PLATEAU OF THE MAGNETIZATION CURVE OF THE

FERROMAGNETIC-FERROMAGNETIC-ANTIFERROMAGNETIC

SPIN CHAIN

Kiyomi Okamoto

Department of Physics, Tokyo Institute of Technology, Meguro, Tokyo 152, Japan

(Received )

I analytically study the plateau of the magnetization curve at (where is the saturation magnetization) of the one-dimensional trimerized Heisenberg spin system with ferromagnetic ()-ferromagnetic ()-antiferromagnetic () interactions at . I use the bosonization technique for the fermion representation of the spin Hamiltonian through the Jordan-Wigner transformation. The plateau appears when , and vanishes when , where the critical value is estimated as . The behavior of the width of the plateau near is of the Kosterlitz-Thouless type. The present theory well explains the numerical result by Hida.

Recently Hida numerically studied the spin chain system in which the interaction between neighboring spins changes as ferromagnetic-ferromag-netic-antiferromagnetic (Fig.1). The Hamiltonian of this model is written by

where and are the magnitudes of the antiferromagnetic and ferromagnetic couplings, respectively. I have introduced the anisotropy parameter for later convenience, although Hida investigated only the case. This model is not too artificial nor a theoretical toy, as noticed by Hida. In fact, the substance , of which magnetization is measured by Ajiro et al. in strong magnetic fields, is known to be a (quasi-)one dimensional magnet to which this model is applied.

Hida investigated this model (only the case) by the numerical diagonalization method for finite systems (up to 24 spins) to find that there was a plateau in the magnetization curve as far as is small, and the width of the plateau decreases as the parameter increases (Fig.2). The location of the plateau was , where is the saturation magnetization. He could not obtain a definite conclusion about the existence of the plateau for large case, because the magnetization varied stepwise for the finite systems. Since this model becomes antiferromagnetic chain model when and due to the formation of an quartet by spins , and , it is expected that there is no plateau in this limit. It is believed that all the half-odd spin chains with antiferromagnetic nearest-neighbor interactions belong to the same universality class as that of .

There arise several questions:

(i) Why does the plateau appear at ? (ii) Does the plateau vanish at a finite value of or persist to ? (iii) If the plateau vanishes at , what is the behavior of the plateau near ?

They are interesting questions not only from the standpoint of the statistical physics but also from that of explaining the properties of the existing materials. I note that the plateau was not observed in the report of Ajiro et al.

In this paper I will analytically study the mechanism for the appearance of the plateau and also discuss whether the critical value exists or not.

Performing the spin rotation around the -axis for the spins located at the sites

Hamiltonian (1) is transformed into the form of a generalized version of the trimerized antiferromagne-tic chain:

where

In case of the model where , we can solve Hamiltonian (4) by the use of the fermion representation through the Jordan-Wigner transformation. The dispersion relation is shown in Fig.3. When the magnetic field is applied, the dispersion curve shifts along the -axis, which explains the existence of the plateau as well as its location . Note that the magnetization is related to the number of occupied states as

where is the total number of spins.

This is a simple explanation for the question why the plateau appears at . However, there still remains questions. Why is there no plateau in the experimental results on ? Is there any essential difference between the case () and the isotropic Heisenberg case ()? How can we explain Hida’s result, especially in case? To answer these questions, we have to consider the interactions between fermions when .

To consider the effect of the trimerization and that of the interaction between fermions simultaneously, I use the method of the bosonization. The bosonization is one of the powerful methods in one-dimensional quantum problems. Here I do not enter into the details of the bosonization procedure. I note that the bosonization is usually done in case of no magnetization (half-filled in the language of the fermion), but in the present case, we have to perform the bosonization near (1/3-filled in the language of the fermion).

