the one-dimensional frustrated heisenberg antiferromagnet

Journal of Magnetism and Magnetic Materials 264 (2003) 175–182

The one-dimensional frustrated Heisenberg antiferromagnet

P.A. Leboeuf, M.E. Gouv#ea*, A.S.T. Pires

Departamento de F!ısica, ICEx, Universidad Federal de Minas Gerais, Fisica Av. Antonio Carlos 6627-CP 702, 3012 3970, Belo Horizonte

MG, Brazil

Received 21 November 2002; received in revised form 13 February 2003

Abstract

The one-dimensional S ¼ 1=2 antiferromagnetic Heisenberg model with nearest- (J1) and next-nearest-neighbor (J2)

interactions is investigated, over the whole temperature region, by employing the double-time Green function method

and performing a decoupling proposed by Kondo and Yamaji for cases without long-range order. The susceptibility is

calculated and compared to other numerical calculations existing in the literature. We also evaluate the critical value for

the J2=J1 ratio and find good agreement with other theoretical estimates.

r 2003 Elsevier Science B.V. All rights reserved.

PACS: 75.50; 75.10

Keywords: Antiferromagnetic; One-dimensional; Frustration; Long-range interactions

1. Introduction

Low-dimensional quantum models have, for many years, been the subject of considerable research,analytical as well as experimental, due to both, the unconventional physics of these models, and thequantum field theoretical methods used in the analysis of the problem [1]. In particular, a great deal ofattention has been devoted to the study of spin models that can lead to a better understanding on thebehavior of compounds like SrCu2O3; VO2P2O7; and CuGeO3 where a spin gap occurs. The theoreticalwork accumulated in the last decades has shown that several mechanisms can be responsible for creating agap in the energy spectrum, including, for example, competing interactions and quantum fluctuations. As aresult, different one-dimensional (1D) quantum antiferromagnetic models have been proposed in order toallow the study of these effects—isolately or not. The properties of quantum systems are known to bestrongly affected by fluctuations that can, for example, drive a 1D Heisenberg antiferromagnet with onlynearest neighbor (nn) interactions to a lattice dimerization characterizing a Spin-Peierls transition [2,3]. Onthe other hand, it has been shown that magnetic dimerization can also be induced—even in the absence oflattice dimerization—in 1D systems by the so-called frustration effect involving competing interactions[4,5]. Experimental investigations realized on quasi-1D materials have provided crucial support for the

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*Corresponding author. Tel.: +553134995669; fax: +553134995600.

E-mail address: [email protected] (M.E. Gouv#ea).

0304-8853/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0304-8853(03)00198-7

interesting physics related to low-dimensional magnetism and raised questions about the role played byeffects like the two just mentioned: competing interactions and quantum fluctuations.

The nn model is, by far, the most used model (for theoreticians), and it can describe, to some extent,several features of the interesting behavior shown by low-dimensional quantum magnets. However, in orderto understand the role played by competing interactions on those magnets, it is necessary to increment themodel including ingredients as long-range interactions. It is important to remark that the lattice structure ofseveral compounds have suggested [6] the existence of a substantial competing interaction. Then, it is stillnecessary to get a better understanding of the role played by lattice and/or magnetic dimerizations ondetermining the behavior of different materials. In this aspect, several numerical works [7–9] focusing onthe different possibilities have given meaningful contributions.

In the context of the spin-Peierls transition, a microscopic model that has been widely studied is the 1DHeisenberg model described by

H ¼ J1

Xi

1 þ ð�1Þid� �

Si � Siþ1 þ J2

Xi

Si � Siþ2; ð1Þ

where the index i runs over the lattice sites. The spin–lattice interaction that leads to lattice dimerizationenters in this model through the parameter d; related to the lattice distortion. J1 and J2 describe,respectively, the nearest-neighbor and next-nearest-neighbor (nnn) exchange interactions with their ratiol ¼ J2=J1 defining the strength of frustration. Above the spin-Peierls transition temperature, d vanishes andfrustration of spin interactions can become the dominant feature. The importance of introducingfrustration in models used to study phenomena like the spin-Peierls transition has been suggested by someexperimental investigations like the one performed by Hase et al. [10], although the compound investigatedby those authors, CuGeO3; is, nowadays, believed to not be properly described as a quasi-1D material, and,thus, is not a candidate for a spin-Peierls transition.

