Finite temperature calculations on spin correlation functions of the Haldane gap compound CsNiCl3

ARTICLE IN PRESS Physica B 404 (2009) 100–104

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Finite temperature calculations on spin correlation functions of the Haldane gap compound CsNiCl3 Wende Liu , Hai Huang State Key Laboratory for Mesoscopic Physics, Peking University, Beijing 100871, China

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a b s t r a c t

Article history: Received 14 May 2008 Received in revised form 2 October 2008 Accepted 10 October 2008

We present the finite temperature calculations on the spin correlation functions of the Haldane gap compound CsNiCl3 . Based on the coupled-chain theory, two methods are used to do the calculations: the large-N expansion and the Ginzburg–Landau approach. The comparisons with the inelastic neutronscattering experiments are discussed. & 2008 Elsevier B.V. All rights reserved.

PACS: 75.10.Jm 75.10.Pq 75.50.Ee Keywords: Heisenberg model Antiferromagnet Coupled-chain theory

1. Introduction As predicted by Haldane an energy gap is expected to exist in the magnetic excitation spectrum of the one-dimensional Heisenberg antiferromagnets with integer spins [1,2]. The first experimental evidence for the Haldane gap was from the spin-1 compound CsNiCl3 [3–5]. This compound crystallizes in a hexagonal crystal structure and the lattice constants at low temperatures are a ¼ 7:14 A and c ¼ 5:90 A. The magnetic moments are carried by Ni2þ ions which interact via a superexchange  interaction involving Cl ions. Since the superexchange between 2þ the Ni ions along the c-axis is much stronger than perpendicular to the c-axis, the spin Hamiltonian of CsNiCl3 is that of a system of coupled spin-1 chains with a strong intrachain interaction J and a much weaker interchain interaction J 0 . This weak interchain coupling produces magnetic order at a temperature of 4.8 K [6]. In the ordered phase neighboring spins are anti-parallel along the chains and at angles of 1201 in the planes. The low-energy behavior of the Haldane gap compounds can be described by the ð1 þ 1Þ-dimensional O(3) non-linear s-model ðNLsMÞ [1,2]. The exact form factors of the O(3) NLsM [7,8] can be used to predict the multi-magnon contributions to the spin

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E-mail addresses: [email protected], [email protected] (W. Liu). 0921-4526/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2008.10.010

correlation functions of CsNiCl3 in the disordered phase. It was found that the three-magnon contribution is extremely broad and weak, and the total intensity is less than 2% of the one-magnon contribution [9,10]. But the inelastic neutron-scattering experiment at T ¼ 6:2 K shows that along the wave-vector transfer (0.81, 0.81, 1), the multi-magnon continuum starts far below three times the one-magnon gap, and the integrated intensity of the continuum is about 12(2)% of the total spectral weight [11]. This result is obviously inconsistent with the one-dimensional field theory prediction. Three-dimensional effect has to be considered. A coupled-chain theory which can be applied in both the Ne´el and disordered phases was developed by Affleck [12]. Although it is not possible to find the exact solutions of this model, several more or less reliable approximations are possible. One approach is the large-N expansion. Zero temperature calculations have already been carried out by one of the authors [13]. But strictly speaking, for CsNiCl3, it is not proper to study the properties of the disordered phase by zero temperature Green’s function methods. Moreover, zero temperature calculations cannot explain the increasing width of the one-magnon peak with temperature shown in the experiment [14]. Another approach is to relax the constraint on the field of NLsM and introduce mass terms and a quartic term for stability [12,15]. Only keeping the quadratic terms, this Ginzburg–Landau (GL) Hamiltonian provides a beautiful explanation of the finite-gap mode in the Ne´el phase, which cannot be understood using spin-wave theory. We present here

ARTICLE IN PRESS W. Liu, H. Huang / Physica B 404 (2009) 100–104

the finite temperature calculations on the spin correlation functions using these two methods. The comparisons with the inelastic neutron-scattering experiments are discussed.

