adjoint irreduc-ible representation of SU�2� group, which is a SO�3� matrix�.Under transformation �→G�, the spin operator Sa

= 12Tr��†Ia�� becomes Sa→RabSb, which means a rotation of

the spin. It is obvious that the particle number constraint Eq.�4� remains unchanged under the action of G.

The other kind belongs to a 2�2 unitary group acting onthe right of � which keeps the spin operator Eq. �3� and thefermionic statistics Eq. �1� invariant. This group reflects thesymmetry properties of the underlying Hilbert space struc-ture in the fermionic representation. We call it an internalsymmetry group. The internal symmetry group is different

for integer and half-integer spins. It is U�1��Z2= �ei�z ,�xe

i�z=e−i�z�x ;�R� for the former and SU�2� forthe latter. We leave the rigorous proof in Appendix A, andshall explain qualitatively the reason behind here. Notice that

C and C are not independent. The operators in the internalsymmetry group will “mix” the two fermion operators in the

same row of C and C, i.e., cS and c−S† , cS−1 and −c−S+1

† , etc.For integer spins, c0 and �−1�Sc0

† will be “mixed.” To keepthe relation �c0 ,c0

†�=1 invariant, there are only two methodsof “mixing:” one is an U�1� transformation, the other is in-terchanging the two operators. These operations form the

U�1��Z2 group. For half-odd-integer spins, the pair �c0 ,�−1�Sc0

†� do not exist, and the symmetry group is the maxi-mum group SU�2�. The difference between integer and half-integer spins is a fundamental property of the fermionic rep-resentation as we shall see more later.

Now let us see how the constraints Eq. �4� transform un-der the symmetry group. For S= 1

2 , the constraint Eq. �4� isinvariant under the transformation �→�W because the righthand side vanishes �due to the particle-hole symmetry of theHilbert space�. For integer spins, if W=ei�z, then W�zW

†

=�z, and Eq. �4� is invariant. If W=�xei�z, then W�zW

†

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=−�z, meaning that the particle picture �+ sign in Eq. �4��and the “hole” picture �− sign in Eq. �4�� are transformed toeach other.

For a half-odd-integer spin with S�3 /2, W�SU�2� is arotation and we need to extend the constraint into a vectorform similar to S= 1

2 case8 so that Eq. �4� becomes

Tr����†� = �0,0, � �2S − 1��T. �5�

Under the group transformation �→�W,

Tr����†� → �R−1��0,0, � �2S − 1��T, �6�

where W�aW†=Rab�b, a ,b=x ,y ,z, i.e., R is a 3�3 matrixrepresenting a three-dimensional rotation in the internal Hil-bert space. The transformed constraint represents a new Hil-bert subspace which is still a �2N+1�-dimensional irreduc-ible representation of the spin SU�2� algebra. Anymeasurable physical quantity such as the spin S remains un-changed in the new Hilbert space. Therefore, for half-odd-integer spins �S�3 /2�, there exists infinitely many ways ofimposing the constraint that gives rise to a Hilbert subspacerepresenting a spin. However, for integer spins, there existsonly two possible constraint representations.

The different representations of constraints for S�12 sys-

tems result in different path-integral representations and dif-ferent mean-field theories. These mean-field theories areequivalent in the sense that they can be transformed to eachother by the internal symmetry group. In the next section, weshall study the Heisenberg model in different representationsand shall construct a mixed path-integral representationwhich restored particle-hole symmetry, and the internal sym-metry group becomes “almost” a gauge symmetry.8 Themixed representation is studied in Sec. IV where a distinctmean-field theory is proposed which recovers Haldane con-jecture at 1D for the Heisenberg model.

III. PATH-INTEGRAL FORMALISM AND MEAN-FIELDTHEORY FOR HEISENBERG MODEL

We shall focus on the antiferromagnetic Heisenbergmodel H=J��i,jSi ·S j with J�0 in the rest of the paper. Westart by presenting some useful algebraic manipulations ofthe Hamiltonian, and then discuss the general path-integralformalism. The different ways of handling constraints andhow they affect the symmetry of the Lagrangian will be dis-cussed in the process.

A. Heisenberg model and an effective Hamiltonian

It is known that for spin-12 the Heisenberg interaction can

be written as8

Si · S j = −1

8Tr:��i

†� j� j†�i�:

=−1

4:�ij

† ij + �ij† �ij�: , �7�

where

ij = Ci†Cj, �ij = Ci

†Cj �8�

are spin-singlet operators. ij and �ij will be extended toarbitrary spin magnitudes with the above definition and willbe used in the following discussions. Interestingly, an expres-sion almost the same as Eq. �7� holds for S=1, but this is nolonger true for larger spins. We shall present precise formulafor S=3 /2 and S=2, and provide a general discussion forhigher spins.

For S=1, the three spin matrices are

I+ = �I−�† = �0 2 0

0 0 2

0 0 0 , Iz = �1 0 0

0 0 0

0 0 − 1 ,

where I�= Ix� iIy. The matrix operator � is given by

� = �CC� = � c1 c−1†

c0 − c0†

c−1 c1† .

The spin operator can be written in form of Eq. �3�, and itcan be shown after some straightforward algebra that theHamiltonian can be written as

H = J��i,j

Si · S j = −J

2 ��i,j

Tr:��i†� j� j

†�i�:

=− J��i,j

:�ij† ij + �ij

† �ij�: . �9�

For S=3 /2, the three spin matrices are

I+ = �I−�† =�0 3 0 0

0 0 2 0

0 0 0 3

0 0 0 0 , Iz =�

3

20 0 0

01

20 0

0 0 −1

20

0 0 0 −3

2

.

and the matrix operator � is given by �= �CC�, where C

= �c3/2 ,c1/2 ,c−1/2 ,c−3/2�T and C= �c−3/2† ,−c−1/2

† ,c1/2† ,−c3/2

† �T. Inthis case the singlet operators ij and �ij alone are notenough to represent the Heisenberg Hamiltonian, and triplethoping and pairing terms are necessary. After some straight-forward but tedious algebra, we find that the Hamiltonian forspin S=3 /2 Heisenberg model can be written as

H = J�i,j

Si · S j

= −J

4�i,j

:�Tr��i†I� j · � j

†I�i� + S2 Tr��i†� j� j

†�i��: .

�10�

It is interesting to note that the above form holds also for S=1.13 However, for S=1, the two terms in the square bracket

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can be transformed from one to another, which does not holdfor S=3 /2. This suggests that S= 1

2 ,1 Heisenberg modelshave a higher “hidden symmetry” compared with S=3 /2.Actually, there is another hidden symmetry for the spin S= 1

2 systems which is absent for spin-1,

:ij† ij: = :�ij

† �ij:− NiNj ,

where Ni=Ci†Ci is the particle number on site i. Our analysis

suggests that smaller spins have higher symmetry when thespin interaction is expressed in the fermionic representation.

For S=2, the Heisenberg Hamiltonian can be written as

H = J��i,j

Si · S j = −J

2 ��i,j

Tr:��i†I� j · � j

†I�i�: .

As in the case of lower spins, the terms Tr: ��i†I� j ·� j

†I�i�:and Tr: ��i

†� j� j†�i�: have finite overlaps and are not com-

pletely independent of each other. For S�2, it is not possibleto represent the Hamiltonian in the above two terms alone.Quintet and higher multipolar hoping and pairing operatorsare needed to represent the Heisenberg Hamiltonian and weare not able to obtain any general expression.

