Analytical and Numerical Studies of Quantum Plateau State in One Alternating Heisenberg Chain

Commun. Theor. Phys. 6 (204) 263 269 Vol. 6, No. 2, February, 204 Analytical and Numerical Studies of Quantum Plateau State in One Alternating Heisenberg Chain JIANG Jian-Jun ( ),, LIU Yong-Jun ( È ), 2 TANG Fei (»é), 3 YANG Cui-Hong ( Ð ), 4 and SHENG Yu-Bo (Ǒ ) 5 Department of Physics, Sanjiang College, Nanjing 2002, China 2 School of Physics Science and Technology, Yangzhou University, Yangzhou 225002, China 3 Department of Electronic and Information Engineering, Yangzhou Polytechnic Institute, Yangzhou 22527, China 4 School of Physics and Optoelectroic Engineering, Nanjing University of Information Science and Technology, Nanjing 20044, China 5 Institute of Signal Processing Transmission, Nanjing University of Postsand Telecommunications, Nanjing 20003, China (Received June 3, 203; revised manuscript received September 22, 203) Abstract By using the coupled cluster method and the numerical density matrix renormalization group method, we investigate the properties of the quantum plateau state in an alternating Heisenberg spin chain. In the absence of a magnetic field, the results obtained from the coupled cluster method and density matrix renormalization group method both show that the ground state of the alternating chain is a gapped dimerized state when the parameter α exceeds a critical point α c. The value of the critical points can be determined precisely by a detailed investigation of the behavior of the spin gap. The system therefore possesses an m = 0 plateau state in the presence of a magnetic field when α > α c. In addition to the m = 0 plateau state, the results of density matrix renormalization group indicate that there is an m = /4 plateau state that occurs between two critical fields in the alternating chain if α >. The mechanism for the m = /4 plateau state and the critical behavior of the magnetization as one approaches this plateau state are also discussed. PACS numbers: 75.0.Jm, 75.60.Ej, 03.65.Ca Key words: alternating spin chain, coupled cluster method, density matrix renormalization group method, magnetization curve Introduction Research relating to one-dimensional quantum antiferromagnetic spin chains has been an attractive field in low-dimensional quantum magnetism. Usually, there are two kinds of uniform antiferromagnetic Heisenberg chains. One case is given by the spin chain with half integer spin, and another case is given by the spin chain with integer spin. The classical magnetic long-range order of these two chains disappears as a result of the strong quantum fluctuations due to the low dimensionality. The half integer spin chain has a gapless spectrum and a power-law decay correlation function. By contrast, Haldane proposed in 983 there is a gap between the ground state and the first excited state for the integer spin chain and that the correlation function decays exponentially. [] There are many theoretical studies that have subsequently validated Haldane s conjecture. For example, the research on the AKLT model shows that each s = spin in the ground state can be viewed as two s = /2 spins in the symmetric triplet state. [2] In Ref. [3], den Nijs and Rommelse proposed that, although the ground state of the integer spin chain is disordered, it possesses a hidden antiferromagnetic order, which can be measured by the string order parameter. It was pointed out by Kennedy and Tasaki that the hidden order is a discrete Z 2 Z 2 symmetry. [4] Numerical results indicate that the spin gap in the s = spin chain is about 0.4J, where J is the strength of the antiferromagnetic interaction. [5] In addition to this evidence provided by theoretical calculations, the experimental evidence is that the m = 0 (m is the magnetization per site) magnetization plateau found in the compound [Ni(en) 2 (NO 2 )]ClO 4, also shows the spin gap of the s = spin chain does exist. [6] The magnetization plateau in the magnetization process, which is caused by the spin gap, is a very interesting quantum phenomenon in the spin chain. In 997, Oshikawa, Yamanaka, and Affleck pointed out that the necessary condition for the appearance of the magnetization Supported by the National Natural Science Foundation of China under Grant Nos. 0804053 and 620347, the Natural Science Foundation of Jiangsu Province under Grant No. BK203428, the Natural Science Foundation of the Jiangsu Higher Education Institutions under Grant No. 3KJD40003, the Scientific Research Foundation of Nanjing University of Posts and Telecommunications under Grant No. NY2008, and Qing Lan Project of Jiangsu Province Corresponding author, E-mail: jianjunjiang@26.com c 203 Chinese Physical Society and IOP Publishing Ltd http://www.iop.org/ej/journal/ctp http://ctp.itp.ac.cn