2. O(3) non-linear s-model and coupled-chain theory The spin Hamiltonian for CsNiCl3 is given by H¼J

chain X

~ Siþ1 þ J0 Si  ~

i

plane X

~ Sj  D Si  ~

X z ðSi Þ2

(1)

i

hiji

where the intrachain coupling J ¼ 2:28 meV was measured with a high-field magnetization measurement of the magnetic saturation field [16]. The interchain coupling, J 0 ¼ 0:04 meV, was determined from the measurement of the spin-wave energies in the antiferromagnetically ordered structure at T ¼ 2 K and the comparison with spin-wave theory [3,4]. The weak easy-axis Ising anisotropy D ¼ 4 meV, which is small enough to make CsNiCl3 a good example of an isotropic Heisenberg antiferromagnet [6]. If we omit the interchain coupling and single-ion anisotropy terms, this is the one-dimensional Heisenberg model. For the antiferromagnetic case, there are important low-energy effect at wave-vectors near 0 and p. Thus we write the spin operator as ~ þ~l ~ Sj ¼ sð1Þj f j j

(2)

~ and ~l are slowly varying on the scale of the lattice. Due where f j j to SU(2) algebra of the spin operators, ~lj can be written as ~l ¼ f ~ P ~j j j

(3)

~ . Based upon the large~ j is the conjugate momentum of f where P j s limit, we can map the low-energy behavior of this Hamiltonian into the O(3) NLsM. The Lagrangian density is

L1 ¼

1 ~ Þ2  v2 ðqz f ~ Þ2  ½ðqt f 2gv

2

~ ¼ 1Þ ðf

(4)

with the velocity v ¼ 2Js and the coupling constant g ¼ 2=s. z is the distance along the c-axis. We now include the nearest-neighbor interchain coupling J 0 -term. Based on the coupled-chain theory developed by Affleck ~ ðzÞ and get [12], we represent the ith chain by the NLsM field f i the Lagrangian of coupled ð1 þ 1Þ-dimensional field theories as 2 3 Z X X 0 2 ~ ~ ~ 4 L¼ dz L1 ðfi ðzÞÞ  J s fi ðzÞ  fj ðzÞ5 (5) i

hiji

with i and j labeling different chains. This model is threedimensional, which makes spontaneous symmetry breaking possible. We will present the finite temperature calculations on spin correlation functions based on this model.

3. The large-N expansion In this section, we try to study the coupled-chain model based on the large-N expansion. We assume that the results obtained ~ field as remain qualitatively correct even at N ¼ 3. So we set f i 1 2 N ~ N components instead of 3, i.e., fi ¼ ðfi ; fi ; . . . ; fi Þ. After we ~ field introduce an auxiliary field li to relax the constraint, the f i can be integrated out and the effective action Seff will be expanded in the order of 1=N. Since there is a factor of N in Seff , to a first approximation, we may ignore the fluctuations of li and evaluate it at the lowest action saddle point. This is obtained by deforming the functional integration contour of li into the complex plane, and a saddle point can be defined at ili ¼ D2 . Then the dispersion relation for three kinds of massive magnons can be written as

101

eðkx ; ky ; kz þ pÞ  e~k ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D2 þ v2 k2z þ 4vJ 0 sf ðkx ; ky Þ

(6)

where v ¼ 2:49J from numerical simulations [17] and pffiffiffi ! pffiffiffi ! 1 1 3 3 kx  kx þ ky þ cos ky þ cos kx f ðkx ; ky Þ ¼ cos 2 2 2 2

(7)

The temperature dependence of the gap parameter D can be determined self-consistently [13]. For example, we have the gap D ¼ 1:24 meV at 6.2 K and D ¼ 1:39 meV at 9 K. The Brillouin zone on the ðkx ; ky Þ plane is plotted in Ref. [15]. Now we try to calculate the spin correlation functions at finite temperature. To the order of 1=N, the frequency and momentum dependent contribution to the self-energy Pðion ; ~ kÞ can be computed as [13]

Pðion ; ~ kÞ ¼

1 X T N m

Z

3 vd ~ 1 p ð2pÞA ðon  om Þ2 þ e~2

2 ˜ ðiom ; ~ P pÞ ~

(8)

kp

with ˜ ðiom ; ~ P pÞ ¼ T

X Z v d3~ q 1 1 2 þ e2 2 2 ð2 p ÞA o ð o  o m l Þ þ e~ ~ l q l

(9)

p~ q

pffiffiffi Here A ¼ 8p2 = 3 is the area of the Brillouin zone and on ¼ 2pnT, n is an integer. After performing the summation on l and m, we get [13] Im Pðion ! o þ id; ~ kÞ " # Z Z 3 1 vd ~ 1 p 1 dy Im ¼ f½1 þ n~k~p þ ny  ˜ ðy; ~ N ð2pÞA e~k~p P pÞ  dðo  e~k~p  yÞ þ ½n~k~p  ny dðo þ e~k~p  yÞg

(10)