Summarizing, the Heisenberg model can be written in thefermionic representation as

H = −J

2 ��i,j

�a�S�Tr:��i†� j� j

†�i�:

+ b�S�Tr:��i†I� j · � j

†I�i�:+ ¯� , �11�

where a�S� ,b�S� , . . . are parameters dependent on the spinmagnitude S and ¯ represents quintet and higher multipletterms. In the following, we shall study in detail a Hamil-tonian which keeps the first two terms only. The Hamiltoniancan be considered as an effective Hamiltonian for construct-ing trial ground-state wave functions in a variational calcu-lation. Notice that the effective Hamiltonian is in fact “exact”up to S=2 if a�S� and b�S� are chosen properly.

B. Path-integral formalism and constraints

In imaginary time path-integral formalism, the partitionfunction is given by

Z = Tr�e−�H� =� D�D�†D e−�0�L��,�†, �d�,

L = �i

1

2�Tr �i��� − i i�z��i

† � i�2S − 1� i� + H , �12�

where i is the Lagrange multiplier field introduced to im-pose the constraint Eq. �4�. The +�−� sign corresponds to the

particle �N=1� and hole �N=2S� representations. The Grass-mann number � satisfy the antiperiodic boundary condition��0�=−����.

Notice that Lagrangian �12� is not invariant under the ac-tion of the internal symmetry group because of the nonin-variant particle number constraint Eq. �4�, i.e., the internalsymmetry group of the spin operator is not a symmetry

group of the Lagrangian. In the following we will employ atrick to restore the symmetry of the Lagrangian. The integer-and half-odd-integer spin models will be discussed sepa-rately because of the intrinsic difference in their internalsymmetry groups.

1. Integer spins

Our trick is to consider an average of all the partitionfunctions with different particle-number constraint represen-tations. In the integer spin case, the average is given by

Z =� D�D�†e−�0���

i1/2Tr��i���i

†�+H�d�

� ��i,��

1

2���Ni − 1� + ��Ni − 2S��

=� D�D�†D e−�0���i�1/2�Tr��i���i

†�+H�d�

� ��i,��

1

2�ei i/2�Tr��i�z�i

†�−2S+1� + ei i/2�Tr��i�z�i†�+2S−1�� .

�13�

Notice that we do not make any approximations in derivingEq. �13� from Eq. �12� and the partition function is a faithfulrepresentation of Heisenberg model except that it averagesover all possible particle and hole representations of con-straints locally. Similar ideas have been applied to generatethe supersymmetric representation14 and the SU�2� represen-tation of the t-J model.15 The fermion particle-hole symme-try is restored in this representation. The Lagrangian corre-sponding to Eq. �13� is

L =1

2�i�Tr��i��� − i i�z��i

†� − 2 ln cos�2S − 1� i

2� + H .

This form of Lagrangian is invariant under the internal sym-

metry group U�1��Z2 and the symmetry group becomes al-most a “gauge symmetry” of the new Lagrangian as we shallsee in the following. We note that the Lagrangian is complexbecause of the multiplier . This problem can be solved bylifting the contour of integration over into complex planevia analytic continuation to find a saddle point in the imagi-

nary axis.16 For convenience, we write i= i i, and integrate

i from −i� to i� �the saddle point of i is real�. Then theLagrangian becomes

L =1

2�i�Tr��i��� − i�z��i

†� − 2 ln cosh�2S − 1� i

2� + H .

�14�

We shall now consider the path-integral representation interms of the effective Hamiltonian �11� keeping only the firsttwo terms. The effective Hamiltonian can be decoupled by aHubbard-Stratonovich matrix field �see Appendix B�

Uij = �i†� j = �ij − �ij

†

�ij − ij† � ,

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Vij = �i†I� j = �vij uij

†

uij vij† � ,

where uij =Ci†ICj, and vij = Ci

†ICj. Note that uij and vij formtwo sets of spin triplet operators, respectively. The Hamil-tonian becomes

H =J

2 ��i,j

�a�S�Tr�Uij† Uij − �Uij

† �i†� j + H.c.��

+ b�S�Tr�Vij† · Vij − �Vij · � j

†I�i + H.c.��� . �15�

The Uij, Vij, and i fields define U�1��Z2 lattice gauge fieldscoupling to the fermionic particles which we shall callspinons in the following. Under the local gauge transforma-tion �i→�iWi, the temporal component and the spacial com-ponent of the gauge fields transform as

i�z → Wi†� i�z −

d

d��Wi,

Uij → Wi†UijWj ,

Vij → Wi†VijWj . �16�

Notice that i changes sign under an uniform Z2 gauge trans-formation, and ij, �ij exchange their roles under a “stag-gered” Z2 transformation where we have �x on one sublatticeand I on the other sublattice. Notice also that the Lagrangianis not invariant under a time-dependent gauge transforma-

tion, because the term 2 ln cosh�2S−1� i

2 is not invariant. Asimilar situation occurs in the supersymmetric representationof Heisenberg model.14 Because of this restriction the inter-nal symmetry group does not generate a complete “gaugesymmetry” in the path-integral formalism.

Integrating out the fermion fields, we get an effective ac-

tion in terms of the Uij, Vij, and i fields. The mean-fieldvalues of the these fields are given by the saddle point of theeffective action determined by the self-consistent equations

U= �Uij, V= �Vij with the mean-field Hamiltonian

Hm = J��i,j

�a�S��− ��Ci†Cj + ��Ci

†Cj + H.c.� + ��2 + ���2�

+ b�S��− �v� · Ci†ICj + u� · Ci

†ICj + H.c.� + �u�2 + �v�2��

+ �i� �Ni −

2S + 1

2� − ln cosh

�2S − 1� 2

� , �17�

where = �Ci†Cj, �= �Ci

†Cj, u= �Ci†ICj, v= �Ci

†ICj, and is determined by the condition

�Ni −2S + 1

2=

2S − 1

2tanh

�2S − 1� 2

.

Notice that the averaged particle number satisfies 1� �Ni�2S because we are mixing the particle and hole pictures inthe constraint. If the ground state does not break the particle-

hole symmetry, then =0 and we get the half-filling condi-

tion �Ni= �2S+1� /2.

2. Half odd integer spins

We first examine how the two particle number constrainsTr���z�

†�= � �2S−1� are transformed into each other by theSU�2� internal symmetry group. We consider the vector con-straint Eq. �5� for S�

12

M =1

2S − 1Tr����†� = �0,0,1�T,

where

Mx =1

2S − 1Tr�CC† + CC†� ,

My =i

2S − 1Tr�CC† − CC†� ,

Mz =1

2S − 1Tr�CC† − CC†� .

The Mx and My components indicate that the one-site pair-ings cSc−S , . . . and c1/2c−1/2 are equal to 0 and the Mz compo-nent imposes the constraint that there is exactly one fermionper site. Notice that the first two components of the con-straint is automatically satisfied if the third component issatisfied rigorously. Under the SU�2� transformation �→�W, the constraint becomes 1

2S−1Tr����†�=R−1�0,0 ,1�T,where W�aW†=Rab�b, R is an SO�3� rotation. Since the con-straint is invariant under an U�1� gauge transformation gen-erated by �z, the continuum space formed by different con-strains is the surface of a sphere SU�2� /U�1�=S2. Inparticular, for W=ei�x�/2, the north pole of the sphere whichcorresponds to the positive sign in Eq. �4� is transformed tothe south pole, R−1�0,0 ,1�T= �0,0 ,−1�T, corresponding tothe negative sign in Eq. �4�.

