Long range chiral order in the antiferromagnetic S = 12xy model on the triangular lattice

~ ~

Solid State Communications, Printed in Great Britain.

Vol. 67, No. 3, pp, 229-232,

LONG RANGE CHIRAL ORDER IN THE ANTIFER/KAMAGNETIC

1988.

0038-1098/88 $3.00 + .00 Pergamon Press plc

S = 1/2 XY MODEL ON THE TRIANGULAR LATTICE

F. Matsubara and S. Inawashiro Department of Applied Physics, Tohoku University,

Sendal 980, Japan

( Received 6 May 1988 by T. Tsuzuki

)

The antiferromagnetic S = 1/2 xy model on finite triangular lattices with lattice sites N ~ 21 is studied by using a Monte Carlo method of random sampling of states. The peak height of the specific heat is shown to increase remarkably with N and the long range chirality is shown to increase rapidly below a finite temperature. These results strongly suggest the occurrence of a phase transition related with the long range chiral order in the model. The transition temperature is estimated to be kTch/J = 0.39__. 0.03.

sublattice magnetization [4,5], whereas the non-zero sublattice magnetization is observed in nature. Moreover, the ground state properties do not reveal the nature of the phase transition, if it exists. Both efforts of treating larger lattices and of studying the model at finite temperatures would be desirable to obtain a reliable answer of this quest ion. In this commmanication, we study the properties of the model at finite temperatures using a Monte Carlo method proposed by Imada and Takahashi [ 6]. We do not treat such large lattices as treated in NN. Nevertheless results tell us much about whether the long range chiral order really exists or not. We consider the model on a finite lattice with lattice sites N whose Hamiltonian is described by

Quantum spin systems on two dimerusional lattices have been a current topic in recent years. One of attractive problems is a spin structure of an antiferromagnetic S = 1/2 xy model on the triangular lattice. Although the model has no long range order of spin, it has the possibility of the long range chiral orx~er similar to that found in a classical rotator model on the triangular lattice [i]. In fact, Fujiki and Betts [2] studied properties of the ground state of the model on finite lattices with lattice sites N = 3, 9, 12 and 21 and conjectured the presence of the long range chiral order at low temperatures. Hereafter we refer to ref. 2 as FB. Immediately after that, Nishimori and Nakanishi [3 ] carried out calcu lations of the ground state of larger lattices with lattice sites N = 24 and 27 and showed that an extrapolated value of the long range chirality for N = oo by using data for N = 9, 21 and 27 is a few per cent smaller than that for N = 3, 9 and 21 obtained in FB. Hereafter, we refer to ref. 3 as NN. Based on the result, they gave the different conjecture that the long range chiral order in the classical model ( S = eo ) is destroyed by quantum fluctuation in the S = I/2 model. The conjecture is re*7 attractive, since it breaks an opinion widely accepted in statistical physics. That is, the nature of the phase transition does not depend much on the magnitude of spin. The conjectuI~ as well as the former one, however, should ~ examined carefully because of the following reasons. The lattices treated so far are not large enough to exclude boundary effects and the difference in the extrapolated value of the long range chirality between FB and NN is not large enough to allow us to give the different conjectures. Another reason is that these conjectures are made based only on the properties of the ground state which do not necessarily coincide with properties at finite temperatures. For example, in an antiferromagnet on a three dimensional lattice, the ground state of the model is singlet with zero

x x H = 2JZ (S:Sjl + sYsY)'I j

(I)

where the sum runs over all pairs of nearest neighbor lattice sites. The thermal average of a physical quantity A is given as: =• /Z , (2) i i where the sum runs over all 2 N states of an arbitrary complete set and B =i/k~T. Here we choose ',i> as Ising states. In s~ead of calculating (2), we consider the following quantity [6 ] : M

M

=Z < ~ k : A e x p ( - B H ) : ~ > / Z k "" where the sum runs over M which is given by 2N L "~k

: J(61M

_ 1

~

<~ : e x p ( - ~ H ) : ~ k >, k k states each of

i>.