Here the Bose factor is defined as 1 n~q  ne~q ¼ e =T e ~q  1

(11)

Then the finite-T spin correlation functions can be calculated from [18] Sðo; ~ kÞ ¼

2 gvz2 Im Pðo þ id; ~ kÞ 1  eo=T ðo2  e2 Þ2 þ ðz Im Pðo þ id; ~ kÞÞ2

(12)

~ k

where z is the wave-function renormalization parameter. In principle, we must get the exact form of z based on the large-N expansion as well. It has been proved to be necessary in the problems such as the explanation of the tunnel-current anomalies in superconducting lead [19,20]. Here we simply take z as a constant. From the exact results of the ð1 þ 1Þ-dimensional O(3) NLsM and the small ratio J 0 =J  2%, we expect z  1 [9]. To evaluate the high-dimensional integral in Eq. (10), we perform Monte Carlo simulations with 105 points in a p simulation box defined ffiffiffi pffiffiffi within the region 4p=3pkx p4p=3, 2p= 3pky p2p= 3. Only those points within the first Brillouin zone will be selected for the calculations. We also replace the Dirac d-function by the Lorentz function with the full width at half maximum 0.1 meV. Compared with the experimental resolution 0.35 meV [11], we expect it will be a good estimate. For the wave-vector transfer (0.81, 0.81, 1), we plug in N ¼ 3 and compute the spin correlation function from Eq. (12). We add a flat background into our theoretical curves due to the nonmagnetic contribution shown in the experimental data [11]. The results are shown in Figs. 1 and 2. (The one-magnon contributions are plotted as the dashed lines for comparison, which overlap with the solid lines around the peaks.) The theoretical curves roughly fit the experimental data and the integrated intensity of the threemagnon continuum is 8.0(8)% of the total spectral weight at 6.2 K,

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which is much better than the ð1 þ 1Þ-dimensional NLsM prediction.

Lagrangian [12,15]:

LGL

4. The GL approach In this section, we study the coupled-chain model based on ~ and replace it by the GL approach. We relax the constraint on f i mass terms and a quartic term, then get the following GL

8 2 3 ! !
(13)

hiji

~ field is rescaled by 1=pffiffiffi where f g already. Setting l ¼ 0, we get i the same dispersion relation as Eq. (6). Now we try to calculate the spin correlation functions at finite 2 temperature based on this GL Lagrangian. To the order of l , all the one-particle-irreducible contributions to the self-energy are shown in Fig. 3. Since the frequency and momentum dependent part is only from diagram (c), we will focus on this term. Then the self-energy can be computed as

2 Pðion ; ~ kÞ ¼ 160l v2

0

Z

v d~ k1 v d~ k2 2 T ð2pÞA ð2pÞA

1 1 1 1 A @ 2 2 2 2 2 2 n1 ;n2 on1 þ e~ on2 þ e~ on3 þ e~ X

k1

k2

(14)

k3

with ~ k3 ¼ ~ k ~ k1  ~ k2 and on3 ¼ on  on1  on2 . After performing the summation on n1 and n2 by the contour integration, we get Fig. 1. The large-N calculation of the spin correlation function at wave-vector transfer (0.81, 0.81, 1) vs the experimental data at the temperature 6.2 K. The experimental points are from Ref. [11].

2 Pðion ; ~ kÞ ¼ 20l v2

Z

v d~ k1 v d~ k2 1 ð2pÞA ð2pÞA e~k e~k e~k

1 2 3 " ðn1 þ 1Þðn2 þ 1Þðn3 þ 1Þ  n1 n2 n3  e~k þ e~k þ e~k  ion 1

þ

2

3

n1 ðn2 þ 1Þðn3 þ 1Þ  ðn1 þ 1Þn2 n3 e~k þ e~k þ e~k  ion 1

2

3

ðn1 þ 1Þn2 ðn3 þ 1Þ  n1 ðn2 þ 1Þn3 þ e~k  e~k þ e~k  ion 1

2

3

ðn1 þ 1Þðn2 þ 1Þn3  n1 n2 ðn3 þ 1Þ þ e~k þ e~k  e~k  ion 1

2

3

n1 n2 ðn3 þ 1Þ  ðn1 þ 1Þðn2 þ 1Þn3 þ e~k  e~k þ e~k  ion 1

2

3

n1 ðn2 þ 1Þn3  ðn1 þ 1Þn2 ðn3 þ 1Þ þ e~k þ e~k  e~k  ion 1

2

3

ðn1 þ 1Þn2 n3  n1 ðn2 þ 1Þðn3 þ 1Þ þ e~k  e~k  e~k  ion 1

2

3

n1 n2 n3  ðn1 þ 1Þðn2 þ 1Þðn3 þ 1Þ þ e~k  e~k  e~k  ion 1

Fig. 2. The large-N calculation of the spin correlation function at wave-vector transfer (0.81, 0.81, 1) vs the experimental data at the temperature 9 K. The experimental points are from Ref. [11].