In this paper, we will be interested on the frustrated Heisenberg antiferromagnet described by the non-dimerized version of Hamiltonian (1), that is,

H ¼X

i

J1Si � Siþ1 þ J2Si � Siþ2f g: ð2Þ

For lo0; the nnn coupling strengthens the correlations produced by the nn coupling reinforcing theantiferromagnetic order. However, for l > 0; the nn and nnn couplings compete with each other because thelatter interaction frustrates the ordering tendency of the former. It is also known that there is a critical valuefor the frustration parameter, lc; defining regions of different behavior for systems described byHamiltonian (2): for 0ololc the Hamiltonian flows to the isotropic fixed point [11–13], the spectrum isgapless, and the correlation functions decay as power laws. For l > lc [4,14], a gap develops in the spectrumand the spin correlations become very short ranged.

The classical version of Hamiltonian (2) has been studied by Harada and Mikeska [15] using thenumerical transfer matrix method to investigate a wide range of exchange constants. Those authorspresented results for the heat capacity and susceptibility but their main concern was the helical short-rangeorder existing for l > 0:25: For the quantum problem, analytical results for the thermodynamics of themodel are available for l ¼ 0 only. The phase diagram of Hamiltonian (2) was obtained by Haldane [11]who transformed the spin-Hamiltonian into an effective Hamiltonian using the Jordan–Wignertransformation and by taking the continuum limit. Applying the renormalization group method, Haldanediscussed the fluid-dimer transition obtaining lc ¼ 1=6: Kuboki and Fukuyama [16] transformed Haldane’seffective Hamiltonian into a phase Hamiltonian and, by using some arguments, obtained lc ¼ 1=3: Castillaet al. [8] performed numerical diagonalization of finite systems described by Eq. (2) with chain lengths up toL ¼ 18 obtaining lc ¼ 0:241270:0001: Obviously, Kuboki and Fukuyama’s estimate for lc are much closer

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P.A. Leboeuf et al. / Journal of Magnetism and Magnetic Materials 264 (2003) 175–182176

to the value obtained by numerical diagonalization than the Haldane’s estimate. However, both theoreticalestimates are indeed very approximate and it seems that lc can only be obtained numerically.

In order to have a better knowledge of the properties of the system we are interested in, it is important tohave some analytical results. Although an exact analytical solution for arbitrary values of l is not known,some universal features of the physical system are usually properly captured by field theoretical studies andseveral papers [4,14,17–19] focusing on models such as the one described by Eq. (2) have been published.Despite being important to know the phase diagram and the asymptotic behavior of the correlationfunctions, it is still necessary to investigate details governed by the physics at short length scales whichremains out of range of applicability of the field theory methods. Moreover, those methods work only atvery low temperatures. If we want to interpret experimental data for a given substance we should haveresults for, say, the susceptibility and specific heat for a wide range of temperatures. The two-time Greenfunction method is well known as a standard method that can give reasonable results over the whole rangeof temperature. In the application of the method, the so-called random-phase or Tyablikov decouplingapproximation is usually introduced to obtain an approximate solution from an infinite set of coupledequations. However, the theory of Tyablikov cannot be applied to Hamiltonian (2), because, in thatapproach, the spin-wave excitation is proportional to magnetization, which vanishes in the present case,with no applied magnetic field. Self-consistent decoupling of the Green function equation of motion atsecond order, rather than at first, was proposed by Kondo and Yamaji [20] and Knapp and ter Haar [21]for the spin-1/2 chain and by Scales and Gersch [22] for arbitrary spin for one-, two-, and three-dimensionallattices. These decouplings lead to spin waves with excitation energies proportional to the correlationfunctions, so that the existence of spin waves does not depend on having magnetization [23].

