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Spin-Peierls mechanism for the non-trivial magnetization plateaux in two-leg spin ladders
Physica B 329–333 (2003) 1207–1208
Spin-Peierls mechanism for the non-trivial magnetization plateaux in two-leg spin ladders K. Okamotoa,*, T. Sakaib a
Department of Physics, Tokyo Institute of Technology, Oh-okayama 2-12-1, Meguro-ku, Tokyo 152-8551, Japan b Department of Physics, Tohoku University, Aramaki, Aoba-ku, Sendai 980-8578, Japan
Abstract We propose the spin-Peierls mechanism for the non-trivial magnetization plateaux at Ms =4 and ð3=4ÞMs (Ms is the saturation magnetization) of the two-leg S ¼ 1 spin ladder, bearing in mind the recent experiment on the BIP-TENO. r 2003 Elsevier Science B.V. All rights reserved. Keywords: Spin ladder; Spin gap; Magnetization plateau; Spin-Peierls transition
The magnetization plateaux of low-dimensional magnets have been extensively studied in these years. A new organic tetraradical, 3; 30 ; 5; 50 -tetrakis(N-tert-butylaminoxyl)biphenyl, abbreviated as BIP-TENO, has been synthesized recently [1]. This material can be regarded as an S ¼ 1 antiferromagnetic two-leg spin ladder, and the magnetization plateau at Ms =4 (Ms is the saturation magnetization) was observed between 45 and 65 T [2,3]. This plateau should be non-trivial because this must occur with a two-fold degeneracy in the ground state, associated with the spontaneous breakdown of the translational symmetry, based on the necessary condition of the magnetization quantization [4]. We have already proposed the ‘‘dimer mechanism’’ for this plateau due to the third-neighbor interactions [5,6] and succeeded to explain the magnetization curve semiquantitatively. However, the required strength of the third neighbor interactions is fairly larger. In consideration of these situations, we propose another new mechanism for the Ms =4 plateau, the spin-Peierls mechanism. We note that this mechanism can be applicable not only to the Ms =4 plateau of the S ¼ 1 ladder, but also to many other plateaux, as will be stated. The S ¼ 1 ladder (simplified model for BIP-TENO [1]) is shown in Fig. 1. Both coupling constants are
*Corresponding author. Fax: +81-3-5734-2739. E-mail address: [email protected] (K. Okamoto).
supposed to be antiferromagnetic. In experimental estimation, it seems J> bJ1 : The Hamiltonian of Fig. 1 is expressed as H ¼ J1
L X X
S l;j S l;jþ1 þ J>
l¼1;2 j¼1
h
L X X
L X
S 1;j S 2;j
j¼1 z Sl;j ;
ð1Þ
l¼1;2 j¼1
where S denotes the spin-1 operator, j the rung number and l ¼ 1; 2 the leg number. The last term is the Zeeman energy in the magnetic field h: In the limit J> bJ1 we can use the degenerate perturbation theory [7]. When J1 ¼ 0; all the rung spin pairs are mutually independent. Thus, at M ¼ Ms =4; half of the rung spin pairs are in the state + + +! 0 k 1 m ð2Þ þ cð0; 0Þ ¼ pffiffiffi 0 m 3 k and the remaining half-pairs are in the state + +! 0 1 m cð1; 1Þ ¼ pffiffiffi ; m 2 0
ð3Þ
z where cðStot ; Stot Þ is the wave function of with the z quantum numbers Stot and Stot : These two wave functions have the lowest energies in the subspace of z z Stot ¼ 0 and Stot ¼ 1; respectively. The Ms =4 state is highly degenerate as far as J1 ¼ 0; because there is no restriction for the configurations of these two states.
0921-4526/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4526(02)02127-0
K. Okamoto, T. Sakai / Physica B 329–333 (2003) 1207–1208
1208
J1
S1, j
S1, j+1
S2, j
S2, j+1
J
Fig. 1. Simplified model of BIP-TENO. Open circles represent S ¼ 1 spins.