2

# (15)

3

Here ni ði ¼ 1; 2; 3Þ is defined as the Bose factor n~k . Then the i imaginary part of the real-frequency self-energy can be written as

Fig. 3. All the one-particle-irreducible diagrams to the second order.

ARTICLE IN PRESS W. Liu, H. Huang / Physica B 404 (2009) 100–104

103

Im Pðo þ id; ~ kÞ Z v d~ k1 v d~ k2 p 2 ¼ 20l v2 ð2pÞA ð2pÞA e~k e~k e~k 1

2

3

 f½ðn1 þ 1Þðn2 þ 1Þðn3 þ 1Þ  n1 n2 n3 dðo  e~k  e~k  e~k Þ 1

2

3

þ ½n1 ðn2 þ 1Þðn3 þ 1Þ  ðn1 þ 1Þn2 n3 dðo þ e~k  e~k  e~k Þ 1

2

3

þ ½ðn1 þ 1Þn2 ðn3 þ 1Þ  n1 ðn2 þ 1Þn3 dðo  e~k þ e~k  e~k Þ 1

2

3

þ ½ðn1 þ 1Þðn2 þ 1Þn3  n1 n2 ðn3 þ 1Þdðo  e~k  e~k þ e~k Þ 1

2

3

þ ½n1 n2 ðn3 þ 1Þ  ðn1 þ 1Þðn2 þ 1Þn3 dðo þ e~k þ e~k  e~k Þ 1

2

3

þ ½n1 ðn2 þ 1Þn3  ðn1 þ 1Þn2 ðn3 þ 1Þdðo þ e~k  e~k þ e~k Þ 1

2

3

þ ½ðn1 þ 1Þn2 n3  n1 ðn2 þ 1Þðn3 þ 1Þdðo  e~k þ e~k þ e~k Þ 1

2

3

þ ½n1 n2 n3  ðn1 þ 1Þðn2 þ 1Þðn3 þ 1Þdðo þ e~k þ e~k þ e~k Þg 1

2

3

(16)

On evaluation of the integral in Eq. (16), we perform Monte box Carlo simulations with 109 points in a simulation pffiffiffi pffiffiffidefined within the region 4p=3pkx p4p=3, 2p= 3pky p2p= 3. (Since the integrand in Eq. (10) contains a multiple integral in its denominator for each given ~ p, it takes much more CPU time for the evaluation of Eq. (10) than Eq. (16). Therefore we can have the total number of the randomly generated points much larger here.) For each effective point selected, we obtain the corresponding o values which give finite contributions to the integral. In our computation program, we divide the interested range of o ð0214 meVÞ into 1000 intervals. It corresponds to a computing resolution of 0.014 meV. Based on Eqs. (16) and (12), we can calculate the finite-T spin correlation functions for the wave-vector transfer (0.81, 0.81, 1). We choose the parameter l ¼ 0:4 meV at 6.2 K and l ¼ 0:42 meV at 9 K to fit the experimental data. We also add a flat background into our theoretical curves. The results are plotted in Figs. 4 and 5. The GL results drop fast compared with the large-N expansion. The integrated intensity of the three-magnon continuum is 7.0(4)% of the total spectral weight at 6.2 K. We use two different methods and obtain qualitatively similar but quantitatively different results in this paper. The large-N expansion gives better fit to the neutron-scattering data than the GL theory at high energy transfer. The ratio of the three-magnon continuum to the total spectral weight is 8.0(8)% for the large-N theory, which is larger than 7.0(4)% for the GL theory. But the numerical errors in the Monte

Fig. 5. The GL results of the spin correlation function at wave-vector transfer (0.81, 0.81, 1) vs the experimental data at the temperature 9 K. The experimental points are from Ref. [11].