The so-called Kondo-Yamaji (KY) theory has been quite successfully used in studying the low-dimensional Heisenberg model over a wide temperature region [24–29]. The advantages of the KY theorylie mostly on the following points: (1) it is based on a clear physical picture; (2) it gives results for the wholerange of temperature; (3) it satisfies zeroth-, first-, and second-moment relaxation-function sum rules forthe spin-wave excitation energy.

In Section 2, we outline the method and give the basic self-consistent equations for the spin-paircorrelations and decoupling parameter as a function of temperature. The self-consistent equations aresolved numerically and the results are discussed and displayed in Section 3. Finally, our conclusion ispresented in Section 4.

2. Green function formalism

The two-time Green function is defined as

GABðtÞ ¼ �iyðtÞ/½AðtÞ;Bð0ÞS ¼ 0AðtÞ;Bð0ÞT; ð3Þ

where yðtÞ is the step function ½A;B is the commutator of operators A and B; and /?S denotes a thermalaverage. The time-Fourier transform of Eq. (2)

0A;BTo ¼1

2p

Z þN

�N

GABðtÞeiot dt ð4Þ

satisfies the equation

o0A;BTo ¼1

2p/½A;BSþ0½A;H;BTo; ð5Þ

and 0½A;H;BTo obeys an equation similar to Eq. (5), with a higher-order Green function appearingon the right side. In this way, an infinite set of coupled equations is generated. In the KY procedure,in order to cut the set having second-order Green functions as the highest degree, we use the following

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P.A. Leboeuf et al. / Journal of Magnetism and Magnetic Materials 264 (2003) 175–182 177

decoupling rule

0ABC;DTo ¼ a/ABS0C;DTo þ a/ACS0B;DTo þ a/BCS0A;DTo; ð6Þ

where the decoupling parameter a has been introduced [20] to guarantee that the sum rule of the correlationfunction is obeyed, and is determined so as to satisfy some physical consideration like, for example/ðSz

i Þ2S ¼ 1=4 in the S ¼ 1=2 Heisenberg model.

The correlation function can then be obtained by the spectral representation theory which gives

/BAS ¼ iZ

N

N do 0A;BToþie �0A;BTo�ie

� � 1

ebo � 1; ð7Þ

where b ¼ ðkBTÞ�1:Following the steps briefly described above, we obtain the equation of motion corresponding to

Hamiltonian (2), in the KY approximation,

0Sþ0 ;S�

n To o2 � J21 ð1 þ *c2Þ � J2

2 ð1 þ *c4Þ � 2J1J2ð*c1 þ *c3Þ� �

¼J1 *c1

2p�2dn;0 þ dn;�1 þ dn;1

� �þ

J2 *c2

2p�2dn;0 þ dn;�2 þ dn;2

� �

�1

2J2

1 1 þ *c1 þ *c2ð Þ þ J1J2 *c2 þ *c3ð Þ� �

0Sþ�1 þ Sþ

1 ;S�n To

þ1

2J2

1 *c1 þ J22 1 þ *c2 þ *c4ð Þ þ J1J2 3*c1 þ *c3ð Þ

� �0Sþ

�2 þ Sþ2 ;S�

n To

þJ1J2

2*c1 þ *c2ð Þ0Sþ

�3 þ Sþ3 ;S�

n To þJ2

2 *c2

20Sþ

�4 þ Sþ4 ;S�

n To; ð8Þ

where S7 ¼ Sx7iSy; as usual, and we have adopted the notation *cn ¼ acn ¼ a/Sz0Sz

nS: We have also usedthe special property for spin S ¼ 1=2 operators

sxsy ¼ isz: ð9Þ

Eq. (8) can easily be solved in the reciprocal space if we define the spatial Fourier transform as