This degeneracy is lifted up by the introduction of J1 : To investigate the effect of J1 ; we introduce the pseudospin T with T ¼ 1=2: The j * S and j + S states of the T spin correspond to cð1; 1Þ and cð0; 0Þ; respectively. Neglecting the other seven states for the rung spin pairs, the effective Hamiltonian can be written as, through straightforward calculations X xy y x z z Heff ¼ fJeff ðTjx Tjþ1 þ Tjy Tjþ1 Þ þ Jeff Tjz Tjþ1 g ð4Þ j
in the lowest order of J1 ; where xy Jeff ¼ ð8=3ÞJ1 ;
z Jeff ¼ ð1=2ÞJ1 :
ð5Þ
Thus, the Ms =4 plateau problem of the original model in a magnetic field is mapped onto the M ¼ 0 problem of the T ¼ 12 antiferromagnetic XXZ spin chain with the nearest-neighbor interaction in the absence of magnetic field (Fig. 2). Let us introduce the bond alternation to Heff ; coming from the lattice distortion to Heff ; as xy;z xy;z ) Jeff f1 þ ð1Þj dg; Jeff
ð6Þ
where d is the magnitude of the bond alternation. The energy gain of the T-system due to the bond alternation is proportional to da where [8,9] a ¼ 4=ð4 ZÞ;
Z ¼ 2=½ð1 þ ð2=pÞ sin1 Deff
respectively. In this case, we see Deff ¼ 1=4;
a ¼ 1:76o2;
ð10Þ
which also results in the possibility of the spinPeierls mechanism. A similar analysis can be applied also to the Ms =2 plateau of the two-leg S ¼ 1=2 spin ladder. We have proposed the spin-Peierls mechanism for the Ms =4 and ð3=4ÞMs plateaux of the S ¼ 1 two-leg ladder. The spin-Peierls mechanism and the third neighbor interaction may work in a cooperative way to form singlet dimer pairs of the T picture. However, more detailed investigations may be required for the full understanding of the magnetization curve of the BIPTENO. The present mechanism is applicable to many other plateaux in various systems. This research was supported by a Grant-in-Aid for Scientific Research from Ministry of Education, Culture, Sports, Science and Technology of Japan.
ð7Þ
xy z Jeff =Jeff :
Since the energy loss of the lattice with Deff distortion is proportional to d2 ; the condition for the occurrence of the spontaneous lattice distortion is ao2: This mechanism is essentially the same as that of the spin-Peierls transition. Thus, we call our mechanism ‘‘spin-Peierls mechanism’’. In our case, we can see a ¼ 1:81o2;
Fig. 2. Physical picture of the spin-Peierls mechanism. The distance between neighboring two spins on a leg changes alternatingly. This is the origin of the bond alternation. The rectangle denote the effective singlet pair of two T spins.
ð8Þ
which means that the spin-Peierls mechanism is possible from the consideration of the energy. Our preliminary numerical calculation based on the S ¼ 1 model shows that a is very close to 1.8 as conjectured in Eq. (8). We can develop a similar consideration for the ð3=4ÞMs plateau case, where j + S and j * S should be modified to + + +! m 0 1 m ; ð9Þ ; j*S¼ j + S ¼ pffiffiffi m m 0 2
References [1] K. Kato, Y. Hosokoshi, K. Inoue, T. Goto, J. Phys. Soc. Japan 69 (2000) 1008. [2] T. Goto, M.I. Bartashevich, Y. Hosokoshi, K. Kato, K. Inoue, Physica B 294–295 (2001) 43. [3] T. Sakai, N. Okazaki, K. Okamoto, K. Kindo, Y. Narumi, Y. Hosokoshi, K. Kato, K. Inoue, T. Goto, this conference 26CP9. [4] M. Oshikawa, M. Yamanaka, I. Affleck, Phys. Rev. Lett. 78 (1997) 1984. [5] K. Okamoto, N. Okazaki, T. Sakai, J. Phys. Soc. Japan 70 (2001) 636. [6] N. Okazaki, K. Okamoto, T. Sakai, J. Phys. Chem. Sol., in press. [7] K. Totsuka, Phys. Rev. B 57 (1998) 3454. [8] T. Nakano, H. Fukuyama, J. Phys. Soc. Japan 50 (1981) 2489. [9] H. Nishimori, K. Okamoto, J. Phys. Soc. Japan 56 (1987) 1132.