264 Communications in Theoretical Physics Vol. 6 plateau is n(s m) = integer, () where n is the period of the ground state. [7] One typical example that satisfies the above condition is the mixed spin chain. For example, the mixed spin (, /2) Heisenberg chain possesses the m = /4 plateau state and it can be used to describe the properties of the actual oxamato compound NiCu(pba)(D 2 O) 3 2D 2 O. [8 9] Another example is the bond alternating Heisenberg chain. Due to the different spatial structures, different alternating Heisenberg chains possess various magnetization plateaus. [0 6] For instance, the spin-/2 tetrameric Heisenberg chain with alternating coupling ferromagnetic-ferromagneticantiferromagnetic-antiferromagnetic, which has been realized in the compound Cu(3-Clpy) 2 (N 3 ) 2, has the m = /4 magnetization plateau. [] In the present paper, we study the behaviour of the magnetization plateau of an alternating Heisenberg spin chain by using the analytical coupled cluster method (CCM) and numerical density matrix renormalization group (DMRG) method. [5] The model is shown in Fig. and the Hamiltonian is N/4 H = J (S 4i 3 S 4i 2 + S 4i 2 S 4i + αs 4i S 4i i= + S 4i S 4i+ ) h N Si z, (2) i= where S 4i+j (j = 3, 2,, 0, and ) is a spin-/2 operator. J and αj (α ) denote the antiferromagnetic couplings. When α =, the model (2) reduces to the isotropic Heisenberg chain in an external magnetic field. In that case, the phase diagram of the model is exactly known and it only possesses the classical saturation plateau. [7] When α, according to Eq. (), model (2) may exhibit the m = 0 and m = /4 plateau states besides the m = /2 plateau state. In the following discussion, our main goal is to study the effect of modulating parameter α on the properties of the m = 0 and m = /4 plateau states that are caused by quantum fluctuations. We now assume, without losing generality, that J = throughout the rest of this paper. We now describe the organization of this paper. In the next section, the details of the application of CCM formalism to model (2) are described. In Sec. 3, the application of the DMRG to the alternating chain is given. In Sec. 4, the results of CCM and DMRG are presented. A summary is given in the final section. 2 The Coupled Cluster Method Applied to the Alternating Heisenberg Chain In recent years, a quite new method called CCM has been successfully applied to different quantum spin chains. [8 30] Detailed descriptions of the CCM applied to quantum spin systems have been given in papers. [8,20 2] We just focus on the application of CCM to m = 0 plateau state of the alternating Heisenberg chain. When m = 0, Hamiltonian (2) is equivalent to the following Hamiltonian N/4 H = J (S 4i 3 S 4i 2 + S 4i 2 S 4i + αs 4i S 4i i= + S 4i S 4i+ ). (3) The critical field h c of model (2), at which the m = 0 plateau state disappears, equals to the spin gap of Hamiltonian (3). So, we can use Hamiltonian (3) to investigate the properties of the m = 0 plateau state. The starting point of any CCM calculation is to choose a model state φ and this is often a classical spin state. So we choose the Neel state ( ) as the model state for model (3). Then we perform a rotation of the local axes of the up spins by 80 about the y-axis such that all spins in the model state align along the negative z-axis. After this rotation, the CCM parameterization of the ket ground state of model (3) is given by [20 2] ψ = e S φ, S = N l= i,i 2,i l S i,i 2,i l s + i s + i 2 s + i l. (4) The CCM formalism is exact if all spin configurations in the S correlation operator are considered, but it is impossible in practice. In this paper, a quite general approximation scheme called LSUBn is used to truncate the expansion of the operator S. [20 2] In the LSUBn approximation, only the configurations including n or fewer correlated spins which span a range of no more than n contiguous lattice sites are retained. The fundamental