Now we construct a Lagrangian which is invariant underthe SU�2� internal symmetry group. In the path-integral for-malism, the particle number constraint 1

2S−1Tr��i��i†�

= �Mix ,Mi

y ,Miz�T= �0,0 ,1�T can be realized by introducing a

vector Langrange multiplier field �i

��Mix���Mi

y���Miz − 1�

= � 1

2��3� d i

xd iyd i

z

�exp�i� ixMi

x + iyMy + i

z�Miz − 1��� . �18�

The vector �0,0 ,1�T changes into R−1�0,0 ,1�T= �nx ,ny ,nz�T

= n under SU�2� gauge transformation, where n is an unitvector on the surface of the sphere S2. As in the integer spincase, a SU�2� invariant path-integral formalism can be ob-tained by averaging over all possible particle number con-straints

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���Mix − nx���Mi

y − ny���Miz − nz��n�

=1

4�� 1

2��3� d2n� d3 i exp�i�i · �Mi − n��

= � 1

2��3� d3 i exp�i��i · Mi − i ln

sin i

i�� , �19�

where i=� ix�2+ � i

y�2+ � iz�2. Introducing �i= i�i, we ob-

tain a Lagrangian with “almost” SU�2� gauge symmetry

L =1

2�i�Tr��i��� − �i · ���i

†� − �2S − 1�lnsinh i

i

� + H .

�20�

As in the case of integer spins, we replace H by the effectiveHamiltonian Eq. �11� with only the first two terms. Introduc-ing the Hubbard Stratonovich field as in the integer spin case�see Appendix B for details�,

Uij = �i†� j = �ij �ij

†

�ij − ij† � ,

Vij = �i†I� j = �vij − uij

†

uij vij† � ,

then Uij, Vij, and �i form an SU�2� lattice gauge field cou-pling to the spinons. �Notice the sign difference in the matrix

elements of U and V between integer and half-odd-integerspins.� Under gauge transformation �i→�iWi�Wi�SU�2��the gauge fields transform as,

�i · � → Wi†��i · � −

d

d��Wi,

Uij → Wi†UijWj

Vij → Wi†VijWj ,

and the Lagrangian Eq. �20� is invariant only under time-independent gauge transformations as in the integer spin case

because of the noninvariant term �2S−1�lnsinh i

i. This is dif-

ferent from the S= 12 case, where the Lagrangian is invariant

under time-dependent gauge transformations. The mean-fieldeffective Hamiltonian corresponding to Eq. �20� is

Hm = J��i,j

�a�S��− ��Ci†Cj + ��Ci

†Cj + H.c.� + ��2 + ���2�

+ b�S��− �v� · Ci†ICj + u� · Ci

†ICj + H.c.� + �u�2 + �v�2��

− �i

2S − 1

2 �� · Mi + lnsinh

� �21�

where the parameters ,� ,u ,v are solved self-consistently

by the mean-field equations U= �Uij, V= �Vij, and � is de-termined by

�Mi +�

2� coth − 1� = 0.

The averaged particle number per site is again given by

�Ni= �2S+1� /2 if the ground state respects particle-hole

symmetry ��=0�.

C. An important difference between integer andhalf-odd-integer spins

It can be proved �see Appendix B� that for integer spins, ji=ij

† , � ji=−�ij, u ji=uij, and v ji=vij† whereas for half-odd-

integer spins, ji=ij† , � ji=�ij, u ji=−uij, and v ji=vij

† . Noticethat the parity of the paring terms �ij and uij are different forinteger and half-odd-integer spins. These operators are cen-tral to the mean-field theory, because the ground states arecompletely determined by their expectation values. In par-ticular, for a mean-field theory with � ,�0 and u ,v=0, themean-field ground state is a BCS spin-singlet pairing statewhere the order parameter ��ij has even parity for half-odd-integer spins, but has odd parity for integer spins. We shallsee how this important difference leads to different excitationspectrums between integer and half-odd-integer spin systemsin the fermionic mean-field theory. Similar result exists forstates with � ,=0 and u ,v�0. In this case the mean-fieldground state is a BCS spin-triplet pairing state and the parityof the order parameters are reversed.

We note that as in the S= 12 case, the mean-field Hamil-

tonian should be viewed as a trial Hamiltonian for theground-state wave function of the spin systems afterGutzwiller projection. For the mean-field theory withparticle-hole symmetry the Gutzwiller projection is rathernontrivial. The state is a coherent superposition of stateswhich allows sites with both one fermion and with 2S fermi-ons. The numerical analysis of such a state is complicatedand we shall not go into details in this paper.

IV. FERMIONIC MEAN-FIELD THEORY IN 1DAND HALDANE CONJECTURE

In this section we apply our mean-field theory to the an-tiferromagnetic Heisenberg model in one dimension. We firstconsider the cases of spin S=1 and S=3 /2 where two ver-sions of mean-field theories based on different methods ofimplementing the constraints will be discussed. The mean-field results are summarized in Sec. IV C where the case ofgeneral spin S and Haldane conjecture will be discussed.

To simplify our analysis we make use of the Wagner-Mermin theorem which asserts that a continuous symmetrycannot spontaneously break in one dimension.17 Thereforewe assume in our mean-field theory that the ground state is aspin singlet �spin-liquid state� and the expectation values of�uij and �vij are zero, since the rotational symmetry will bebroken otherwise. As a result, we keep only the a�S� term inthe trial Hamiltonian. For simplicity we shall also restrictourselves to translational invariant solutions of the mean-field theory in this paper.

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A. Integer spin: S=1

We choose Htr to be the same as Eq. �9�, i.e., a�1�=−1and b�1�=0. For convenience of discussion, we introducetwo different fermions cs ,ca which are the symmetric andantisymmetric combinations of c1 and c−1,

csi = �c1i + c−1i�/2,

cai = �c1i − c−1i�/2. �22�

The mean-field Hamiltonian Eq. �17� becomes completelydecoupled in terms of cs, ca, and c0. In Fourier space, it canbe written as

Hm = �k

k�csk† csk + cak

† cak + c0k† c0k�

− �k��k

�

2�cs−kcsk − ca−kcak − c0−kc0k� + H.c.�

+ JN���2 + ���2� − N�3

2 + ln cosh

2� , �23�

where k= −2J cos k, �k�=2iJ� sin k �since the phases of

and � are unimportant in 1D, we choose and � to be realnumbers� and N is the length of the chain. The mean-fieldHamiltonian can be diagonalized by a Bogoliubov transfor-mation

�sk = ukcsk + vkcs−k† ,

�ak = ukcak − vkca−k† ,

�0k = ukc0k − vkc0−k† . �24�

The coefficients uk ,vk satisfy the Bogoliubov-de Gennes�BdG� equations,

Ekuk = kuk − �k�vk

Ekvk = − �kuk − kvk. �25�

Solving the equations we obtain

Ek = � − 2J cos k�2 + �2J� sin k�2, �26a�

uk = cosk

2, vk = i sin

k

2, �26b�

where k is given by tan k=�k

�

ikand the diagonalized Hamil-

tonian is

Hm = �k�0

Ek��sk† �sk + �ak

† �ak + �0k† �0k� + E0,

where E0=�k�k−Ek�+JN�2+�2�−N� 32 +ln cosh

2 � is theground-state energy. Minimizing E0, we obtain the self-consistent mean-field equations at zero temperature,