)

k H e r e C. i s a r a n d o m n u m b e r o f - 1 N C < 1. 1 . We c a n r e a d i l y show that, changing oNer of t h e s u m m a t i o n s o f ' i ' a n d ~'l~' , <"~> --* f o r M ~ c o , b e c a u s e ( 6 / M ) Z kC'i'U'~,--* ~ i , i . , for M

229

230

ANTIFERROMAGNETIC

S = l / 2 XY MODEL ON THE TRIANGULAR LATTICE

The main advantage of using the formula (3) is that we can obtain an approximate value of the average '-:Z<~klAl~k>/ <~k,~k , 151 k where ~> = exp(-BH/2)',~k > [7]. The c a l c u l a t i o n 5 f e x p ( - • H/2 ) ', ~ k > i s made by e x p a n -

Vol. 67, No. 3

0.1

ding exp(-BH/2) in terms of power series of B H/2 a n d by o p e r a t i n g e v e r y t e r m t o ',$ k >. The convergence of the series is, of course, more rapid for a higher temperature T O ( or smaller B 0). We start at some higher tenrperature T =VT O ( or some smaller ~ o )" Once the I~ ~> for T = T n is obtained by operating exp(-~/2)to :~> defined by eq,(4), we can obtain ',~ .> f6r T = To/2 by operating ,.~r again exp(-~n~/2) to , ~> ~or T = T~. In that way, we 5an readily "bbtain step ~ y step , ~ / > for T = TO, TN/2, T~/3, .... using raplclly converging Dperat~r exp(-~ ~iH/2). By the use of the method, we treat the model on finite parallelogrammatic lattices with lattice sites up to N = 21. The boundary condition is chosen enabling spins to take the 120 ° structure of the classical model. In Fig. I, we show the lattice with N = 15 as an example. We choose M = 200 for N = 9, 12 and 15, M = 20 for N = 18 and M = I0 for N = 21. For every lattices, we calculate quantities of interest for three or four runs by starting with different sets of initial states given by eq. (4). Although we do not take so many initial states, obtained values of those quantities for different runs agree well with each other. Error bars presented in figures given below only mean deviations of the values for different runs.

~l

o

0.2

o.t.

r

1

0.6

0.8

[

I kT/J

Fig. 2. Temperature dependence of the specific heat for various lattices. kT~ 0.7-

0.6

0.5

Zlh 0.3 I oo

•

t' t

t

21 18 15

t

t

12

9

Fig. 3. Plots of peak-temperature

V:TVVV% %

Fig. i. The unit cell with lattice sites N = 15 treated in this paper.

T

m

I/N 2

v s 1/N 2 .

We calculate various quantities such as the energy, specific heat, susceptibility, long range chirality and etc. Here we only present results for the specific heat and the long range chirality, since we are currently interested in whether the long range chiral order occurs or not. The specific heats for various lattices are shown in Fig. 2. The temperature dependence of the specific heat has a different trend whether N is even or odd. However, for both the cases of N being even and odd, the peak height of the specific heat increases with N suggesting the occurrence of a divergent singularity. We estimate the temperature Tch at

ANTIFERROMAGNETIC

Vol. 67, No. 3

which the specific heat will diverge at the thermodynamic limits. In Fig.3, we plot the peak temperature of the specific heat T as a function of N 2. In the case of odd N , m t ~ points lie aln~st on a straight line. The intersection of the line with the T axis, m which gives the value of T_L , agrees well wlth that of the llne connectln~ two polnts for even N. The value of Tch is thus estimated as kT . / J = 0 . 3 9 ± 0 . 0 2 . c n The c h i r a l i t y o p e r a t o r X (R) i n a t r i a n g u l a r p l a q u e t t e a t R i s d e f i n e d by X (R)=(I/24-3) (SIXS2+S2xS3+S3xSI) ,

(6)

and the operator of the long range c h i r a l i t y by x

= ExZ(R),

(7)

R

where the sum runs over all upright triangular plaquettes of the lattice [2]. We c~Iculate the average of the square of it, < X - > , and show it in Fig. 4. In the lattices of N = 9, 12 and 15, the values of for different, runs agree well for all temperatures. In the larger lattices, however, those deviate not a little. This is because the number of initial states taken in (3) is not large enogh, i.e., M = 20 for N = 18 and M = I0 for N = 21. We a g ~ n find that the temperature dependences of < X ~ > for even and odd N are different to each other. For odd N, they increase monotonically as the temperature is decreased, whereas for even N they have a maximum at some finite temperature. Irrespective of whether N being _~ven or odd, increases rather rapidly ~round a finite temperature and saturates ( or @ e c ~ a s ~ s for even N ). We note the value of .'X ">/N- for N = 9 and 12 at very low tempera5urea agrees with rigorous values in FB wit-

:X2>I 0.8 N=9

0.6: £;; :_:;~ - ~ ' - ' - ~ b - ~

23]

S=I/2 XY MODEL ON THE TRIANGULAR LATTICE

....

h i n a n a c c u r a c y o f 0.05%. The v a ~ u e ~ o r N = 15 a t v e r y low t e m p e r a t u r e s is /N- = 0.638 ± 0.002, which also agrees well with t h a t i n t e r p o l a t e d by u s i n g a n e x t r a p o l a t i o n function given in FB: /N2 = 0 . 5 5 6 0 + l . 2 0 5 0 / N + 0 . 3 8 0 7 / N 2.

