Quadrupole phase transition of a planar classical Heisenberg model1

Physica 93A (1978) 526-530 (~) North-Holland Publishing Co.

Q U A D R U P O L E PHASE T R A N S I T I O N OF A PLANAR C L A S S I C A L H E I S E N B E R G MODEL* K.G. CHENt, H.H. CHEN and C.S. HSUE Institute of Physics, National Tsing Hua University, Hsinchu, Taiwan

Received 13 April 1978

The quadrupole phase transition of a planar classical Heisenberg model with biquadratic interactions is investigated by the high-temperature series expansion method. From the quadrupole susceptibility series the critical temperatures and the critical exponents are determined for cubic lattices.

Consider the planar classical H e i s e n b e r g model with competing bilinear and biquadratic interactions. The Hamiltonian is given by

= -- ~

[2J1Si " S i +

2J2(Si

•

$i)2],

where Si = (Six, Sir) is a two-dimensional unit vector at the lattice site labelled i, Jl and J2 are respectively the bilinear and biquadratic interaction constants. When Ji > J2 t> 0, the bilinear term is dominant. The phase transition of the system is associated with the ordering of the dipoles, i.e., (EiSix) = (E~ cos 0i) ¢ 0. In this case critical properties of the system extrapolated f r o m the high-temperature series expansions have been reported by the authors in a previous paper1). In the present paper we consider the other case that the biquadratic interactions are more important than the bilinear ones. At high t e m p e r a t u r e s the system is in a disordered phase. Thermal averages of the dipole and the quadruople m o m e n t s are zero. As we a p p r o a c h the stability limit of the disordered phase, the thermal fluctuation of the dipole m o m e n t or that of the quadrupole m o m e n t will diverge at the critical * Work supported by the National Science Council of the Republic of China. t Present address: Department of Physics, Soochow University, Taipei, Taiwan. 526

QUADRUPOLE PHASE TRANSITION OF HEISENBERG MODEL

527

temperature. When the biquadratic term is dominant, it is the thermal fluctuation of the quadrupole m o m e n t that diverges. The system undergoes a quadrupole phase transition. Below the critical temperature, T o, the quadrupoles of the system order. Most of the previous studies on the quadrupole phase transitions are based on the mean-field approximation2-4). In this approximation the critical t e m p e r a t u r e and the critical exponent are independent of the ratio J1/J2. For the spin-1 Heisenberg model with the biquadratic interactions high-temperature series expansions for the quadrupole susceptibility have been derived to order T -4 by Chen and LevyS). The series were too short for estimating the critical exponents. For the present model we are able to derive the hight e m p e r a t u r e series expansions for the quadrupole susceptibility to order T -7. Reliable estimates of the critical temperature and the critical exponent are obtained. For a three-dimensional spin system the quadrupole m o m e n t has five components2'3). For the planar spin model Sz = 0, the quadrupole m o m e n t has only two components. T h e y are E~ (Sj2 - S~) = Y-i cos 20~ and ~,i 2Si,,Siy = E~ sin 20i. As the Hamiltonian is an isotropic one, it is sufficient to investigate the quadrupole phase transition for one of the two components, e.g., Y~ cos 20~. We apply a fictitious quadrupole field to the system. The Hamiltonian is then written as - ~ (211 cos

0o + J2 cos 2Oij) - ~i HQq cos 2Ol,

where 0o = O~- Oj, H o is a quadrupole field and q is the quadrupole m o m e n t per spin. The zero-field quadrupole susceptibility is defined as Xo = Lim ~

t9 2

HQ~O O-r~ Q

= (q2

[kT In Tr exp(-~/kT)] cos 20~

-

cos 20~

.

N o t e that the quantity between brackets {} is nothing but the thermal fluctuation of the quadrupole m o m e n t El cos 20i. At high t e m p e r a t u r e s it is convenient to express XQ in the f o r m

Xo = (Nq2/2kT)[l + ~= en(~)(J2lkT)n], where N is the number of spins in the system, and the coefficients en are functions of ~ = JJJ2. For general values of ~ we have derived the coefficients el-e7 for the cubic lattices. The technique used for deriving these coefficients

528

K.G. CHEN, H.H. CHEN AND C.S. HSUE

is t h e s a m e as the o n e w h i c h w e h a v e u s e d f o r d e r i v i n g the d i p o l e s u s c e p t i b i l i t y s e r i e s . T h i s t e c h n i q u e w a s d e v e l o p e d , a n d d e s c r i b e d in d e t a i l b y Joyce6). F o r the f.c.c, l a t t i c e e~(sr) = 6, e2(~'l = 33 + 6~ 2, w3(~') = 174.75 + 66~"2 + 48~ "3, e4(¢) = 906 + 524.25¢ 2 + 528~"3 + 425~"4, es(~') = 4634.1875 + 3624¢ 2 + 4194~"3 + 5027.75~ "4+ 3728¢ 5, e6(~') = 23485.3594 + 23170.9375~ "2 + 28992~ 3 + 41992~"4 + 47364~"~

+ 32782.25~r 6, e7(~') = 118221.713 + 140912.156~ "2 + 185355.5sr 3 + 302519.917~ "4 + 413786~ 5 + 448568.8~ "6 + 291731.333~ "7, F o r the b.c.c, l a t t i c e e,(~') = 4, e2(~r) = 14 + 4~r2, e3(~') = 48.5 + 28~"2, e4(~') = 1 6 2 + 145.5~'2 + 155.333~"4,

es(sr) = 538.458333 + 648~"2 + 1230.5sr 4, e6(~') = 1763.17708 + 2692.29167¢ 2 + 6662.6667~ "4 + 4968.1667~ "6, e7(~) = 5757.03255 + 10579.0625~ "2 + 31241.9444~ "4 + 46336.5333~ "6.

