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Magnetic properties of S=1 spin chains with alternating interactions
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 272–276 (2004) 888–889
Magnetic properties of S ¼ 1 spin chains with alternating interactions I. Harada*, M.M. Rahman, A. Hamaguchi, Y. Nishiyama Department of Physics, Faculty of Science, Okayama University 3-1-1 Tsushima-naka, Okayama 700-8530, Japan
Abstract Magnetic properties of S ¼ 1 antiferromagnetic spin chains with alternating nearest-neighbor and competing nextnearest-neighbor interactions are studied, using the strong coupling expansion as well as numerical methods. Characteristic features of this system such as magnetization plateau, field-induced long-range ordering and incommensurate behavior are discussed in connection with the dimer gap, focusing on a novel role of the anisotropy. r 2003 Elsevier B.V. All rights reserved. PACS: 75.10.Jm; 75.30.Kz Keywords: Dimer gap; Magnetization plateau; Field-induced long-range ordering
1. Introduction It is known that S ¼ 1 antiferromagnetic spin chains with alternating nearest-neighbor (nn) and competing next-nearest-neighbor (nnn) interactions exhibit characteristic behavior, such as the magnetization plateau [1], the field-induced long-range ordering [2] and the incommensurate (IC) behavior [3]. These are usually discussed in connection with quantum fluctuations with the dimer gap, as in the case of S ¼ 12 spin chains. We, however, stress that, in the case of the S ¼ 1 spin chain, different kinds of the spin gap, the Haldane or the dimer gap, appear [4] and, the spin-anisotropy plays a novel role. It is the purpose of this paper to reveal the characteristics mentioned above, using the strong coupling expansion (SCE) (see for instance Ref. [5]) from the dimer limit as well as the numerical methods, the density matrix renormalization group (DMRG) and the matrix-product wave function (MPWF) [6] methods. We especially focus our attention on the physics behind the characteristics. To this end, SCE for the ground state *Corresponding author. Tel.: +81-86-251-7808; fax: +8186-251-7808. E-mail address: [email protected] (I. Harada).
and the low-energy excited states as well as for the thermodynamic quantities of the system is quite instructive. Now, let us introduce the Hamiltonian describing the S ¼ 1 spin chain: H=J ¼ H0 þ H1 ; where H0 ¼
X
ð1Þ
z 2 z fS2i S2iþ1 þ d½ðS2i Þ þ ðS2iþ1 Þ2
i
h ðS2i þ S2iþ1 Þg; H1 ¼
X
ðaS2iþ1 S2iþ2 þ jSi Siþ2 Þ:
ð2Þ ð3Þ
i
Here the parameters, J; a; j; d and h, being positive, denote, respectively, the antiferromagnetic exchange constant which defines the unit of energy, the alternating exchange constant, the competing nnn exchange constant, the anisotropy constant which makes the xy plane as an easy-plane, and the magnetic field, of which direction is crucial for our discussions. Firstly, we note the existence of the so-called disorder line [4] in this system, on which a direct product of each spin-singlet state is an exact ground state. By virtue of this, SCE works quite effectively near the disorder line.
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.1030
ARTICLE IN PRESS I. Harada et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 888–889
Fig. 1. Magnetization curves. The circles and diamonds are the experimental results [1].
Another merit of SCE is to take the anisotropy and the Zeeman terms into account in the zeroth order. The ground state energy within the second order SCE is written as Eg ¼ 2f1 þ ða 2jÞ2 ð1 h2 =3Þ=3ð1 h2 Þg for d ¼ 0; which follows quite reasonably the numerical results, although we do not show it here. Further, we point out that, in the low-energy excitation spectra, the IC behavior appears in the second-order process. Secondly, we discuss the magnetization process based on SCE. We can map the spin dimer system onto the Fermion system, which classifies the magnetization plateau into two types: a band insulator and a Mott insulator with site order. The former type of plateau occurs at m ¼ 12 m0 ; where m0 denotes the saturation magnetization, due to one-body effects while the latter at m ¼ 14 m0 and 34 m0 are the consequences of many-body interactions, controlled by the nnn interactions. We show in Fig. 1 the magnetization curves with the experimental results, which depend crucially on the field direction. The physics is understandable if we analyze the energy spectra of the zeroth order SCE and take into consideration an important role of the anisotropy. Thirdly, the field-induced long-range ordering is discussed within the mean-field approximation for interchain interactions. As already mentioned in the Haldane chain [7], a quite different physics in the critical behavior can be observed in the present case, changing the field direction relative to the anisotropy axis. In fact, the critical temperatures for two cases are very different, as is seen in Fig. 2, where these have been determined by the same equation as that in the Haldane chain [7] with appropriate critical fields.