Spin-Peierls mechanism for the non-trivial magnetization plateaux in two-leg spin ladders K. Okamotoa,*, T. Sakaib a
Department of Physics, Tokyo Institute of Technology, Oh-okayama 2-12-1, Meguro-ku, Tokyo 152-8551, Japan b Department of Physics, Tohoku University, Aramaki, Aoba-ku, Sendai 980-8578, Japan
Abstract We propose the spin-Peierls mechanism for the non-trivial magnetization plateaux at Ms =4 and ð3=4ÞMs (Ms is the saturation magnetization) of the two-leg S ¼ 1 spin ladder, bearing in mind the recent experiment on the BIP-TENO. r 2003 Elsevier Science B.V. All rights reserved. Keywords: Spin ladder; Spin gap; Magnetization plateau; Spin-Peierls transition
The magnetization plateaux of low-dimensional magnets have been extensively studied in these years. A new organic tetraradical, 3; 30 ; 5; 50 -tetrakis(N-tert-butylaminoxyl)biphenyl, abbreviated as BIP-TENO, has been synthesized recently [1]. This material can be regarded as an S ¼ 1 antiferromagnetic two-leg spin ladder, and the magnetization plateau at Ms =4 (Ms is the saturation magnetization) was observed between 45 and 65 T [2,3]. This plateau should be non-trivial because this must occur with a two-fold degeneracy in the ground state, associated with the spontaneous breakdown of the translational symmetry, based on the necessary condition of the magnetization quantization [4]. We have already proposed the ‘‘dimer mechanism’’ for this plateau due to the third-neighbor interactions [5,6] and succeeded to explain the magnetization curve semiquantitatively. However, the required strength of the third neighbor interactions is fairly larger. In consideration of these situations, we propose another new mechanism for the Ms =4 plateau, the spin-Peierls mechanism. We note that this mechanism can be applicable not only to the Ms =4 plateau of the S ¼ 1 ladder, but also to many other plateaux, as will be stated. The S ¼ 1 ladder (simplified model for BIP-TENO [1]) is shown in Fig. 1. Both coupling constants are
*Corresponding author. Fax: +81-3-5734-2739. E-mail address: [email protected] (K. Okamoto).
supposed to be antiferromagnetic. In experimental estimation, it seems J> bJ1 : The Hamiltonian of Fig. 1 is expressed as H ¼ J1
L X X
S l;j S l;jþ1 þ J>
l¼1;2 j¼1
h
L X X
L X
S 1;j S 2;j
j¼1 z Sl;j ;
ð1Þ
l¼1;2 j¼1
where S denotes the spin-1 operator, j the rung number and l ¼ 1; 2 the leg number. The last term is the Zeeman energy in the magnetic field h: In the limit J> bJ1 we can use the degenerate perturbation theory [7]. When J1 ¼ 0; all the rung spin pairs are mutually independent. Thus, at M ¼ Ms =4; half of the rung spin pairs are in the state + + +! 0 k 1 m ð2Þ þ cð0; 0Þ ¼ pffiffiffi 0 m 3 k and the remaining half-pairs are in the state + +! 0 1 m cð1; 1Þ ¼ pffiffiffi ; m 2 0
ð3Þ
z where cðStot ; Stot Þ is the wave function of with the z quantum numbers Stot and Stot : These two wave functions have the lowest energies in the subspace of z z Stot ¼ 0 and Stot ¼ 1; respectively. The Ms =4 state is highly degenerate as far as J1 ¼ 0; because there is no restriction for the configurations of these two states.
0921-4526/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4526(02)02127-0
K. Okamoto, T. Sakai / Physica B 329–333 (2003) 1207–1208
1208
J1
S1, j
S1, j+1
S2, j
S2, j+1
J
Fig. 1. Simplified model of BIP-TENO. Open circles represent S ¼ 1 spins.
