Physics Letters A 373 (2009) 3075–3078

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Physics Letters A

www.elsevier.com/locate/pla

Fermionic representation of a symmetrically frustrated SU(3) model:Application to the Haldane-gap antiferromagnets

Peng Li a,∗, Shun-Qing Shen b

a Department of Physics, Sichuan University, Chengdu 610064, Chinab Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 26 May 2009Received in revised form 17 June 2009Accepted 23 June 2009Available online 1 July 2009Communicated by R. Wu

PACS:75.10.Jm75.40.Gb73.43.Cd

Keywords:Haldane gapBond operatorMean field

The one-dimensional spin 1 bilinear–biquadratic model is re-expressed in a symmetrically frustratedSU(3) model, which facilitates us to introduce a fermionic representation and related bond-operatormean-field theory. By analyzing the gap and the static spin susceptibility, we shows that this treatmentcan easily capture the commensurate and incommensurate Haldane gap phases.

© 2009 Elsevier B.V. All rights reserved.

One-dimensional quantum antiferromagnets exhibit fascinatingproperties in many situations [1]. The spin 1 bilinear–biquadraticchain is extensively studied in the context of Haldane gap phase[2–8]. Intensive studies based on both theoretical analysis and nu-merical simulations have been able to reveal their delicate prop-erties and the well-known properties have become the testingground for various methods. In a previous study, we had proposeda class of symmetrically frustrated SU(N) models for quantummagnets, and developed a corresponding bond-operator mean-fieldtheory to solve it [9]. Although the theory is not restricted to thedimensionality and we had applied it to the two-dimensional anti-ferromagnets [10], but how good is such a mean-field type theoryfor one-dimensional systems? Here we apply it to the famous Hal-dane chain problem to show the main properties of the system iswell described by this simple mean-field theory.

The generalized frustrated SU(N) model reads

H = J1

∑〈i j〉,μν

J μν (ri)J ν

μ(r j) − J2

∑〈i j〉,μν

J μν (ri)J μ

ν (r j), (1)

where J1 and J2 are two coupling constants for nearest neigh-bors, J μ

ν (ri)’s are the N2 −1 generators of SU(N) group and satisfythe algebra [J α

β (ri), J μν (r j)] = δi j(δ

αν J μ

β (ri) − δμβ J α

ν (ri)). The first

* Corresponding author. Tel.: +86 28 8541 2323; fax: +86 28 85412322.E-mail addresses: lipeng@scu.edu.cn (P. Li), sshen@hku.hk (S.-Q. Shen).

0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2009.06.041

term exhibits the SU(N) symmetry [11]. While the second termexhibits SU(N) symmetry only on bipartite lattices. The existenceof both terms makes the model deviate from the SU(N) symme-tries [9]. One can choose J μ

ν (ri) as the fundamental representationwith a single box in Young tableau. We introduce a set of creationand annihilation operators to rewrite the Hamiltonian in the sec-ond quantization representation. In the SU(N) representation eachsite has N quantum states |μ〉 so that we may introduce N pairs ofoperators f †

iμ and f iμ: |i,μ〉 = f †iμ| 〉 with the vacuum state | 〉 at

site i. In this way we can construct the operator, J μν (ri) ≡ f †

iν f iμ ,

with a constraint for single occupancy,∑N

μ=1 f †iμ f iμ = 1, on each

site. This is the so-called hard-core condition even if the particlesare bosons. Interestingly it is found that the generators satisfy theSU(N) algebra for either bosonic or fermionic representation. Sothe Hamiltonian is rewritten as

H = J1

∑〈i j〉

Pij − J2

∑〈i j〉

B†i j Bi j +

∑i

λi

(∑μ

f †iμ f jμ − 1

), (2)

where Pij ≡ ∑μν J μ

ν (ri)J νμ(r j) serves as the permutation opera-

tor, the bond pairing operator Bij = ∑μ f jμ f iμ and the Lagrangian

multipliers λ j are introduced to realize the constraint of singleoccupancy. The permutation operator can be expressed as Pij =∑