=2S + 1

2N�

k

cos k�1 −k

Ek� , �27a�

� =2S + 1

2N�

k

sin k�k

Ek, �27b�

� 2S + 1

2− Ni� =

2S + 1

2N�

k

k

Ek= −

2S − 1

2tanh

�2S − 1� 2

�27c�

with S=1. The above equations are solved numerically

where we obtain =�=3 /4 and =0 as shown in the firstrow of Table I. The averaged particle number of the spinonsper site is 3/2, indicating that the ground state respectsparticle-hole symmetry. The ground-state energy is E0=−J�2+�2�, and the excitation spectrum is flat �i.e., k inde-pendent� with a finite energy gap 3J /2. The spin-spin corre-lation function is

�Si · Si+r =3

2N2�p,q

ei�q−p�r��1 +p

Ep��1 −

q

Eq� −

�p�

Ep

�q

Eq�

and is nonzero only for r=0 and r=1, with �Si ·Si=3 /2 and�Si ·Si+1=3 /4. The correlation function is zero for r�2, in-dicating that the mean-field theory describes a short-rangedvalence-bond-solid state.

We have also studied the mean-field theory based on theLagrangian Eq. �12� for comparison. It gives rise to a dual of

mean-field theories with particle number constraints �Ni=1

and �Ni=2, respectively. There is a one-to-one correspon-dence between the solutions of the two mean-field theories

and we will only consider the particle representation �Ni=1. The mean-field equations are the same as Eq. �27� exceptthat Eq. �27c� becomes

�Ni =3

2N�1 − �k

k

Ek� = 1.

The solution is also summarized in Table I. The excitationspectrum is gapped but no longer dispersionless. We notethat the mean-field solution Eq. �27� with particle-hole sym-metry has a better ground-state energy.

TABLE I. Solutions of two versions of mean fields for spin-1 AFM Heisenberg chain. E is the ground-state energy per site and Egap is the excitation gap.

�Ni � E�J� Egap�J� �Si ·Si

3/2 3/4 3/4 0 −9 /8 3/2 3/2

1 0.7625 0.5671 0.7709 −0.9032 0.3634 4/3

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B. Half-odd-integer spin: S=3 Õ2

For the spin-3/2 Heisenberg chain, we consider the effec-tive Hamiltonian �21� with b� 3

2 �=0 such that the groundsstate is a singlet state. We shall also take a� 3

2 �= 1516 for reason

we shall see later. In momentum space, the effective Hamil-tonian becomes

Hm = �k

k�c3/2k† c3/2k + c1/2k

† c1/2k + c−1/2k† c−1/2k + c−3/2k

† c−3/2k�

+ �k

��k��c−3/2−kc3/2k − c−1/2−kc1/2k� + H.c.� + const,

where k= z− 158 J cos k and �k

�= x− i y − 158 J�� cos k. We

shall set and � to be real numbers in the following.As in the S=1 case, the Hamiltonian can be diagonalized

by Bogoliubov transformations and lead to the self consistentmean-field equations at zero temperature,

=2S + 1

2N�

k

cos k�1 −k

Ek� ,

� =2S + 1

2N�

k

cos k�k

Ek,

�M+ =2S + 1

N�2S − 1��k

�k

Ek=

�1 − coth � +

2,

�Mz =2S + 1

N�2S − 1��k

k

Ek=

�1 − coth � z

2, �28�

where S=3 /2, M+=Mx+ iMy, += x+ i y, and

Ek =� z −15

8J cos k�2

+ � + −15

8J� cos k�2

�29�

is the spinon dispersion.

Solving the equations we find that �=0 �i.e., the groundstate does not break particle-hole symmetry� and there existsinfinite degenerate solutions for and � satisfying2+ ���2=1.2732 �see Table II�. This is a direct conse-quence of the SU�2� gauge symmetry for half-odd integerspins we mentioned in Sec. III B 2, and all these solutionsare equivalent. Notice that the ground-state energy E0

=− 15NJ16 �2+ ���2� is the same as the expectation value of

Heisenberg Hamiltonian �10�, which is why we shouldchoose a� 3

2 �= 1516 in the effective Hamiltonian. As a result of

the particle-hole symmetry ��=0�, the spinon energy disper-

sion Eq. �29� is gapless at Fermi points k= �� /2. The spin-spin correlation is given by

�Si · Si+r =15

4N2�p,q

ei�q−p�r��1 +p

Ep��1 −

q

Eq� −

�p�

Ep

�q

Eq�

and decays at large distance as �Si ·Si+r�r−2 because of lin-earized spectrum around Fermi surface. This is also con-firmed directly by numerical calculation.

Similar to the S=1 case, we have also solved the mean-

field theory with particle number constraint �Ni=1 or

equivalently �M= �0,0 ,1�T. The solution is listed in Table

II. Since z�0 in this case, the spinon dispersion Ek

= 15J8

� z /2− cos k�2+ �� cos k�2 breaks particle-holesymmetry and has a finite gap over the whole Brillouin zone.The particle-hole symmetric solution is also found to has alower mean-field ground-state energy.

C. Haldane’s conjecture and edge states

Comparing the mean-field energy dispersion Eqs. �26a�and �29�, we find that the excitation spectrum of spin-1Heisenberg model is gapped whereas the excitation spectrumfor the spin-3/2 Heisenberg model is gapless in the mean-field formulation with particle-hole symmetry. This differ-ence between integer spin and half-odd-integer spin persistsfor any spin S, if we consider the trial Hamiltonian �17� or�21� with a�S��0, b�S�=0, i.e., if we consider BCS spin-singlet ground-state wave functions. The mean-field equa-tions for arbitrary S are the same as Eq. �27� or Eq. �28� forarbitrary S, and the dispersion is qualitatively the same asEq. �26a� or Eq. �29�, as long as we consider particle-holesymmetric mean-field solutions which do not break transla-tional invariance. The main effect of changing S in mean-field theory is to change the number of fermionic spinonspecies. This result is consistent with the Haldaneconjecture,3 which asserts that the integer spin Heisenbergchains have singlet ground state with finite excitation gapsand exponentially decaying spin-spin correlation functions,while half-odd-integer spin chains have singlet ground stateswith gapless excitation spectrums and power law decayingcorrelations. Our particle-hole symmetric mean-field resultsagree well with Haldane conjecture.

Haldane noticed that the time-reversal operator satisfiesT2=1 for integer spin and T2=−1 for half-odd-integer spin,and this results in different Berry phase contributions fromthe topological excitations �skymion or instanton� in thepath-integral formulation given by ei2�SQ, where

TABLE II. Solutions of two versions of mean fields for spin-3/2 Heisenberg chain. In the first row � ,��means the real combinations satisfying 2+ ���2=1.2732.

�Ni � z E�J� Egap�J� �Si ·Si

2 � 0 −1.5198 0 15/4

1 0 1.1490 0.7181 −1.2377 0.6732 45/16

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Q =� dtdx1

4�n · ��tn � �xn� �30�

is the Skyrmion number. For Q=odd integer, the Berry phase�ei2�S�Q is 1Q=1 for integer spins and �−1�Q=−1 for half-odd-integer spins. The difference in the �1 factor betweeninteger spin and half-odd-integer spin is the origin ofHaldane’s conjecture. The situation is quite similar in our

mean-field theory. Noticing that �ij = Ci†Cj is a singlet formed

by two spin-S particles. The Clebsch-Gordan coefficients im-plies that � ji=−�ij for integer S and � ji=�ij for half-odd-integer S �see Appendix B�. This sign or parity differenceresults in appearance of � sin k in Eq. �26a� for S=1 and� cos k in Eq. �29� for S=3 /2, which leads to different sym-metries of ground-state wave functions and different excita-tion spectrums between integer and half-odd-integer spinchains.