After the bosonization, Hamiltonian (3) is transformed into a generalized sine-Gordon Hamiltonian:

where is the distance between neighboring spins. The effect of the trimerization appears in the and terms. If the term and/or the term are relevant in the sense of the renormalization group, the spectrum of has a gap, which brings about the plateau in the magnetization curve. If both of them are irrelevant, the spectrum of is gapless, which results in no plateau. Which case is realized? — It depends on the magnitudes of and and also on the parameter

The renormalization group calculation shows that the term and/or the term are relevant as far as .

The value of is slightly shifted through the bosonization procedure. The expressions of and in eq.(8) is considered to be the lowest order expansions with respect to . When (i.e. ), the system becomes ferromagnetic. Therefore should diverge to when , although it seems to diverge at from eq.(8).

The bosonized Hamiltonian (7) has the same form as that of the generalized version of the dimerized model.

In fact, if we perform the bosonization for Hamiltonian (12), we obtain

where

Also in this case, the expression of and should be considered to be the lowest order expansions near . In fact, in the absence of the dimerization, the exact form of is obtained from the application of the exact solution of the eight-vertex model as

If we expand eq.(16) near , we can see that it agrees with the expression of from eq.(11) and (14).

has the same form as with the identification of the parameters

Then we can use the knowledge on the dimerized model. The phase diagram of the dimerized model when is shown in Fig.4. This phase diagram was obtained by the renormalization group calculation, by the high temperature series expansion after mapping onto the finite-temperature classical 2D model (modified Ashkin-Teller model), and by the numerical diagonalization of the original spin Hamiltonian for finite systems. When , a naive consideration leads to the effective dimerization parameter

because is related to and , whereas to . However, the renormalization group method and the variational method bring about

From eqs.(5), (17) and (19), we obtain the mapping of the present model onto the dimerized model, as shown in Fig.4. The case of the present model corresponds to the case of the dimerized model, and the case to the case . Therefore there exists the critical value , where the transition from the plateau state to the no-plateau state. This transition is of the Kosterlitz-Thouless type, as known from the critical properties of the sine-Gordon Hamiltonian.

The above discussion is based on the bosonization method, which make it difficult to estimate the value of itself. It is because the parameters are slightly shifted through the bosonization procedure, as already explained. A rough estimation of the value of is

I have analytically investigated the plateau in the magnetization curve of the ferromagnetic-ferromagnetic-antiferromagnetic spin chain, which is first pointed out by Hida by the use of the numerical diagonalization. The present analytical study semi-qualitatively explains the numerical result of Hida.

References

1. K. Hida, J. Phys. Soc. Jpn. 63, 2359 (1994).

2. Y. Ajiro, T. Asano, T. Inami, H. Aruga-Katori and T. Goto, J. Phys. Soc. Jpn. 63, 859 (1994).

3. for instance, E. Fradkin, Field Theories of Condensed Matter Systems, Chap.4, Addison Wesley, Redwood City (1991).

4. K. Okamoto, D. Nishino and Y. Saika, J. Phys. Soc. Jpn. 62, 2587 (1993).

5. J. D. Johnson, S. Krinsky and B. McCoy, Phys. Rev. A 8, 2526 (1973).

6. K. Okamoto and T. Sugiyama, J. Phys. Soc. Jpn. 57, 1610 (1988).

7. M. Kohmoto, M. den Nijs and L. P. Kadanoff, Phys. Rev. B 24, 5229 (1981).

8. S. Yoshida and K. Okamoto, J. Phys. Soc. Jpn. 58, 4367 (1989).

9. for instance, J. B. Kogut, Rev. Mod. Phys. 51, 659 (1979).

Figure Captions:

Fig.1. Ferromagnetic-ferromagnetic-antiferromagnetic spin chain. Solid lines and dotted lined represent ferromagnetic couplings and antiferromagnetic couplings , respectively. Three spins in an ellipse make an quartet when and .

Fig.2. Sketch of Hida’s numerical result for the magnetization curve. The case (a) is for smaller and (b) for larger .

Fig.3. Dispersion relation of of eq.(4) in case of . (a): case. (b): case.

Fig.4. Mapping of the present model onto the dimerized chain. The open circle corresponds to the case and the closed circle to the case.