μν f † f iν f † f jμ = ς :F † Fij : with Fij = ∑μ f † f iμ , where : : de-

iμ jν i j jμ

3076 P. Li, S.-Q. Shen / Physics Letters A 373 (2009) 3075–3078

notes normal ordering of operators, and ς = 1 for bosons and −1for fermions. For N = 2,3,4, the model above is equivalent to thespin 1/2 X X Z model, the spin 1 bilinear–biquadratic model, andthe SU(2) × SU(2) spin–orbital model [12], respectively. Here wefocus on the case of N = 3, i.e. we study the Haldane-gap phase ofthe one-dimensional spin 1 bilinear–biquadratic model. We choosefermionic representation, i.e. { f iμ, f †

jν} = δi jδμν , and the reasonwill be discussed appropriately later.

For spin 1, each site has three states |mi〉 with mi = −1,0,+1according to the eigenvalues of Sz

i . We reorganize the three statesand define three operators,

f †i1|0〉 = i√

2

(|mi = −1〉 + |mi = 1〉), (3)

f †i2|0〉 = |mi = 0〉, (4)

f †i3|0〉 = 1√

2

(|mi = −1〉 − |mi = 1〉). (5)

In terms of f operators, the three spin operators can be written as

Sxi = ψ

†i Ω

xψi, (6)

S yi = ψ

†i Ω

yψi, (7)

Szi = ψ

†i Ω

zψi, (8)

where

ψ†i = (

f †i1 f †

i2 f †i3

), (9)

Ωx =( 0 −i 0

i 0 00 0 0

), Ω y =

(0 0 00 0 −i0 i 0

),

Ω z =( 0 0 i

0 0 0−i 0 0

). (10)

Then, up to a constant, the above model is transformed to thespin 1 bilinear–biquadratic model,

H = Jφ∑〈i j〉

[cosφ Si · S j + sinφ(Si · S j)

2], (11)

with the couplings transformed as J1 = Jφ cosφ and J2 = Jφ ×(cos φ − sin φ). Its ground state phase diagram has been estab-lished in great detail [7]. The gapped phase of Eq. (11) can bedivided into two intervals: the “commensurate” Haldane phase for−π/4 < φ < φVBS and the “incommensurate” Haldane phase forφVBS < φ < π/4, where φVBS = tan−1 1/3 is the valence-bond-solid(VBS) point [13]. It can be analyzed by the shifting peak of thestatic spin susceptibility χ(q,ω = 0).

As we mentioned above, we choose fermionic representation.This choice is related to the mean fields and the decompositionscheme we are going to introduce. We define the two mean fieldsas the thermodynamic average of the bond operators, F = 〈Fij〉 andB = i〈Bij〉, i = √−1. Now that we have non-negative decomposi-tions,

:F †i j F i j: � 0, (12)

B†i j Bi j � 0, (13)

and we are dealing with the Haldane gap phase with J1, J2 > 0,we had to take ς = −1 (i.e. the fermionic representation) to makesure that

Pij = ς J1:F †i j F i j: � 0, (14)

− J2 B† Bij � 0, (15)

i j

so that nonzero mean fields are reasonable to mimic the lowenergy sectors of the Hamiltonian (2). In this case, we say thedecomposition scheme is semi-negative and the mean fields arenontrivial if nonzero mean fields solutions are presented in theend.

The Hubbard–Stratonovich transformation [14] is performed todecouple the Hamiltonian into a bilinear form. The chemical poten-tial λi is taken to be site-independent, λi = λ, which can be alsoregarded as a mean field. In the momentum space the mean-fieldHamiltonian is

H =∑k,μ

ε(k) f †kμ fkμ − 1

2

∑k

�B(k)(

fkμ f−kμ + f †−kμ f †

kμ

)− λNΛ + NΛ J1 F 2 + NΛ J2 B2, (16)

where ε(k) = λ − �F (k), NΛ is the total number of lattice sites,and we have defined

�F (k) = 2 J1 F cos k; (17)

�B(k) = 2 J2 B sin k. (18)

The Hamiltonian (16) is of a Bardeen–Cooper–Schrieffer (BCS) typeand needs to be diagonalized. By performing the Bogoliubov trans-formation,