The topological term Eq. �30� leads to exponentially lo-calized edge states for open integer spin chains which isabsent for half-odd-integer spin chains.18 This important dif-ference between integer spin and half-odd-integer spinchains is also reflected in our mean-field theory where zero-energy Majorana edge fermions exist at the open boundariesfor integer spin chains which are absent for half-odd-integerspin chains. The existence of Majorana edge fermions forinteger spin chains is a direct consequence of the odd-pairingsymmetry for integer spin chains.19 The details of the Majo-rana edge fermions is discussed in Appendix C.

V. 2D: S=1 SPIN LIQUIDS ON TRIANGULAR LATTICE

We show in the previous section that our mean-fieldtheory is able to capture the essential physics of spin-liquidstates in 1D antiferromagnetic quantum spin chains. In thissection we shall apply our mean-field theory to the 2D J1−J3 Heisenberg model on triangular lattice. We shall showthat our mean-field theory admits spin-liquid solutions notexplored before. Some results of this model have been re-ported in a previous paper,20 where particle number con-

straint is treated in the “particle representation” ��Ni=1�. Inthis paper, we shall revisit this model with the particle-holesymmetric constraint. The Hamiltonian of the J1−J3 model is

H = ��i,j

J1Si · S j + J3��i,j�

Si · S j , �31�

where �i , j denotes nearest neighbor �NN� and �i , j� thenext-next-nearest neighbors �NNNN�. For S=1, there are twochannels of decoupling the spin interaction, namely, Si ·S j

=− J2 :Tr��i

†� j� j†�i�: or Si ·S j =− J

2 :Tr��i†I� j ·� j

†I�i�:. Sincethere is no Mermin-Wigner theorem to protect us at zerotemperature at 2D, we have to keep both terms in the effec-tive Hamiltonian Eq. �17�. We shall choose b=1−a, with theweight a determined by minimizing the ground-state energy.

Next we introduce the mean-field parameters 1, �1, u1,and v1 �for NN� and 3, �3, u3, and v3 �for NNNN� to de-couple the Hamiltonian. We assume that u and v are parallelso we only keep the z component of these vectors, and notethem simply as u1,3 and v1,3 �which are set as real numbers�.

As in 1D case we consider here only spin-liquid solutionsthat respect translation and rotational symmetries. Since thepairings have odd parity ��ij=−�� ji, only p-wave andf-wave like pairings are allowed. Solving the mean-fieldequations we find two kinds of solutions �see Fig. 2�, whichhave f-wave and px+ ipy-wave pairing symmetries, respec-tively. We shall concentrate on these two kinds of solutionsin the following.

In momentum space �Fig. 1�, the mean-field HamiltonianEq. �17� can be written as

Hm = �k

k�c1k† c1k + c0k

† c0k + c−1k† c−1k�

− �k��k

��c−1−kc1k −1

2c0−kc0k� + H.c.�

+ �k

vk�c1k† c1k − c−1k

† c−1k� − �k

uk��c−1−kc1k + H.c.�

�32�

with

k = − aZ�J11�k + J33�2k� ,

�k� = iaZ�J1�1�k + J3�3�2k� ,

vk = − �1 − a�Z�J1�kv1 + J3�2kv3� ,

uk� = �1 − a�Z�J1�ku1 + J3�2ku3� ,

where is the Lagrange multiplier and

�k =1

3�cos kx + cos�−

kx

2+

3ky

2� + cos�−

kx

2−

3ky

2�� ,

�kf =

1

3�sin kx + sin�−

kx

2+

3ky

2� + sin�−

kx

2−

3ky

2�� ,

FIG. 1. �Color online� The first Brillouin zone of the triangularlattice. b1= �2� , 2�

3� and b2= �0, 4�

3� are two reciprocal vectors. The

Brillouin zone is divided into three regular pieces by the three newbases u, v, and w.

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�kp+ip =

1

3�sin kx + ei�/3 sin� kx

2+

3ky

2� + ei2�/3

�sin�−kx

2+

3ky

2�� .

The pairing symmetries can be verified easily by expandingthe pairing terms at small k: �k

f �kx�kx2−3ky

2� and �kpx+ipy

�kx+ iky. There are three lines of zeros for the f-wave pair-ing function as shown in Fig. 2.

Introducing a vector Ck= �c1k ,c−1−k† ,c0k ,c0−k

† �T, the Hamil-tonian Eq. �32� can be written in a matrix form

Hm = �k

Ck†HkCk + �

k�3

2k − vk� , �33�

where

Hk =�k + vk − uk − �k 0 0

− uk� − �k

� − k + vk 0 0

0 0k

2

�k

2

0 0�k

�

2−

k

2

.

Hk can be diagonalized by the Bogoliubov transformation,

Ak = cosk

2c1k − sin

k

2ei�kc−1−k

† ,

B−k† = sin

k

2e−i�kc1k + cos

k

2c−1−k

† ,

Dk = cos�k

2c0k + sin

�k

2ei�kc0−k

† , �34�

where tan k=�uk+�k�

k, ei�k =

uk+�k

�uk+�k�, tan �k=

��k�k

, and ei�k =�k

��k�.

The corresponding eigenvalues are given by vk�E1k and�E0k, where

E1k = k2 + ��k�2 + uk

2, E0k = k2 + ��k�2,

and the self-consistent mean-field equations are

1,3 =1

N�k

�k,2k�3

2−

k

E1k−

k

2E0k� ,

�1,3 =1

N�k

�k,2k� ��k�E1k

+��k�2E0k

� ,

u1,3 =1

N�k

�k,2kuk

E1k,

v1,3 = 0,

�Ni −3

2� =

1

2tanh

2, �35�

where we have adopted the particle-hole symmetric con-straint in writing down the mean-field equations. We have

also checked the mean-field solutions with constraint �Ni=1 �Ref. 20� and found that the excitation spectrums arequalitatively the same as those in the particle-hole symmetrictheory. The ground-state energy is lowest for vanishingu1,3 ,v1,3, i.e., a=1, uk=vk=0 and the ground state is a spinsinglet. The excitations are characterized by three branchesof fermionic spinons with Sz=0, �1 and identical dispersionEk=k

2+ ��k�2.Our mean-field theory for the J1−J3 model contains two

regimes of spin-liquid states for both f and px+ ipy pairingsymmetries as a function of J1 /J3. A first-order phase transi-tion occurs between the two regimes at J1 /J3�1 �see Fig. 3�.When J1 dominates, the spin-liquid state is characterized by1,3�0 and �1,3�0 �consequently �Si ·Si+1�0 and�Si ·Si+2�0�; while when J3 dominates, 1=�1=0 and 3�0, �3�0 �consequently �Si ·Si+1=0, �Si ·Si+2�0�. Thepx+ ipy states remain lower in energy in both regimes.

FIG. 2. �Color online� Two different pairing symmetries for thesinglet pairing �ij. We indicate the relative phase of the nearest-neighbor pairing ��ij=�1ei�ij. �a� f-wave pairing. The dished linesare the zero lines of the pairing function �in lattice space or mo-mentum space�. �b� px+ ipy-wave pairing. There are no zero lines,the pairing function vanishes only at the � point.