889
Fig. 2. Phase diagram. The circles and diamonds are the experimental results [2].
Lastly, the IC behavior is discussed. For this purpose, MPWF method is appropriate and gives rise to the IC behavior in the static spin structure factor (SSSF) [8]. It is interesting to note that the line jc ¼ 0:14ð1=SÞ þ 0:25 can interpolate three jc points for S ¼ N; 1; and 12; above which SSSF shows a peak at IC wave vectors. In conclusion, we have revealed the interesting behaviors of the S ¼ 1 spin chain with alternating nn and competing nnn interactions: the magnetization plateau, the field-induced long-range ordering and the IC behavior have been discussed in connection with the dimer gap as well as with the novel role of the anisotropy, for which SCE is very effective to make clear the physics behind such kinds of the quantum phenomena. The authors thank Dr. Hagiwara for fruitful discussions on his experimental results.
References [1] Y. Narumi, et al., J. Magn. Magn. Mater. 177–181 (1998) 685. [2] N. Tateiwa, et al., Physica B 329–333 (2003) 1209. [3] T. Tonegawa, et al., J. Phys. Soc. Jpn. 61 (1992) 2890. [4] S. Pati, et al., J. Phys.: Condens. Matter 9 (1997) 219. [5] M. Reigrotzki, et al., J. Phys.: Condens. Matter 6 (1994) 9235. [6] S. O.stlund, S. Rommer, Phys. Rev. B 55 (1997) 2164. [7] I. Harada, Y. Nishiyama, Progr. Theor. Phys. Suppl. 145 (2002) 107. [8] A. Hamaguchi, et al., unpublished.
Journal of Magnetism and Magnetic Materials 272–276 (2004) 888–889
Magnetic properties of S ¼ 1 spin chains with alternating interactions I. Harada*, M.M. Rahman, A. Hamaguchi, Y. Nishiyama Department of Physics, Faculty of Science, Okayama University 3-1-1 Tsushima-naka, Okayama 700-8530, Japan
Abstract Magnetic properties of S ¼ 1 antiferromagnetic spin chains with alternating nearest-neighbor and competing nextnearest-neighbor interactions are studied, using the strong coupling expansion as well as numerical methods. Characteristic features of this system such as magnetization plateau, field-induced long-range ordering and incommensurate behavior are discussed in connection with the dimer gap, focusing on a novel role of the anisotropy. r 2003 Elsevier B.V. All rights reserved. PACS: 75.10.Jm; 75.30.Kz Keywords: Dimer gap; Magnetization plateau; Field-induced long-range ordering
1. Introduction It is known that S ¼ 1 antiferromagnetic spin chains with alternating nearest-neighbor (nn) and competing next-nearest-neighbor (nnn) interactions exhibit characteristic behavior, such as the magnetization plateau [1], the field-induced long-range ordering [2] and the incommensurate (IC) behavior [3]. These are usually discussed in connection with quantum fluctuations with the dimer gap, as in the case of S ¼ 12 spin chains. We, however, stress that, in the case of the S ¼ 1 spin chain, different kinds of the spin gap, the Haldane or the dimer gap, appear [4] and, the spin-anisotropy plays a novel role. It is the purpose of this paper to reveal the characteristics mentioned above, using the strong coupling expansion (SCE) (see for instance Ref. [5]) from the dimer limit as well as the numerical methods, the density matrix renormalization group (DMRG) and the matrix-product wave function (MPWF) [6] methods. We especially focus our attention on the physics behind the characteristics. To this end, SCE for the ground state *Corresponding author. Tel.: +81-86-251-7808; fax: +8186-251-7808. E-mail address: [email protected] (I. Harada).