configurations contained in the LSUBn approximation can be reduced by using the lattice symmetries and furthermore by the imposing that restriction that Stol z = N Si z = 0. i= We note that the ground state of model (3) lies in the subspace Stol z = 0. The number of LSUBn configurations is given in Table. Table Number of fundamental configurations of the LSUBn approximation with n = {0, 2, 4} N F denotes the number of the fundenmental configurations for the ground state with S z tol = 0, N Fe denotes the number of the fundenmental configurations for the excited state with S z tol =. N F N Fe LSUB0 283 420 LSUB2 987 584 LSUB4 3559 6006 Once the correlation coefficients contained in the operator S have been found then one can use ψ to calculate the ground state energy E g of Hamiltonian (3) by using the following formula E g = φ e S H e S φ. (5)

No. 2 Communications in Theoretical Physics 265 To obtain the correlation coefficients contained in the operator S, one needs to solve the ket-state equations, which are given by [20 2] φ s i s i 2 s i l e S H e S φ = 0. (6) After these coupled equations are solved, the correlation coefficients retained in the LSUBn approximation can be obtained. We can use the CCM ket-state correlation coefficients to calculate the ground state energy using formula (5). As the derivation of the coupled equations for higher orders of approximation is extremely tedious, we have developed our own programme by using Matlab to automate this process according to the method discussed in Ref. [2]. The Matlab code with double precision was performed on a private computer. Then, by applying an excitation operator X e linearly to the ket-state wave function (4), an excited state wave function ψ e of model (3) can be obtained [20] ψ e = X e e S φ, N X e = χ i,i 2,...i l s + i s + i 2 s + i l. (7) i,i 2,...i l l= Analogously to the ground state, we also use the LSUBn approximation scheme to truncate the expansion of the, n,,, n 2 n k, n 2, n 2 2, n p, n 2 p,, n k 2,, n k p n n, n 2 n 2 n2, n 2 2 n p np, n 2 p operator X e. One can find the fundamental excited state configurations retained in the LSUBn approximation by using the lattice symmetry and the restricted condition Stol z = + or. Table also gives the number of such fundamental configurations. To get the excitation energy ε e (ε e is the difference between the excited state energy and the ground state energy), one can use the method introduced in Ref. [20] to obtain the LSUBn eigenvalue equations. Those equations with eigenvalues ε e and corresponding eigenvectors χ e i,i 2,...i l are as follows ε e χ e i,i 2,...i l = φ s i s i 2 s i l e S [H, X e ] e S φ = 0. (8) The excitation energy gap can be obtained from the lowest eigenvalue of Eq. (8). Since the LSUBn approximation becomes exact in the limit n, we need to extrapolate the LSUBn results to the limit n. [20 2] For the ground state energy, we use the following well-tested formula E g (n) = a 0 + a ( n 2 ) + a 2 ( n 2 ) 2. (9) For the gap of the lowest-lying excitations, we use a matrix given by n,, n l n n2,, n l 2 n2 np,, n l p np a 0 a a k b b l = n n2 np, where k + l + = p and the extrapolated value of n is given by a 0. In this paper, we use n = {0, 2, 4} for both extrapolation schemes described above. 3 The Application of the Density Matrix Renormalization Group to the Alternating Heisenberg Chain In this section, we briefly describe how to treat the alternating Heisenberg chain with the method of DMRG. Similar to the DMRG treatment of the dimerized spin chain, [3] the superblock of Hamiltonian (2) in our DMRG calculation is composed of four blocks S a E b, where S and E are the system and environment blocks, while a and b denote the elementary blocks respectively added to S and E in each DMRG iteration. Blocks S and b are linked because we use periodic boundary condition in this paper. In the first iteration of the DMRG calculation, both the system and environment blocks only contain one unit cell (4 sites). After one iteration, the system size of the superblock grows by 8 sites as we take one unit cell as the elementary block. 4 sweeps were done to obtain accurate results. The DMRG programme code was also written in the Matlab environment and performed on a private computer. The number of the state kept per block is 80 240 and the highest errors are in the order of 0 7. 4 Results 4. The Magnetization Curve of the Alternating Heisenberg Chain To find the plateau state of model (2), we calculate the magnetization curve of that model by employing DMRG. Figure 2 displays the magnetization curve for a system size N = 64 when α =.5. Using the method introduced in Ref. [32], we also present the magnetization curve in the thermodynamic limit in Fig. 2. It can be seen that, besides the m = 0 plateau state in the parameter regime h h c, a finite plateau appears at m = /4. In the following discussion, we focus on the properties of the above