FIG. 3. �Color online� Phase diagram for the J1−J3 Heisenberg

model with J1+J3=1. We use the constraint �Ni−32 = 1

2 tanh

2 . Thered dotted line indicate energy per site for the px+ ipy state and theblack square line is for the f-wave state. Dirac nodes exist for thef-wave state. The number of nodes is 6 when J1 is dominating and24 when J3 is dominating.

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The f-wave pairing solution respects particle-hole sym-metry and has �Ni= 3

2 . The excitation is gapless with severalDirac cones in the first Brillouin zone. A cut of the spinondispersion is shown in Fig. 4 where the particle-hole symme-try is obvious. The mean-field solution with �Ni=1 has simi-lar properties, except that the particle-hole symmetry is lostand the position of the Dirac nodes are shifted.20

For the px+ ipy-wave pairing, we find that the ground state

breaks particle-hole symmetry and �0. The excitationspectrum is fully gapped. Similar to integer spin chains, wefind that the px+ ipy-wave state is topologically nontrivial

�see also Appendix C�. The solution has � ��Z�J11+J33�so that k can be either positive or negative, depending on k.Therefore the Bogoliubov spinor space described by the vec-tor �

Im�k

Ek,

Re�k

Ek,

k

Ek�T �which is S2� covers the Brillouin zone �k

space, which is also S2� at least once. In other words, thetopological �Skyrmion� number of mapping from the k spaceto the spinor space is nonzero m�0. However, in the

vacuum where the spinon density is zero, is very big andk can only take positive values, which gives a zero topo-logical number.19 Since the bulk and vacuum belong to dif-ferent topological sectors, the boundary defines a domainwall between the two phases, and there should exists gapless�chiral� Majorana edge states in the px+ ipy state followingthe analysis of Read and Green.19 In this sense, thepx+ ipy-state describes a time-reversal symmetry-breakingtopological spin liquid.

Next we consider the specific heat and spin susceptibilityfor the f and px+ ipy states. For the px+ ipy-wave pairingstates, since the excitations are fully gapped as in BCS su-perconductors, the specific heat and spin susceptibility willshow exponential behavior21 at low temperature. For thef-wave pairing states, both the specific heat and the spinsusceptibility show power law behavior due to the Diraccone structure �see Fig. 4�. The energy for the system is E

=3�kEk

e�Ek+1in mean-field theory. At low temperature, the spe-

cific heat is dominate by the low-energy excitations near thenodes of the fermi surface, which is given approximately by

Ek=vFk, where vF is the fermi velocity �the anisotropy of thedispersion can be removed by a rescaling of momentum andwill not affect the result qualitatively�. The magnetic specificheat is given by

CM =�E

�T

= 3�

�T�

0

k� vFk

evFk/T + 12�kdk

= 3�

�T�

0

vFk�/T→� 1

ex + 12�x2 T3

vF2 dx � n�T2/vF

2 , �36�

where n is the number of Dirac cones. The magnetic suscep-tibility can be calculated from the linear response theory with

z�k,i�� =� d2r� d�T��Sz�r,��Sz�0,0�ei���−kr�

=2

N2� d2r� d�ei���−kr� �q,q�,p,p�

ei�q�−q�r

� �T��c1†�q,��c1�q�,��c1

†�p,0�c1�p�,0�

− T��c1†�q,��c1�q�,��c−1

† �p,0�c−1�p�,0�� .

�37�

The static susceptibility is given by z=limk→0 z�k ,0��T

v f2 .

The detailed calculation is given in Appendix D.Recently, a spin-1 material NiGa2S4 was discovered,

where the S=1Ni2+ ions form a triangular lattice with anti-ferromagnetic �AFM� interaction. The system has a Curie-Weiss temperature �80 K, and is found to exhibit no con-ventional magnetic order down to 0.35 K. Several plausibleground states have been proposed for this system,22–28 in-cluding an antiferronematic state,24 a ferronematic state,23

and a KT phase driven by vortices.28 Spin liquid is also pro-posed to be a plausible ground state. In our mean-fieldtheory, the f-wave pairing spin liquid has coherent gaplessexcitation with specific heat scaling as T2 at low temperaturewhich is consistent with the experimental result.22 Further-more, the Knight shift data in Ref. 23 shows that the intrinsicsusceptibility decreases with decreasing temperature �but thevery low-temperature data are missing.23� This matchesqualitatively with our mean-field result Eq. �37�. We there-fore proposed that the f-wave pairing spin-liquid state is aplausible ground state for the material.

VI. COMMENTS AND CONCLUSION

Summarizing, we show in this paper how the fermionicrepresentation for spin S= 1

2 systems can be generalized tospin systems with S�

12 . The symmetry group of the spin

operator is SU�2� group for integer spin and U�1��Z2 groupfor half-odd-integer spin. Different path-integral formula-tions and mean-field theories are developed corresponding todifferent ways of handling the constraints. In 1D, we showthat the particle-hole symmetric mean-field theory for S=1and S=3 /2 spin chains are consistent with Haldane’s conjec-

FIG. 4. �Color online� The spinon dispersion the f-wave pairing

state with constraint �Ni−32 = 1

2 tanh

2 for J1=0.6 and J3=0.4. Thefigure on the right shows the dispersion along the line linking two Kpoints of the Brillouin zone �see the blue lines on the left�. Thenegative energy part is filled and the excitation is gapless with sixDirac nodes in the first Brillouin zone. The dispersion for the mean-

field solution with �Ni=1 is similar to the above one except that thelocations of the nodes are shifted a little bit.

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ture, and we argue that the difference reflects a fundamentaldifference between integer and half-odd-integer spin chains.We also study 2D spin-1 AFM on triangular lattice where wefind two spin-liquid states, one is a gapless f-wave spin liq-uid and the other is a topological px+ ipy spin-liquid state.We propose that the gapless f-wave spin liquid is a plausibleground state for the material NiGa2S4. Our approach can beapplied to any other spin models and provides a distinct ap-proach to spin-liquid states for S�

12 spin systems.

We note that because of limitation in scope we have ad-dressed only a very limited number of issues in the study ofspin-liquid states in this paper. Within the mean-field theorywe have restricted ourselves to spin-liquids solutions ofHamiltonian �31� with spin rotational symmetry �1D� andlattice translational symmetry �1D and 2D�. We note thatsolutions which break spin rotational symmetry exist in ourtheory which may serve as ground states of Hamiltonian�31�.20 For example, we show in Ref. 20 that state with long-ranged magnetic order exists as lowest energy state of themean-field theory of the J1−J3 model and antiferronematicorder may exist when the biquadratic �K�Si ·S j�2� spin-spininteraction exists in the Hamiltonian. For simplicity we onlyconsidered the mean-field states without breaking the trans-lational symmetry. We note that states that break translationalsymmetry �such as dimerized states� are believed to beground states of some 1D or 2D spin models.29

Another very important issue is whether the spin-liquidstates we find are stable against gauge fluctuations. In 1D,the gauge fluctuations can be removed by a time-dependentRead-Newns gauge transformation and thus have no effect tothe low-energy properties.6 The situation is very different intwo or higher dimensions where the stability of the mean-field state depends on dimensionality and the �gauge� struc-ture of the gauge field fluctuations. For instance, Z2 spinliquid is believed to be stable at 2D �Ref. 30� and gaplessU�1� Dirac fermionic spin liquids is stable at 2D in thelarge-N limit.31 In our case of spin-liquid solutions for theS=1 Heisenberg model, the U�1� gauge fluctuation is gappedby the spinon-pairing term via the Anderson-Higgs mecha-nism. The ground state is stable against the low-energygauge fluctuations which is described by an effective Z2gauge theory. We note that the emergence of gauge-fieldstructure is a direct consequences of particle number con-straints and a reliable answer to the question of stability ofspin-liquid states can be obtained only if we can handle theparticle number constraint reliably. Gauge field theory canonly handle long distance, low-energy gauge fluctuations anda more satisfactory answer to the question of stability of thespin-liquid states can be obtained only after the GutzwillerProjection wave functions are studied carefully.