and the low-energy excited states as well as for the thermodynamic quantities of the system is quite instructive. Now, let us introduce the Hamiltonian describing the S ¼ 1 spin chain: H=J ¼ H0 þ H1 ; where H0 ¼
X
ð1Þ
z 2 z fS2i S2iþ1 þ d½ðS2i Þ þ ðS2iþ1 Þ2
i
h ðS2i þ S2iþ1 Þg; H1 ¼
X
ðaS2iþ1 S2iþ2 þ jSi Siþ2 Þ:
ð2Þ ð3Þ
i
Here the parameters, J; a; j; d and h, being positive, denote, respectively, the antiferromagnetic exchange constant which defines the unit of energy, the alternating exchange constant, the competing nnn exchange constant, the anisotropy constant which makes the xy plane as an easy-plane, and the magnetic field, of which direction is crucial for our discussions. Firstly, we note the existence of the so-called disorder line [4] in this system, on which a direct product of each spin-singlet state is an exact ground state. By virtue of this, SCE works quite effectively near the disorder line.
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.1030
ARTICLE IN PRESS I. Harada et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 888–889
Fig. 1. Magnetization curves. The circles and diamonds are the experimental results [1].
Another merit of SCE is to take the anisotropy and the Zeeman terms into account in the zeroth order. The ground state energy within the second order SCE is written as Eg ¼ 2f1 þ ða 2jÞ2 ð1 h2 =3Þ=3ð1 h2 Þg for d ¼ 0; which follows quite reasonably the numerical results, although we do not show it here. Further, we point out that, in the low-energy excitation spectra, the IC behavior appears in the second-order process. Secondly, we discuss the magnetization process based on SCE. We can map the spin dimer system onto the Fermion system, which classifies the magnetization plateau into two types: a band insulator and a Mott insulator with site order. The former type of plateau occurs at m ¼ 12 m0 ; where m0 denotes the saturation magnetization, due to one-body effects while the latter at m ¼ 14 m0 and 34 m0 are the consequences of many-body interactions, controlled by the nnn interactions. We show in Fig. 1 the magnetization curves with the experimental results, which depend crucially on the field direction. The physics is understandable if we analyze the energy spectra of the zeroth order SCE and take into consideration an important role of the anisotropy. Thirdly, the field-induced long-range ordering is discussed within the mean-field approximation for interchain interactions. As already mentioned in the Haldane chain [7], a quite different physics in the critical behavior can be observed in the present case, changing the field direction relative to the anisotropy axis. In fact, the critical temperatures for two cases are very different, as is seen in Fig. 2, where these have been determined by the same equation as that in the Haldane chain [7] with appropriate critical fields.
889
Fig. 2. Phase diagram. The circles and diamonds are the experimental results [2].
Lastly, the IC behavior is discussed. For this purpose, MPWF method is appropriate and gives rise to the IC behavior in the static spin structure factor (SSSF) [8]. It is interesting to note that the line jc ¼ 0:14ð1=SÞ þ 0:25 can interpolate three jc points for S ¼ N; 1; and 12; above which SSSF shows a peak at IC wave vectors. In conclusion, we have revealed the interesting behaviors of the S ¼ 1 spin chain with alternating nn and competing nnn interactions: the magnetization plateau, the field-induced long-range ordering and the IC behavior have been discussed in connection with the dimer gap as well as with the novel role of the anisotropy, for which SCE is very effective to make clear the physics behind such kinds of the quantum phenomena. The authors thank Dr. Hagiwara for fruitful discussions on his experimental results.
References [1] Y. Narumi, et al., J. Magn. Magn. Mater. 177–181 (1998) 685. [2] N. Tateiwa, et al., Physica B 329–333 (2003) 1209. [3] T. Tonegawa, et al., J. Phys. Soc. Jpn. 61 (1992) 2890. [4] S. Pati, et al., J. Phys.: Condens. Matter 9 (1997) 219. [5] M. Reigrotzki, et al., J. Phys.: Condens. Matter 6 (1994) 9235. [6] S. O.stlund, S. Rommer, Phys. Rev. B 55 (1997) 2164. [7] I. Harada, Y. Nishiyama, Progr. Theor. Phys. Suppl. 145 (2002) 107. [8] A. Hamaguchi, et al., unpublished.