266 Communications in Theoretical Physics Vol. 6 two quantum plateau states. should find the critical point α c. Fig. The structure of the alternating Heisenberg spin chain. Fig. 3 The GS energy per site e versus α using CCM and DMRG. Fig. 2 The magnetization m as a function of h when α =.5. The line with circles represents the magnetization curve for the finite chain of N = 64. The bold line is the result for the infinite chain. 4.2 The m = 0 Plateau State Fig. 4 The second derivative of e, d 2 e/dα 2 as a function of α. In this section, we discuss the behaviour of the m = 0 plateau state, which is caused by the spin gap of Hamiltonian (3). The width of that state equals to the value of the critical field h c and it is given by h c = = E (Stol z = ) E g (Stol z = 0), () where E and E g are the energies of the lowest-lying state with Stol z = and Sz tol = 0. When α =, the spectrum of Hamiltonian (3) is gapless. Thus, model (2) does not possess the m = 0 plateau state. In the case of large α or small α, the m = 0 plateau state belongs to the dimerized state because the spin pairs S 4i 3 and S 4i 2 and the spin pairs S 4i and S 4i tend to form singlet dimmers under that condition. One can reasonably predict that a quantum phase transition at a critical point α c between the gapless spin-liquid state and the gapped dimerized state occurs in Hamiltonian (3). As the m = 0 plateau state only appears in the dimerized state parameter regime, we Fig. 5 The spin gap versus α using CCM and DMRG. It is known that the ground state energy density e or its derivative may exhibit special property at the critical point. [33] We search firstly for quantum phase transition existing in the alternating chain by using the ground state energy of Hamiltonian (3). Figure 3 shows the ground state energy per site e = Eg/N obtained from CCM and DMRG. The energies calculated by DMRG are extrapolated to the thermodynamic limit by using the following

No. 2 Communications in Theoretical Physics 267 formula with N = 32, 40, 48, 56, and 64 spins [34] exp( N/c 2 ) f(α, N) = f(α) + c N p, (2) where p = 2. When α =, the energies per site obtained from DMRG and CCM are 0.443 46 and 0.443 38 respectively. They are both close to the exact result 0.443 47. [20] As shown in Fig. 3, the ground state energy decreases as a result of increasing α and the results given by DMRG and CCM are in good agreement with each other. Since e changes continuously when α varies, its derivative is calculated. Figure 4 presents the second derivative of e obtained from DMRG. As is apparent in this figure, d 2 e/dα 2, all of the results for various lattice sizes, N, display a peak near α =. Indeed, we see that the location of the peak moves nearer to α = and that the peak gets higher and shaper as N increases. Therefore, the second derivative of e can be used to detect the quantum phase transition in the alternating chain. magnetic field when α. A similar phenomenon can also be observed in the dimerized spin chain. [36] 4.3 The m = /4 Plateau State We now discuss the effect of the parameter α on the properties of the m = /4 plateau state by using DMRG. That state occurs between two critical magnetic fields h c2 and h c3. The critical fields h c2 and h c3 are respectively given by h c2 = lim [E(N, N Sz tol = N/4) E(N, Stol z = N/4 )], h c3 = lim [E(N, N Sz tol = N/4 + ) E(N, Stol z = N/4)], (3) where E(N, S z tol ) is the lowest energy in the subspace Sz tol. Fig. 6 Finite size scaling of the spin gap for lengths: N = 48, 56, 64. Next, the spin gap of the alternating chain was also calculated by DMRG and CCM because the existence of a spin gap is an indication of a dimerized state. In order to extrapolate the results of DMRG, we use formula (2) with p =. Figure 5 shows the results of the spin gap as a