F o r the s.c. l a t t i c e el(~') = 3, e2(~') = 7.5 + 3~"2, e3(~') = 18.375 + 15~"2, e4(~') = 43.5 + 55.125~r2 + 41.5~ "4, es(~') = 102.34375 + 174~r2 + 248.875~ "4, e6(~') = 237.054688 + 511.71875 ~-2+ 988~.4 + 722.125 ~-6,

e7(~') = 546.946289 + 1422.32813~ "2 + 3371.58333~ "4 + 5234.4~ "6. W h e n J~ = 0, J2 = 2J, t h e p r e s e n t s e r i e s r e d u c e to the h i g h - t e m p e r a t u r e s e r i e s f o r the d i p o l e s u s c e p t i b i l i t y f o r the u s u a l p l a n a r c l a s s i c a l H e i s e n b e r g modep'7). T h i s is e x p e c t e d b e c a u s e f o r t h e p l a n a r s p i n s the p u r e l y b i q u a d r a t i c i n t e r a c t i o n s w i t h spin a n g l e s r a n g i n g f r o m 0 to 2zr h a v e the s a m e p r o p e r t i e s as t h e p u r e l y b i l i n e a r i n t e r a c t i o n s w i t h spin a n g l e s r a n g i n g f r o m 0 to 47r.

QUADRUPOLE PHASE TRANSITION OF HEISENBERG MODEL

529

The above series have been analysed for a singularity of the form Xo (T - To)-vQ by using the ratio and Pad6 methodsS). It is important to note that for a given set of values of J1 and J2, one must compare the critical temperature, To, obtained from the quadrupole susceptibility series and the critical temperature, To, obtained from the dipole susceptibility seriesg). The system undergoes a quadrupole phase transition at To only if T o > Tc and if the transition is of second order. This point has been discussed in detail by Chen and LevyS). The critical temperatures kTQ/J2 and the critical exponents yQ for several values of 2Jl/J2 0.7 we find that To < T~ for all of the cubic lattices. The system doesn't undergo a quadrupole phase transition for 2~" > 0.7. From table I we see that when the strength of the bilinear interaction, J~, increases from 0 to 0.35J2, the critical temperature To increases, while the exponent 3'o decreases. The present calculation is the first one in which reliable estimates of the quadrupole susceptibility exponent, yQ, are obtained. The range of 3'0 for the present model is about 1.2-1.3. It is very close to the values of the dipole susceptibility exponent 3'. TABLE I Critical temperature

kTo/J2 and

critical exponents 3'0 for cubic

lattice¢ f.c.c,

b.c.c,

s.c.

2J,[J2

kTQ/J2

yQ

kTo[J2

To

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

4.82 4.83 4.84 4.86 4.90 4.95 5.04 5.17

1.31 1.31 1.31 1.30 1.29 1.27 1.24 1.19

3.12 3.13 3.14 3.15 3.18 3.22 3.28 3.36

1.31 1.31 1.31 1.30 1.29 1.28 1.25 1.20

kTQ/J2 2.20 2.21 2.22 2.23 2.25 2.28 2.33 2.40

yQ 1.32 1.32 1.32 1.31 1.30 1.28 1.26 1.21

t The uncertainties in the estimates of kTQIJ2and Yo are within -+0.01 and -+0.03, respectively.

Acknowledgement The authors would like to thank Professor F.Y. Wu for helpful discussions.

530

K.G. CHEN, H.H. CHEN AND C.S. HSUE

References 1) 2) 3) 4) 5) 6) 7) 8)

K.G. Chen, H.H. Chen, C.S. Hsue and F.Y. Wu, Physica 87A (1977) 629. H.H. Chen and P.M. Levy, Phys. Rev. B7 (1973) 4267. J. Sivardi6re, A.N. Berker and M. Wortis, Phys. Rev. B7 (1973) 343. J. Lajzerowicz and J. Sivardi~re, Phys. Rev. A l l (1975) 2079. H.H. Chen and P.M. Levy, Phys. Rev. B7 (1973) 4284. G.S. Joyce, Phys. Rev. 155 (1967) 478. R.G. Bowers and G.S. Joyce, Phys. Rev. Lett. 19 (1967) 630. D.S. Gaunt and A.J. Guttmann, Phase Transitions and Critical Phenomena, 3, C. Domb and M.S. Green, eds. (Academic Press, London, 1974) p. 181. 9) In ref. 1 the term 6.0625-06 in a6(-0) for the s.c. lattice should read 6.0625-05. 10) K.G. Chen, unpublished.