This degeneracy is lifted up by the introduction of J1 : To investigate the effect of J1 ; we introduce the pseudospin T with T ¼ 1=2: The j * S and j + S states of the T spin correspond to cð1; 1Þ and cð0; 0Þ; respectively. Neglecting the other seven states for the rung spin pairs, the effective Hamiltonian can be written as, through straightforward calculations X xy y x z z Heff ¼ fJeff ðTjx Tjþ1 þ Tjy Tjþ1 Þ þ Jeff Tjz Tjþ1 g ð4Þ j
in the lowest order of J1 ; where xy Jeff ¼ ð8=3ÞJ1 ;
z Jeff ¼ ð1=2ÞJ1 :
ð5Þ
Thus, the Ms =4 plateau problem of the original model in a magnetic field is mapped onto the M ¼ 0 problem of the T ¼ 12 antiferromagnetic XXZ spin chain with the nearest-neighbor interaction in the absence of magnetic field (Fig. 2). Let us introduce the bond alternation to Heff ; coming from the lattice distortion to Heff ; as xy;z xy;z ) Jeff f1 þ ð1Þj dg; Jeff
ð6Þ
where d is the magnitude of the bond alternation. The energy gain of the T-system due to the bond alternation is proportional to da where [8,9] a ¼ 4=ð4 ZÞ;
Z ¼ 2=½ð1 þ ð2=pÞ sin1 Deff
respectively. In this case, we see Deff ¼ 1=4;
a ¼ 1:76o2;
ð10Þ
which also results in the possibility of the spinPeierls mechanism. A similar analysis can be applied also to the Ms =2 plateau of the two-leg S ¼ 1=2 spin ladder. We have proposed the spin-Peierls mechanism for the Ms =4 and ð3=4ÞMs plateaux of the S ¼ 1 two-leg ladder. The spin-Peierls mechanism and the third neighbor interaction may work in a cooperative way to form singlet dimer pairs of the T picture. However, more detailed investigations may be required for the full understanding of the magnetization curve of the BIPTENO. The present mechanism is applicable to many other plateaux in various systems. This research was supported by a Grant-in-Aid for Scientific Research from Ministry of Education, Culture, Sports, Science and Technology of Japan.
ð7Þ
xy z Jeff =Jeff :
Since the energy loss of the lattice with Deff distortion is proportional to d2 ; the condition for the occurrence of the spontaneous lattice distortion is ao2: This mechanism is essentially the same as that of the spin-Peierls transition. Thus, we call our mechanism ‘‘spin-Peierls mechanism’’. In our case, we can see a ¼ 1:81o2;
Fig. 2. Physical picture of the spin-Peierls mechanism. The distance between neighboring two spins on a leg changes alternatingly. This is the origin of the bond alternation. The rectangle denote the effective singlet pair of two T spins.
ð8Þ
which means that the spin-Peierls mechanism is possible from the consideration of the energy. Our preliminary numerical calculation based on the S ¼ 1 model shows that a is very close to 1.8 as conjectured in Eq. (8). We can develop a similar consideration for the ð3=4ÞMs plateau case, where j + S and j * S should be modified to + + +! m 0 1 m ; ð9Þ ; j*S¼ j + S ¼ pffiffiffi m m 0 2
References [1] K. Kato, Y. Hosokoshi, K. Inoue, T. Goto, J. Phys. Soc. Japan 69 (2000) 1008. [2] T. Goto, M.I. Bartashevich, Y. Hosokoshi, K. Kato, K. Inoue, Physica B 294–295 (2001) 43. [3] T. Sakai, N. Okazaki, K. Okamoto, K. Kindo, Y. Narumi, Y. Hosokoshi, K. Kato, K. Inoue, T. Goto, this conference 26CP9. [4] M. Oshikawa, M. Yamanaka, I. Affleck, Phys. Rev. Lett. 78 (1997) 1984. [5] K. Okamoto, N. Okazaki, T. Sakai, J. Phys. Soc. Japan 70 (2001) 636. [6] N. Okazaki, K. Okamoto, T. Sakai, J. Phys. Chem. Sol., in press. [7] K. Totsuka, Phys. Rev. B 57 (1998) 3454. [8] T. Nakano, H. Fukuyama, J. Phys. Soc. Japan 50 (1981) 2489. [9] H. Nishimori, K. Okamoto, J. Phys. Soc. Japan 56 (1987) 1132.