ACKNOWLEDGMENTS

We thank Michael Ma for the discussion about 1D mod-els. We also thank Patrick. A. Lee and Naoto Nagaosa forsuggesting the px+ ipy ansatz for AFM on triangular latticeand thank Cheung Chan and Xiao-Yong Feng for helpfuldiscussions. Z.X.L. and T.K.N. are supported by RGC grantof HKSAR under Project No. HKUST3/CRF/09. Y.Z. is sup-

ported by National Basic Research Program of China �973Program, Grant No. 2011CB605903�, the National NaturalScience Foundation of China �Grant No. 11074218�, and theFundamental Research Funds for the Central Universities inChina.

APPENDIX A: SYMMETRY GROUP OF THE SPINOPERATORS AND THE PAIRING PARITY

We shall look for allowed transformations �→�W, whichkeep the spin operator Eq. �3� invariant. At first glance, thiscondition is satisfied as long as W�U�2�. However, it is nottrue because cm’s and cm

† ’s are not independent. By checkingeach row of � directly, it is easy to see that�cm , �−1�S−mc−m

† � transforms to �cm , �−1�S−mc−m† �W and

�cm† , �−1�S−mc−m�→ �cm

† , �−1�S−mc−m�W�. Noticing that�cm

† , �−1�S−mc−m�= �−1�S−m�c−m , �−1�S−mcm† ��x, we obtain

�− 1�S−m�c−m,�− 1�S−mcm† ��x → �− 1�S−m�c−m,�− 1�S−mcm

† ��xW�

or

�c−m,�− 1�S−mcm† � → �c−m,�− 1�S−mcm

† ��xW��x. �A1a�

On the other hand, �c−m , �−1�S+mcm† � transforms to

�c−m,�− 1�S+mcm† �W .

Noticing that

�c−m,�− 1�S+mcm† � = �c−m,�− 1�−S−mcm

† �

= �c−m,�− 1�S−mcm† ���z�−2S

and ��z�−2S= ��z�2S, we obtain

�c−m,�− 1�S−mcm† ���z�2S → �c−m,�− 1�S−mcm

† ���z�2SW

or

�c−m,�− 1�S−mcm† � → �c−m,�− 1�S−mcm

† ���z�2SW��z�2S.

�A1b�

Equations �A1a� and �A1b� impose the condition

�xW��x = ��z�2SW��z�2S. �A2�

For integer spin, Eq. �A2� becomes W�=�xW�x, which re-

sults in W=ei�z or W=ei�z�x, i.e., W�U�1��Z2, whereU�1�= �ei�z� is the usual gauge symmetry group and Z2= �I ,�x� is the particle-hole symmetry group. Notice that thetwo groups don’t commute, thus the total symmetry group isa semidirect product of the U�1� group and the Z2 group

U�1��Z2= �ei�z ,�xei�z=e−i�z�x ;�R�. For half-odd-

integer spin, Eq. �A2� implies W�=�yW�y, which is equiva-lent to det W=1 since W†W=1. This indicate that W�SU�2�. Thus the symmetry group is SU�2� group.

APPENDIX B: PROOF OF SOME ALGEBRAICRELATIONS

First we introduce a �2S+1�� �2S+1� matrix B, which isessential in our proof

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B =�1

− 1

1

¯

. �B1�

The matrix is the CG coefficients combining two spin S to a

spin-singlet state, i.e., �0,0=Bmm��S ,m1�S ,m�2=C†C�vac.Thus C can be expressed as C=BC�=B�cS

† ,cS−1† , . . . ,c−S

† �T.Notice that for integer spin B−1=B while for half-odd-integerspin B−1=−B.

The following are some common properties of the B ma-

trix. Suppose R is an rotation operation, and D�R� is its�2S+1�� �2S+1�-dimensional matrix representation, then

we have RC=D�R�C, and

RC = D�R�C = D�R�BC�.

On the other hand, RC�=D�R��C� so

RC = R�BC�� = BRC� = BD�R��C�.

Comparing these two, we have D�R�B=BD�R�� or

B−1D�R�B = D�R��. �B2�

This means that B is the transformation matrix which trans-fers the representation matrix of a rotational operator to itscomplex conjugate. In general, D�R� can be written asD�R�=eiI·R, where R are the SU�2� group parameters �real

numbers� for the operator R, and consequently D�R��

=e−iI�·R. From Eq. �B2� we have

B−1D�R�B = B−1eiI·RB = eiB−1IB·R = D�R�� = e−iI�·R.

So we have

B−1IB = − I� = − IT.

Noticing B−1= �B, above equation is equivalent to BIB−1

=−I�=−IT.Now let us consider the spin algebra. The spin interaction

is decomposed by the following two operators:

�i†� j = �Ci

†Cj Ci†Cj

Ci†Cj Ci

†Cj

� , �B3�

�i†I� j = �Ci

†ICj Ci†ICj

Ci†ICj Ci

†ICj

� . �B4�

We define ij =Ci†Cj, �ij = Ci

†Cj and uij = Ci†ICj, vij =Ci

†ICj.The following discussions distinguish integer- and half-odd-integer spins and prove the results in Sec. III C.

1. Integer spins (B†=BT=B−1=B)

The second term on the first row of Eq. �B3� can be writ-ten as

Ci†Cj = Ci

†BCj� = − �Ci

†BCj��T = − Cj

†BCi� = − Cj

†Ci = − �ij†

In second step, we have used the fact that the transverse of ascalar operator is equivalent to reversing the order of theconstituting components. Commuting two fermionic opera-tors we obtain a minus sign. On the other hand, � ji

†

= �Cj†Ci�†=Ci

†Cj, so � ji=−�ij.The second term on the second row of Eq. �B3� is

Ci†Cj = Ci

TBBCj� = − �Ci

TCj��T = − Cj

†Ci = − ij† .

It is obvious that ji=ij† .

To simplify Eq. �B4�, we will use the following relations:BIB=−IT, BI=−ITB, and IB=−BIT. The second term on thefirst row of Eq. �B4� is

Ci†ICj = Ci

†IBCj� = − Ci

†BITCj = �Ci†BITCj

��T

= Cj†IBCi

� = Cj†ICi = �Ci

†ICj�† = uij† . �B5�

On the other hand, above operator can be written into an-

other form, Ci†ICj = �Cj

†ICi�†=u ji† . Comparing with Eq. �B5�

we get uij =u ji.The second term on the second row of Eq. �B4� is

Ci†ICj = Ci

TBIBCj� = − Ci

TITCj� = �Ci

TITCj��T = Cj

†ICi = vij† .

