GROUND-STATE PROPERTIES OF THE ONE-DIMENSIONAL SPIN-1/2 HEISENBERG-XY ANTIFERROMAGNET WITH COMPETING INTERACTIONS

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GROUND-STATE PROPERTIES OF THE

ONE-DIMENSIONAL SPIN-1/2 HEISENBERG-XY

ANTIFERROMAGNET WITH COMPETING

INTERACTIONS

T. Tonegawa, I. Harada

To cite this version:

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JOURNAL DE PHYSIQUE

Colloque C8, Supplkment au no 12, Tome 49, decembre 1988

GROUND-STATE PROPERTIES OF THE ONE-DIMENSIONAL SPIN-1/2

HEISENBERG-XY ANTIFERROMAGNET WITH COMPETING INTERACTIONS

T. Tonegawa and I. Harada

Department of Physics, Kobe University, Rokkodai, Kobe 657, Japan

Abstract. - We study the ground-state properties of the one-dimensional spin-112 Heisenberg-XY magnet with antifer-

romagnetic first- and second-neighbor interactions. Extrapolating exact results for finite-size systems of up to 20 spins,

we estimate the behavior at the infinitesize limit. The limiting ground state has a finite gap when the second-neighbor interaction is sufficiently large.

In the past years a lot of work has been devoted to the study of magnetic systems wit6 competing interactions. Most systems treated in this kind of work, however, is restricted t o classical spin systems. In order to examine the quantum effect on the competition of interactions, we consider here the one-dimensional spin-112 system governed by the Hamiltonian,

where periodic boundary conditions (SO+N = SO) are imposed and where SF (a = x, y,z) is the a-

component of the spin-112 operator Sf a t the

t

-th site;

J1and J z are the first- and second-neighbor interaction

constants, respectively; 6 is the parameter describing the anisotropy of the interactions; N is the number of spins in the system. We assume that both interactions are antiferromagnetic (J1

>

0, Jz

2

0), thus competing with each other, and that the anisotropy is of the Heisenberg-XY type (0

5

6

5

1 ) .

This paper aims at exploring the ground-state properties of the system in the limit of N -+ oo. We cal- culate exactly the ground-state energy, the twwspin correlation function, and the singlet-triplet energy gap for finite-size ( N = 6,8,

...,

20) systems by diagonalizing numerically matrices representing the Hamiltonian given by equation (1) and extrapolate them to N --+ co to discuss the limiting properties. The method of the matrix diagonalization as well as that of the extrap- olation to N -,

oo

has been discussed in detail in a previous paper by the present authors [I], where the case of isotropic interactions, i.e., the case of 6 = 1 has been thoroughly studied. It is noted that the magne- tization versus external field curve in the ground state of the present system has been discussed in another paper 121 and that the finite-temperature properties of

the present system has also been discussed in separate papers [3, 41.

We now turn t o the discussion on the results of nu- merical calculation. In figure 1 the limiting ( N -+ m) ground-state energy E, per spin in units of J1 is plotted

as a function of j (= Jz/Jl) for representative values of 6, the result for 6 = 1 obtained in reference [I] being also replotted for comparison. It should be noted that

cg estimated for j = 0 and 6 = 0 and E~ estimated for

j = 0 and 6 = 1 are in excellent agreement with the exact values -2/.rr [5] and (1

-

4 In 2) /2 [6], respectively. Furthermore E, estimated for j = 112 agrees completely with the exact value

-

(2

+

6 ) /4 [7, 81. We expect that E, for any value of 6 has been estimated

0.0 0.5 1.0

Fig. 1. - Plot of sg versws j for representative values of 6.

with accuracy better than 0.003 % for 0

<

j

<

112 and with accuracy better than 0.1 % for j

>

112.

Similarly to the case of 6 = 1 discussed in reference [I], we have estimated with satisfactory accuracy the limiting (N -+ co) values of the first three two- spin correlation functions w" ( I ) , w" (2)

,

and w" (3), where w" (n) is defined by w" (n) =

(ag

ISFS&nI (Pg)

with the ground-state eigenfunction

ag.

Unfortu- nately, however, we do not have enough space to

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C8 - 1412 JOURNAL DE PHYSIQUE

demonstrate explicitly these results. Instead, we discuss here the wave number q," at which the Fourier transform Sa (q) = exp (-iqn) wa (n) of wa (n) has a maximum value. The limiting value of q," estimated for any value of 6 is equal t o n when 0

<

j

<

112 and decreases monotonically from n to n/2 as j in- creases from 112. This means that the limiting ground state of the present system is of an antiferromagnetic character or of an incommensurate (or higher-order commensurate) character depending on whether 0

5

j

5

112 or j

>

112. Figure 2 shows the j-dependence of q," and qg for the case of 6 = 0 together with that of qo (= q: = q;) for the case of 6 = 1 which has been obtained in reference [I]. From this figure we see that

q," little depends both on 6 and on a. In the one- dimensional classical Heisenberg-XY spin system, qg

is independent of 6 and is equal to a when 0

<

j

<

114 and to cos-' (-1/4j) when j

2

114; for comparison this q: is also shown in figure 2.

Fig. 2.

-

Plot of qg versus j for 5 = 0 and 1. The solid

lines and the dotted line show, respectively, the results for

the present spin-112 system and that for the classical spin system.

As in the case of 6 = 1 [I], the limiting (N + oo)

ground state of the present system in the case of 0

5

6

<

1 is also in the gapless or finite-gap phase depending on whether 0

<

j

5

j, or j

>

j,. The value of j, estimated for 6 = 0, 114, 112, and 314 are, respectively, jc=0.35 f 0.01, 0.32 f 0.01, 0.30 f 0.01, and 0.30 0.01. These values together with the value j,=0.30 f 0.01 for 6 = 1 obtained in reference [I] are plotted in figure 3. The line obtained by connecting

the points plotted is a phase boundary line between the

0.50

I

dirner phase I.

spin-fluid phase

1

Fig. 3.

-

Ground-state phase diagram in the j versus 6

plane.

gapless phase and the finite-gap pliase. In the former phase, which is called the spin-fluid phase by Haldane [9], any type of long-range order does not exist, as ex- pected; while the latter phase, called the dimer phase [9],-is characterized by the existeince of spontaneous dimerization, or, in other words, by the existence of dirner long-range order. (See Ref. [I] for more detailed discussion on the case of

6

= 1.)

The authors are indebted t o 1)rs. J. Takimoto, H. Nishimori and Y. Taguchi for the usage of their computer programs for diagonalizing matrices. The present work has been supported i n part by a Grant- in-Aid for Scientific Research from the Ministry of Ed- ucation, Science and Culture.

[I] Tonegawa, T. and Harada, I., J. Phys. Soc. Jpn

56 (1987) 2153.

[2] Tonegawa, T. and Harada, I., t o be published in

Physica B (Proc. of the 2nd Int. Symp. on High

Field Magnetism, Leuven, 1988).

[3] Harada, I., Kimura, T. and Tonegawa, T., J. Phys. Soc. Jpn 57 (1988) 2770.

[4] Harada, I., Kimura, T. and Tonegawa T., to be published in J. Phys. France, Proc. of this conf.

[5] Lieb, E., Schultz, T. and Mattis, D., Ann. Phys.

N.

Y.

16 (1961) 407.

[6] HulthBn, L., Ark. Mat. Astron. Fys. 26A (1938)

1.

171 Majumdar, C . K., J. Phys.

C

3 (1970) 911. [8] Shastry, B. S. and Sutherlanld B., Phys. Rev.

Lett. 47 (1981) 964.