γkμ = uk fkμ − vk f †−kμ; γ

†−kμ = uk f †

−kμ + vk fkμ (19)

with the coherence factors satisfying

u2k = 1

2

[1 + ε(k)

ω(k)

], (20)

v2k = 1

2

[1 − ε(k)

ω(k)

], (21)

2uk vk = �B(k)

ω(k), (22)

one can diagonalize the Hamiltonian as

H =∑k,μ

ω(k)γ†

kμγkμ + E0, (23)

where the spectrum and the ground state energy are

ω(k) =√

ε(k)2 + �2B(k), (24)

E0 = −3

2

∑k

ω(k) + 1

2λNΛ + NΛ J1 F 2 + NΛ J2 B2. (25)

We have N = 3 degenerate spectra for quasi-fermions. By optimiz-ing the free energy

F = − 3

β

∑k

ln(1 + e−βω(k)

) + E0 (26)

with respect to the mean fields F , B , and λ, we obtain a set of themean-field equations,∫

dk

2π

ε(k)

ω(k)tanh

βω(k)

2= 1

3, (27)

3

2

∫dk

2π

−ε(k) cos k

ω(k)tanh

βω(k)

2= F , (28)

3

2

∫dk

2π

�B(k) sin k

ω(k)tanh

βω(k)

2= B. (29)

Thus the mean-field Hamiltonian is solved together with the self-consistent equations for the three types of mean fields.

P. Li, S.-Q. Shen / Physics Letters A 373 (2009) 3075–3078 3077

Fig. 1. The ground state energy per site as a function of the parameter θ . The solidline is the solution for F �= 0 and B �= 0. The dot-dashed line is the solution forF �= 0 and B = 0. The dashed line is the solution for F = 0 and B �= 0. The plotshows the nontrivial mean-field solution stands in a range 0 � θ < 0.92.

Fig. 2. The spectra of the quasi-particle for parameters θ = 0.2 (dashed line), 0.65(solid line), 0.9 (dotted line).

The mean-field equations are solved numerically at zero tem-perature to reveal the ground state properties. We take J1 = J cos θ

and J2 = J sin θ , and nonzero solutions is available in the region0 � θ < π/2. The ground state energy per site is plotted as a func-tion of the parameter θ in Fig. 1 and J is set as the energy unit.To compare with the two limiting cases, the ground-state energiesfor the solutions B �= 0, F = 0 and F �= 0, B = 0 are also plotted inthe same figure. The solution with B �= 0, F �= 0 is smoothly con-nected to the strong-coupling SU(3) gapless point θ = 0 [11,15],and represents the valid lowest ground energy state in the param-eter range, 0 � θ < 0.92. Outside this range, the above mean-fieldscheme does not hold true any more.

More interestingly, the commensurate and the incommensurateHaldane gap phases can be clearly inferred from the nontrivial so-lution. First, let us see the spectrum of the quasi-particle. At theSU(3) point, the spectrum is gapless, ω(k∗) = 0 with k∗ = ±π/3,which reflects the so-called period 3 in accordance with Suther-land’s Bethe Ansatz solution [11]. As θ increases, a gapped spec-trum is developed. We found the gap opens up very slowly asa function of θ , �gap ∼ a exp(−b/θ c) with a ≈ 5.58, b = 1.03,c = 0.99 approximately. There exists a transition point θVBS ≈ 0.65,which is recognized as the VBS point [7]. Because the minimaof the spectrum take an incommensurate value 0 < |kmin| < π/3for 0 < θ < θVBS, while the minimum always stays at kmin = 0 forθVBS < θ < 0.92 (Fig. 2).

Fig. 3. The gap, Eq. (30), as a function of the parameter θ .

As a consequence, the gap of the system behaves differently inthe two intervals (Fig. 3)

�gap ={

2√

(2 J2 B)2 − (2λ J2 B)2

(2 J1 F )2−(2 J2 B)2 (0 � θ < θVBS),

2|λ − 2 J1 F | (θVBS < θ < 0.92).

(30)

The exact value for the VBS point is θexactVBS = tan−1 2/3 ≈ 0.588.

So the estimation here has an error about 10%, which is not badfor a simple mean-field theory.