�B6�

And the relation v ji† =v ji manifests itself.

2. Half-odd-integer spins (B†=BT=B−1=−B)

Repeating the above procedures, it is straightforward toshow that

Ci†Cj = �ij

† ,Ci†Cj = − ij

† ,

Ci†ICj = − uij

† ,Ci†ICj = vij

† �B7�

with ji=ij† , � ji=�ij , u ji=−uij , v ji=vij

† .

APPENDIX C: EDGE STATES IN OPEN INTEGERSPIN CHAINS

It is known that integer spin antiferromagnetic Heisenbergmodel has free edge states with spin magnitude S /2.18 Ma-jorana fermion edge states also exist in our mean-field theoryunder open boundary condition. Our discussion will followthe argument of Ref. 19 for 2D pairing fermions. First, wepoint out the existence of two topologically distinct phasesunder periodic boundary condition and then we discuss thezero-energy edge state solution at an open boundary.

In our mean-field theory, all the properties of the groundstate are completely determined by uk and vk �or k and �k�.Noticing that uk and vk obey �uk�2+ �vk�2=1 so they can beviewed as a spinor. This spionor is equivalent to a vector

�u� ,v����u ,v�T= �0,sin k , cos k�T= �0,�k

�

iEk,

k

Ek�T where we

have used Eq. �26b�. Obviously, the spinor �u ,v� spans a S1

space. Recalling that the first Brillouin zone is also S1, thefunctions uk ,vk describe a mapping from S1 �k space� to S1

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�spinor space�. The mapping degree is characterized by thefirst homotopy group �1�S1�=Z. When −2J� �2J�which is the case for our mean-field theory�, the map isnontrivial because every point of the spinor S1 is coveredonce, in other words, the mapping degree is m=1. This to-pological number defines a phase, we call it A phase. When� ��2J, the mapping is topologically trivial since k is al-ways positive �or negative� and the lower half circle �or up-per circle� of the spinor S1 is never covered so the mappingdegree is m=0. We call this region B phase. Since the map-ping cannot be smoothly deformed from m=0 to m=1, atopological phase transition occurs at � �=2J. We will seethat the existence of zero-energy edge states is tied with aphase boundary between the bulk �A phase� and the vacuum�B phase�.

Next we consider an infinite chain with a single edge. Weassume that the edge is located at x=0, and x�0 is thevacuum where the spinon density is zero. The boundary canbe described by a potential V�x� that is large and positive atx�0 so that the wave function of the spinon vanishes expo-nentially at x�0. Equivalently, we can assume a position-dependent Lagrange multiplier term �x� which becomesvery large and positive �so that the condition � ��2J issatisfied� at x�0. This implies that the vacuum belongs tothe B phase and the boundary is a domain wall between Aphase and B phase.

We now study the edge states in the continuum approxi-mation. It is convenient to introduce ��x�=2J− �x� suchthat � is positive in the bulk, becomes zero near the edge andturns negative outside the edge. Expanding k and �k to firstorder in k near k=0, we obtain k�−� and �k�2iJ�k. Tosee the effects of the domain wall, we consider the BdGequation Eq. �25�, which can be written in position space as

i�u

�t= − �u − 2J�

�v�x

,

i�v�t

= �v + 2J��u

�x, �C1�

where we have replaced E and k by i ��t and −i �

�x , respec-tively. The BdG equation is compatible with u�x , t�=v�x , t��,the spinor �u ,v� satisfying this relation describes Majoranafermions. There is a normalizeable zero-energy solution forthe above equations. Putting u=v, then

2J��u

�x= − �u ,

and the solution is

u�x� = v�x� = � e−1/2J��x��x�dx.

Notice that the BdG equation Eq. �C1�

Eu = − �u − 2J��v�x

,

Ev = �v + 2J��u

�x

at E�0 admits no normalizable bound solutions. This showsthat the zero mode is the only plausible bound state solution.

In the case of a pair of boundaries at x=0 and x=W, wecan replace the two first-order equations by a second-orderequation,

E2�u � v� = ��2 − �2J��2 �2

�x2 � �2J����

�x��u � v� ,

and the energy E is not zero for finite W. However, twobound state solutions with energy going to zero exponen-tially with increasing W exist. The solution u+v is centeredat x=0 and u−v is centered at x=W. Notice that there is onlyone solution for the above BdG equations, and the twomodes u�v are Majorana fermions.

The above discussion is applicable to our mean-fieldtheory for S=1 open antiferromagnetic chain. The discussioncan be generalized to larger integer spins S�1 if we adoptthe effective Hamiltonian �23�.

Unlike the S=1 case, our S=3 /2 mean-field theory pro-duces a mapping from k-space S1 to the �u ,v� space S1 whichis topologically trivial �essentially because of the even parityof the pairing term�, and the Majorana edge states don’t existanymore. This is in agreement with the result that no expo-nentially localized edge state exists for open half-odd-integerspin chains. The existence of power-law localized edge statesin half-odd-integer spin chains18 is more subtle and is notproduced by our mean-field theory.

APPENDIX D: CALCULATION OF THE SUSCEPTIBILITY

Substituting the Bogoliubov transformation Eq. �34� intothe Kubo formula Eq. �37�, we obtain

z�k,i�� =2

V� d�ei���

q

��Cq2GA�q,− �� − Sq

2GB�− q,���

��− Cq+k2 GA�q + k,�� + Sq+k

2 GB�− q − k,− ���

− CqSq�GA�q,− �� + GB�− q,���

� Cq+kSq+k�GA�q + k,�� + GB�− q − k,− ����

=2

�V�q,i�

��Cq2GA�q,i�� − Sq

2GB�− q,− i���

��− Cq+k2 GA�q + k,i� + i�� + Sq+k

2 GB�− q − k,

− i� − i��� − CqSqCq+kSq+k�GA�q,i�� + GB�− q,

− i����GA�q + k,i� + i�� + GB�− q − k,− i�

− i����

with the notations Cq=cosq

2 and Sq=sinq

2 . Using the resultsGA�k , i��=GB�k , i��=G�k , i�� and G�−k , i��=G�k , i��= 1

i�−Ek, we get the static susceptibility

z = limk→0

z�k,0�

= limk→0

2

�V�q,i�

�− �CqCq+k + SqSq+k�2G�q,i��G�q + k,i��

+ �CqSq+k − SqCq+k�2G�q,i��G�q + k,− i���

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= limk→0

2

V�

q�− �CqCq+k + SqSq+k�2nf��q+k� − nF��q�

�q+k − �q�

= − 2� d2q�nF��q�

��q�D1�

which is the Pauli susceptibility for free fermions. At low

temperature, the excitation spectrum can be approximated byDirac cones so we have

z � − 2�0

�

2�q�nF��q�vF � q

dq =4�

vF�

0

�

nF��q�dq �T

vF2 .

�D2�

*yizhou@zju.edu.cn1 J. Van Kranendonk and J. H. Van Vleck, Rev. Mod. Phys. 30, 1

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Si · S j =− J4 : �−Tr��i

†I� j ·� j†I�i�+S2 Tr��i

†� j� j†�i��:. Notice that

the first term in the square bracket has different sign comparingto the expression for S=1 and 3/2. The two terms in the squarebracket are also identical.

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2 spinons. However, for 1D half-odd-integer antiferromagnetic spin chains, elementary excitations arespin-1

2 spinons. So far we do not know how this modified spin-wave theory can be generalized to deal with this situation.

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