Gmðq;oÞ ¼X

n

0Sþm ;S�

n Toeiqn ¼ eiqmG0ðq;oÞ: ð10Þ

Thus, taking the Fourier transform of Eq. (8), we obtain

G0ðq;oÞ ¼1

pJ1c1ð1 � cos qÞ þ J2c2ð1 � cos 2qÞ

o2 � O2; ð11Þ

where

O2 ¼ J21 1 þ *c2 � 1 þ *c1 þ *c2ð Þcos q þ *c1cos 2q½ þ J2

2 1 þ *c4 � 1 þ *c2 þ *c4ð Þcos 2q þ *c2cos 4q½

þ J1J2 2*c1 þ 2*c3 � *c2 þ *c3ð Þcos q � 3*c1 þ *c3ð Þcos 2q þ *c1 þ *c2ð Þcos 3q½ : ð12Þ

The frequency O corresponds to the poles of the Green function and, thus, gives the excitation spectrum ofour model. Eq. (12) predicts no gap at q ¼ 0 for this spectrum—which is consistent with theoretical results[4,5] for Eq. (2) when lo0:25: our result is compatible with the magnetically non-dimerized phase ofHamiltonian (2). However, the above expression predicts a spurious gap at q ¼ p even for J2 ¼ 0; the nnantiferromagnetic Heisenberg model, as has also been found by Kondo and Yamaji [20]: a similar failure isobtained for J1 ¼ 0 and q ¼ p=2: On the other hand, at T ¼ 0 and for J1; J2o0; that is, the ferromagneticHeisenberg model, we obtain

O2 ¼ J21 ð1 � cos qÞ2 þ J2

2 ð1 � cos 2qÞ2 þ J1J2ð2 � cos q � 2cos 2q þ cos 3qÞ; ð13Þ

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which is the exact result predicted by the standard spin-wave theory. As has been pointed out by Kondoand Yamaji, we can then expect that the wavevector range of applicability of the method is wider forferromagnets than for antiferromagnets because the limit q-p is not amenable to antiferromagnets.

Using Eqs. (8), (10), and (7), we finally obtain

*cn ¼ �2

N

Xq

cothbO2

e�iqnJ1 *c1ð1 � cos qÞ þ J2 *c2ð1 � cos 2qÞ

O; ð14Þ

where the sum is performed over the first Brillouin zone. Inserting l ¼ 0 into our expressions, we recoverthe results obtained by Kondo and Yamaji [20] for the S ¼ 1=2 nearest-neighbor Heisenberg model. Thedecoupling parameter a is determined by Eq. (14) for n ¼ 0; imposing *c0 ¼ 4a/ðSz

0Þ2S ¼ a: We then have a

set of five self-consistent equations that can be solved numerically by an iteration technique at arbitrarytemperature. Fig. 1 shows the cn (n ¼ 1;y; 4) for l ¼ 0:1 as a function of temperature T=J:

3. Results and discussion

It is well known that Hamiltonian (2) shows a magnetically dimerized phase for l > lc: A precisedetermination of the value of lc has been the aim of several studies using different methods [5,30,31]. Morerecently, Castilla et al. [8] used the Lanczos algorithm to determine the critical value lc for lattice sizesvarying between N ¼ 8 and 22 sites. Their results gave lc ¼ 0:241270:0001 in quite good agreement withthe result obtained by Okamoto and Nomura [5] using conformal field theory methods. We can expect thatthe correlation between second neighbors is small in the dimerized phase. However, the method we haveadopted in this work is restricted to the lolc region, where the spin wave picture works. Then, in order toknow the behavior of the spin correlation between second neighbors we calculated c2 at T ¼ 0 for severalvalues of l; starting from l ¼ 0 where our theory is valid. The result is displayed in Fig. 2 where we plotc2 � l for T ¼ 0: we see that c2 decreases as l increases and becomes null for lX0:2570:01: In fact, thesolution of Eq. (14) to determine cn becomes quite difficult above l > 0:25: Our interpretation of Fig. 2 andthe lack of solutions for l > 0:25 (for Ta0) is that this region is out of the range of validity of our theory,that is, it corresponds to the dimerized phase. Thus, our estimate for lc is 0:25 which is in surprisingly goodagreement with the estimates of previous works [5,7,8], as mentioned above.