In fact, the commensurate and the incommensurate Haldanegap phases can be well characterized by the static spin suscep-tibility. In order to calculate spin susceptibility we define the Mat-subara Green’s function in the form of a 2 × 2 matrix,

G(k, τ ) =( −〈Tτ fkμ(τ ) f †

kμ(0)〉 −〈Tτ fkμ(τ ) f−kμ(0)〉−〈Tτ f †

−kμ(τ ) f †kμ(0)〉 −〈Tτ f †

−kμ(τ ) f−kμ(0)〉

),

(31)

where Tτ is the imaginary time order operator. The Green’s func-tion of the frequency is worked out as follows,

G(k, iωn) = iωnσ0 + ε(k)σz + �B(k)σx

(iωn)2 − ω2(k), (32)

where ωn = (2n+1)π/β for all integer n. Due to the degeneracy ofthe spectra the Green’s functions are independent of the index μ.

The Green’s function facilitates us to obtain the imaginary timespin–spin correlations,

χαβ(q, τ ) = ⟨Tτ Sα(q, τ )Sβ(−q,0)

⟩= 1

NΛ

∑k

[Tr

(ΩαΩβ

)G11(k + q, τ )G22(−k, τ )

−Tr(ΩαΩβT )

G12(k + q, τ )G21(−k, τ )]

(33)

where Sα(q, τ ) = eτ H Sα(q)e−τ H with Sα(q) = ( 1√NΛ

)∑

j Sαj e−iqR j

(α = x, y, z), ΩβT is a transpose of Ωβ . Due to rotational symme-try, one can verify that χ xx = χ yy = χ zz and others are zero. InMatsubara frequency, we have

χαβ(q, iωn) =β∫

0

dτ eiωnτ χαβ(q, τ ), (34)

and the spin susceptibility is obtained by analytical continuation,χαβ(q, iωn → ω + i0+). The static spin susceptibility χ zz(q,ω = 0)

at zero temperature is shown in Fig. 4. At the SU(3) symmetricpoint, the sharp peaks occur at q = ±2π/3 and is exactly diver-gent, which is explained by the period 3 in the SU(3) chain based

3078 P. Li, S.-Q. Shen / Physics Letters A 373 (2009) 3075–3078

Fig. 4. The static spin susceptibility χ zz(q,ω = 0) for parameters θ = 0.2 (dashedline), 0.65 (solid line), 0.9 (dotted line).

on the fact that it needs 3 sites to form a singlet. And the diver-gence leads to the gaplessness. But this does not mean the systemdevelops a long-range spin order since the static spin structurefactor χ zz(q, τ = 0+) has no divergence (not shown here). As θ

increases, the sharp peaks become non-divergent and shift fromq = ±2π/3 to incommensurate values. As a consequence, a finitegap opens up for the system. When the parameter θ exceeds θVBStill the value 0.92, the broad peaks always stay at q = ±π , i.e.the system enters into the commensurate gapped phase. All theobtained results here are significantly consistent with the knownproperties of the system.

In conclusion, we have applied the formalism of a symmetri-cally frustrated SU(3) model and related bond-operator mean-fieldtheory to the spin 1 bilinear–biquadratic model. We analyze theformation of the Haldane gap phase in the fermionic representa-tion and found the theory can capture the commensurate and in-commensurate Haldane gap phases successfully. Why such a mean-field type theory can give a quite good result? The answer liesin: (i) the introduced bond operators reflect the short-range cor-

relations in the one-dimensional chain adequately; (ii) the strongcoupling in the model is well described by the assumed nonzeromean fields and the subsequent nonperturbative treatment.

Before ending this Letter, we point out that the commensurate–incommensurate phases can also be characterized by the stringorder parameter and edge states [5,8]. These two interesting sub-jects haven’t been touched in above discussion and will be consid-ered in our future work. An intriguing problem is the relationshipbetween the bond-operator mean-field theory used here and theextensively used bosonization technique [7]. Our theory is definedon a discrete lattice and based on a physical picture similar to theresonanting valence bond (RVB) state [9,14]. To find a link betweenthem constitutes another interesting subject.

Acknowledgements

This work was supported by the Research Grant Council ofHong Kong under Grant No. HKU 703804.

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