(8)

m e a n s t h a t w e obtain the same extrapolation function at T = 0 using data for N = 15 instead of that for N = 3. Data for N = 3 at finite temperatures cannot be used to derive the extrapolation function, because in that lattice each spin interacts with the others for three times. Assuming that the extrapolation function still has the same quadratic form of I/N and using data for N = 9, 15 and 21, we determine it at finite temperatures and obtain extrapolated values which are also plotted in Fig. 4. Using the extrapolated values, we try to estimate T , and the critical exponent ~ of deflCnned by

This

/N2

(kTch/J _kT/J )2 ~ .

(9 )

Plots of the values of both the sides of eq. (9) in a log-log scale are shown in Fig. 5. We see that, when we choose kTch/J ~ 0 . 3 8 , almost all points in the figure seem to lie on a straight line with its slope 2 B 4 0 . 3 8 . Hence, we estimate kT _/J = 0 . 3 8 ± 0 . 0 2 and B = 0.19± 0.I0. This valuCen of T _ agrees fairly well with that obtained from ~ e peak temperature of the specific heat. The value of the critical exponent ~ ~ 0.19 is not different much from that of the magnetization of the Ising model B =1/8, which is what we are expected [2]. We have studied the temperature dependences of the specific heat and the long range chirality of the antiferromagnetic S = I/2 xy model on the finite trian~Jlar lattices using a Monte Calro method. We have shown that, in a finite system, the long range chirality does not increase gradually as the temperature is decreased, but does rather rapidly around some finite temperature. In particular, for

1.0 < X2> N2

0.4

.6

.4

~

0.2

h

.2

0.1

0,2

0.3

0.4

kT/J 0,5 28 =0.38

F i g . 4. T e m p e r a t u r e d e p e n d e n c e o f t h e l o n g range chirality, w h e r e p o i n t s a t T -- 0 f o r N = 9, 12, 21 a r e t h o s e g i v e n i n FB, a n d t h a t f o r N = 15 i s o n e o b t a i n e d by u s i n g e q . ( 8 ) . The p o i n t s l a b e l e d by FB a n d NN a r e e x t r a p o l a t e d v a l u e s f o r N = co g i v e n i n FB and NN, r e s p e c tively. Closed circles for T > 0 are extrapolated values obtained in the present; paper.

.I

,oJ

o 0.40

I

I

.02

.OZ,

F i g . 5. L o g - l o g p l o t s r a l i t y and (Tch - T ) .

I

I

t

.06 .08 .1

kT (ch m -) J

.2

for the long range chi-

232

ANTIFERROMAGNETIC S=I/2 XY MODEL ON THE TRIANGULAR LATTICE

even N, the long range chirality has a maximum value at some finite temperature. The specific heat is also shown to have a peak which becomes sharper as the size of the lattice enlarges. These results suggest the existence of a non-zero temperature above and below which the chiral nature of the model changes qualitatively. Having assumed that the long range chiral order exists at low temperatures, we have estimated the transition temperature Tch. The result agrees fairly well with that

Vol. 67, No. 3

estimated by using the peak temperature of the specific heat. It should be emphasized that, although the lattices treated here are not so large and the data for the larger lattices are not accurate enough, all the results are consistent with each other. We conclude, hence, that the present study strongly suggests the occurrence of the long range chiral order in 'the model and the transition temperature is estimated to be kTch/J=_ 0.39± 0.03.

References

i. S. Miyashita and H. Shiba, J. Phys. Soc. Jpn. 53, 114(1984). 2. S. Fujiki and D. D. Betts, Prog. Theor. Phys. Suppl. No.87, 268(1986). 3. H. Nishimori and H. Nakanishi, J. Phys. Soc. Jpn. 57, 626 (1988). 4. W. Marsha/l, Proc. Roy. Soc. A232, 48(1955).

5. G. W. P r a t t , J r . , Phys.Rev.122,489(1961). 6. M. Imada and M. T a k a h a s h i , J . P h y s . Soc. J p n . 55, 3354 ( 1 9 8 6 ) . 7. T h i s i s r i g o r o u s l y h o l e d when {~¢ k >} i s a complete set.