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0.0 0.5 1.0 1.5 2.0T/J

0.00

0.15

0.30

0.45

0.60

c n

|c1|

|c2||c3||c4|

λ=0.10

Fig. 1. Temperature dependence of the correlation functions (absolute values) for l ¼ 0:10:


We can evaluate the static susceptibility by using the expression

wN

¼ � limq-0

Gðq;o ¼ 0Þ; ð15Þ

or explicitly by

wN

¼J1c1 þ 4J2c2

J21 1 � að3c1 � c2Þð Þ þ 4J2

2 1 � að3c2 � c4Þð Þ þ J1J2að3c1 � 8c2 þ 5c3Þ: ð16Þ

Fig. 3 shows our results for l ¼ 0; 0:1 and 0:2 while Fig. 4 shows the comparison of our results to theones obtained by Fabricius et al. [9] for l ¼ 0:1: notice that, in this last figure, the horizontal axis is in unitsof J=2 due to the units used in Ref. [9]. From Fig. 4 we see that the overall comparison is quite good, exceptat low temperatures. In this region, the details of the short wavelength scale becomes more important and,as we have seen, for antiferromagnets, the region q-p is not well described by the method we have used.

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0.00 0.10 0.20 0.30 0.40J2 /J1

0.00

0.05

0.10

0.15

0.20

C2

T=0.0K

Fig. 2. Second neighbors correlation function c2 as a function of l at temperature T ¼ 0:

0.0 1.0 2.0 3.0 4.0 5.0T/J’

0.00

0.02

0.04

0.06

0.08

0.10

χ J′/

(Ng

2 µ B2 )

λ =0.0λ =0.1λ =0.2

J’=J/2

Fig. 3. Susceptibility per site (in units of J1) versus the reduced temperature T=J 0 where J 0 ¼ J1=2 for l ¼ 0; 0:1 and 0:2:


However, we obtain a very good agreement for moderate and high temperatures showing that the Greenfunction method, despite its simplicity, can lead to a good description of thermodynamic properties.

Nevertheless, neglecting the very low temperature region (T=J 0o0:5), we obtain that Tmax; thetemperature at which the susceptibility w has a maximum, is decreasing with l while the value wmax increaseswith l: this behavior is in qualitative agreement to the one obtained by Fabricius et al. [9].

4. Conclusion

In conclusion, using a double-time Green function, we have investigated the quantum spin S ¼ 1=2Heisenberg chain with nearest and next-nearest neighbor interactions throughout a wide temperaturedomain. This model has become quite important since it is believed that the behavior of many magneticcompounds can be determined by the properties of this model. Notice that the theory we have used isrestricted to lo0:25:

The theory developed in this paper agrees with numerical calculations performed by Fabricius et al [9],and it has the advantage of being quite simple to handle—in comparison to techniques like the numericaldiagonalization used in Ref. [9]. Therefore, the theory presented here can be easily applied to analyseexperimental data—when available—obtained in a wide temperature range for frustrated compounds.

Acknowledgements

This work was partially supported by CNPq (Conselho Nacional para o Desenvolvimento da Pesquisa).

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0.0 1.0 2.0 3.0 4.0 5.0T/J’

0.00

0.02

0.04

0.06

0.08

0.10

χJ′/(

Ng

2 µ2 B)

our theory

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J’=J/2

Fig. 4. Susceptibility per site (in units of J1) versus the reduced temperature T=J 0 where J 0 ¼ J1=2 for l ¼ 0:1: the continuous line

represents our results and the dashed line corresponds to the numerical result obtained by Fabricius et al. [14].


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