Carlo simulations are smaller for the GL theory, and the GL approach can be easily generalized to include small anisotropic terms shown in the Hamiltonian of the crystal. At this point, we would like to give a remark on the temperature effect in the spin correlation functions. Zero temperature calculations on the spin correlation function will give a separate one-magnon d-function peak and a three-magnon continuum above the energy of the peak. In order to compare zero temperature results with the experimental data at 6.2 K, it is assumed in Ref. [13] that the temperature effect is small and the one-magnon peak is broadened by the experimental energy resolution 0.35 meV. But this assumption is against the following experimental fact: the one-magnon peak broadens and increases in energy with increasing temperature [14]. Finite temperature calculations clarify this point. The temperature effect actually broadens the one-magnon peak so much that it makes the fitting to the experimental data well. In conclusion, we present the finite temperature calculations on the spin correlation functions of CsNiCl3 . We show that the coupled-chain theory is still a good starting point, and the inelastic neutron-scattering data can be qualitatively understood by the large-N expansion or the GL approach. Our discussion may help to clarify similar problems in other Haldane gap systems like RbNiCl3 .

Acknowledgments We would like to thank Prof. Z.-Z. Gan for helpful discussions. One of the authors (H.H.) would also like to thank Prof. I. Affleck for the guidance, and Dr. A. Schweiger for discussions on Monte Carlo simulations. References

Fig. 4. The GL results of the spin correlation function at wave-vector transfer (0.81, 0.81, 1) vs the experimental data at the temperature 6.2 K. The experimental points are from Ref. [11].

[1] F.D.M. Haldane, Phys. Lett. A 93 (1983) 464; F.D.M. Haldane, Phys. Rev. Lett. 50 (1983) 1153. [2] For a review, see I. Affleck, in: E. Bre´zin, J. Zinn-Justin (Eds.), Fields, Strings and Critical Phenomena, Proceedings of the Les Houches Summer School of Theoretical Physics, 1988, Elsevier, New York, 1990, session XLIX. [3] W.J.L. Buyers, R.M. Morra, R.L. Armstrong, P. Gerlach, K. Hirakawa, Phys. Rev. Lett. 56 (1986) 371. [4] R.M. Morra, W.J.L. Buyers, R.L. Armstrong, K. Hirakawa, Phys. Rev. B 38 (1988) 543. [5] M. Steiner, K. Kakurai, J.K. Kjems, D. Petitgrand, R. Pynn, J. Appl. Phys. 61 (1987) 3953.

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W. Liu, H. Huang / Physica B 404 (2009) 100–104

[6] H. Kadowaki, K. Ubukoshi, K. Hirakawa, J. Phys. Soc. Japan 56 (1987) 751. [7] A. Krillov, F. Smirnov, Phys. Lett. B 198 (1987) 506; A. Krillov, F. Smirnov, Internat. J. Mod. Phys. A 3 (1988) 731. [8] J. Balog, M. Niedermaier, Nucl. Phys. B 500 (1997) 421. [9] M. Horton, I. Affleck, Phys. Rev. B 60 (1999) 11891. [10] F. Essler, Phys. Rev. B 62 (2000) 3264. [11] M. Kenzelmann, R.A. Cowley, W.J.L. Buyers, R. Coldea, J.S. Gardner, M. Enderle, D.F. McMorrow, S.M. Bennington, Phys. Rev. Lett. 87 (2001) 017201. [12] I. Affleck, Phys. Rev. Lett. 62 (1989) 474; I. Affleck, Phys. Rev. Lett. 62 (1989) 1927(E); I. Affleck, Phys. Rev. Lett. 65 (1990) 2477; I. Affleck, Phys. Rev. Lett. 65 (1990) 2835.

[13] H. Huang, Phys. Lett. A 360 (2007) 731. [14] M. Kenzelmann, R.A. Cowley, W.J.L. Buyers, D.F. McMorrow, Phys. Rev. B 63 (2001) 134417. [15] I. Affleck, G. Wellman, Phys. Rev. B 46 (1992) 8934. [16] A.A. Katori, Y. Ajiro, T. Asano, T. Goto, J. Phys. Soc. Japan 64 (1995) 3038. [17] E. Sorensen, I. Affleck, Phys. Rev. Lett. 71 (1993) 1633; E. Sorensen, I. Affleck, Phys. Rev. B 49 (1994) 15771. [18] S. Doniach, E.H. Sondheimer, Green’s Functions for Solid State Physicists, Imperial College Press, London. [19] J.R. Schrieffer, D.J. Scalapino, J.W. Wilkins, Phys. Rev. Lett. 10 (1963) 336. [20] J.M. Rowell, P.W. Anderson, D.E. Thomas, Phys. Rev. Lett. 10 (1963) 334.