DISORDER LINE IN A QUANTUM SPIN CHAIN WITH COMPETING INTERACTIONS

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DISORDER LINE IN A QUANTUM SPIN CHAIN

WITH COMPETING INTERACTIONS

I. Harada, T. Kimura, T. Tonegawa

To cite this version:

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JOURNAL DE PHYSIQUE

.Colloque C8, SupplBment au no 12, Tome 49, dbcembre 1988

DISORDER LINE IN A QUANTUM SPIN CHAIN WITH COMPETING

INTERACTIONS

I. Harada, T. Kimural and T. Tonegawa2

Institut fGr Theoretische Physik, Universitit Hannover, Appelstrasse 2, 3000 Hannover 1 , F.R. G.

Abstract. - The spin-112 Heisenberg chain with antiferromagnetic first- and second-neighbor interactions is studied by the cluster transfer-matrix method. It is found that this quantum model exhibits a disorder temperature above which the spin correlation function decays with an incommensurate oscillation while below which it decays antiferromagnetically.

It is well known that, in a classical spin chain with competing interactions, the two-spin correlation function shows not only monotonic exponential decay but exponential decay modified by an incommensurate oscillation [I, 2, 31. There is a definite temperature called a disorder point at which the nature of short- range order changes as described above.

In this paper, we report the results of our calculations for the two-spin correlation function in the one- dimensional spin-112 magnet with antiferromagnetic first- and second-neighbor interactions. We express the Hamiltonian describing the systepl as

with periodic boundary conditions s ~ + i = s;, where

s; is the spin-112 operator a t the site i; Jt and J 2 are the first- and the second-neighbor interaction constant, respectively, and are assumed to be positive (antiferromagnetic); 7 is the parameter describing the Ising anisotropy (0 5 7

5 1)

; N is the number of spins in the chain. A portion of our work on the ground state properties has been reported in separate papers [4, 51. The method used in this paper is the cluster transfer-matrix method based on the Suzuki-Trotter theorem [6]. Details of the method for the present system have been given,in our previous paper [71. To make this paper self-contained, however, we briefly summa- rize the results.

According to the Suzuki-Trotter theorem [6], we ap- proximate the partition function as

(see Fig. la); m is the so-called Trotter number. It is noted that, when either n or m is infinite, Z (n, m) yields the exact partition function. In this paper, we perform the numerical calculations for n = 4, 6, 8 and 10 with m = 1, aiming to take into account effects of the competition properly. We can find the two- dimensional Ising system whose partition function is equivalent to that given by equation (2). The Ising system for m = 1 and n = 10 is depicted in figure lb.

P real-space direction

( b )

Fig. 1. - Schematic illustration of (a) the 10-spin cluster decomposition and (b) the corresponding Ising system on a checkerboard-like lattice in the case of m = 1. At the bottom the transfer matrices along the real-space direction,

Tn and TB, are indicated.

Applying the transfer matrix method to this Ising system, we obtain

2 (n, m) =

TI^

{exp (-pH; (n) /m)

x

(n, m) = (XI) N/2(n-2) 9 (3) xexp

lm)

11

'

(2) _{where hl is the largest eigenvalue of the matrix}_T₌ where ,B = l / k ~ T ; H: (n) and H: (n) are the wspin TnTs (see Fig. 1). w e note that in the case of m = 1, cluster Hamiltonian such that T*=Ts holds for any value of n. In a similar way, the correlation function (sf st+k) in the k * oo limit is H = Ck

(HIP

(n)

+

H: (n)) expressed in terms of the largest eigenvalue A1 and the

second largest eigenvalue X2 of T :

'NEC Corporation, Kawasaki, Kanagawa 211, Japan.

2 ~ e p t . of Phys., Kobe Univ., Kobe 657, Japan. (sIsI+k) 0: exP (-k/5‘) cos (qk)

,

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C8 - 1410 JOURNAL DE PHYSIQUE

where the inverse correlation length 1/< and the wavenumber q are given by

I/< = In (XI/ IX21) /2 (n

-

21, (5) q = tan-' {f Im (X2) /Re (X2)) 12 (n

-

2)

.

₍₆₎ Note that in our units, the lattice constant a = 1.

Now, we present our numerical results of 1/< and q for m = 1 and n = 10. We compare the results for the isotropic case 7 = 1 (the solid curves) to those for the Ising case 7 = 0 (the dashed curves). Note that the results in the Ising case are exact. In figure 2

interactions in the case of 0

5

7

5

1.. The effect of the XY type anisotropy is left for a future study.

Fig. 2.

-

Constant q curves on the t versus j plane for m = 1 and n = 10. The solid curve and the dashed curve, labeled by q = .rr-, denote the disorder line.

the constant q curves are shown in the temperature t (= ]CBT/J~) versus j (= Jz/Jl) plane. At high temperatures the curves for both cases behave similarly, but a t low temperatures each curve for 7 = 1 tends to its own value at t -+ 0 [4] while for 7 = 0 all curves merge into the point j = 1/2 at t = 0. We find in the quantum spin chain a definite temperature, which is very similar to the disorder point in the Ising model [I]. We call it a disorder temperature and depict its trace by the solid curve labeled by q = 7 ~ - . The disorder temperature is seen as a kink in a q versus t curve in figure 3. For j

>

1/2 there is no disorder temper* ture and the curve reaches its own value at t = 0. On the other hand, 1/< also shows a kink at the disorder temperature, as is shown in figure 4. As t -, 0,

115

for y = 1 seems to reach a finite value, while it tends to zero for y = 0. Such behavior is consistent with the ground state properties [4].

In summary, we have found a disorder temperature for a quantum spin chain with competing exchange

Fig. 3. - Temperature dependence of .the wavenumber for m = l a n d n = 1 0 .

Fig. 4.

-

Temperature dependence of the inverse carrel* tion length for m = 1 and n = 10.

[I] Stephenson, J., Can. J. Phys. 48 (1970) 1724. [2] Harada, I., J. Phys. Soc. Jpn ti3 (1984) 1643. [3} Harada, I. and Mikeska, H. J., 2. Phys. B 72

(1988) 391.

[4] Tonegawa, T. and Harada, I., .T. Phys. Soc. Jpn 56 (1987) 2153.

[5] Tonegawa, T. and Harada, I., J. Phys. France, Proc. of this conf.

[6] Suzuki, M., Phys. Rev. B 31 (11985) 2957. [7] Harada, I. Kimura, T. and Tonegawa, T., J. Phys.