function of α. From this graph, one can see the results of CCM are in qualitative agreement with those of DMRG. Moreover, the spin gap increases as the parameter α increases, corresponding to the singlet dimer ground state. As shown in Fig. 6, we plot N β for different chain sizes N as a function of α in order to determine the critical point α c precisely. We use numerical results obtained from DMRG and we find that all the curves cross at α =, with the choice of a suitable exponent β. This phenomenon means that the critical point of the alternating chain extracted from the operator is α c =. [35] From the above analysis, one can conclude that the ground state of Hamiltonian (3) evolves from a spin-liquid state to a dimerized state when the parameter α changes across the critical point α c. Therefore, it possesses an m = 0 plateau state with width in the presence of a Fig. 7 The critical magnetic fields h c2, h c3 and the width of the plateau D obtained from DMRG as a function of α. Fig. 8 The scaled gap ND as a function of α. Once these two critical fields have been calculated, the width of the plateau may be determined by using: D = h c3 h c2. We have calculated the above two critical fields for system sizes of N = 32, 40, 48, 56, and 64 by using DMRG. By using formula (2) with p =, the bulk limit of the finite size results is obtained. The numerical results of the critical field and the width of the plateau

268 Communications in Theoretical Physics Vol. 6 are plotted in Fig. 7. As can be seen from this figure, the critical field h c2 decreases as α increases. By contrast, the critical field h c3 increases as α also increases. One sees also that the plateau occurs only when this parameter exceeds. To find the critical point at which the plateau emerges precisely, we display the scaled gap ND of finite systems (N = 40 64) in Fig. 8. It is apparent that a finite plateau exists in the infinite system when α > because ND increases with increasing N in this parameter region. [37] Fig. 9 The expectation values S z i for N = 64 spins when α = 0. N = 64 system when α = 0, in Fig. 9. As is apparent in Fig. 9, the behaviour of Si z obtained from DMRG favors the above mechanism for the m = /4 plateau. Lastly, we investigate the interesting critical behavior of the magnetization at m = /4. In the vicinity of the critical fields h c2 and h c3, the behavior of the magnetization m has the form [38 39] m /4 (h h c3 ) /δ+, h > h c3, /4 m (h c2 h) /δ, h < h c2, (5) where δ + and δ are the critical exponents, which can be used to describe the universality class of the phase transition induced by the field. One can estimate the critical exponents by using two quantities f + (N) and f (N), which are defined as [39] f ± (N) ±[E(N, N/4 ± 2) + E(N, N/4) 2E(N, N/4 ± )]. (6) Once f + and f are obtained, δ + and δ can be determined from the slope of the lnf + lnn and lnf lnn plot respectively. To avoid the large finite size effect in the vicinity of the critical point, we only calculate the critical exponents when α.2. Figure 0 shows the results of lnf ± versus ln N. From this figure, we see that the calculated points fit to a line very well. Thus, by a numerical fitting, δ + and δ can be estimated. The results of δ + and δ are presented in Fig.. From this figure, one can reasonably draw the conclusion that δ + = δ = 2, as expected for conventional one-dimensional gapped spin chains. [38] Fig. 0 ln f +(lnf ) is plotted versus lnn. To understand the mechanism for the m = /4 plateau of the alternating chain, it is instructive to examine the case of large α limit. In this limit, two nearest spins connected by the αj interaction form a singlet dimer and the other two spins in a unit cell linked by J interaction form a triplet dimer in subspace Stol z = N/4. Thus, in the m = /4 plateau state, the expectation values Si z should be ( {, 2, ) ( 2, 0, 0, 2, ) ( 2, 0, 0, 2, ) } 2, 0, 0,. (4) To check this hypothesis, we present the expectation values Si z, which are obtained by using DMRG for the Fig. The critical exponents δ + and δ as a function of α. 5 Conclusions In summary, we have investigated the properties of the quantum plateau state of the alternating Heisenberg chain by using CCM and DMRG methods. As the m = 0 plateau state of Hamiltonian (2) is caused by the spin gap existing in Hamiltonian (3), we first study the critical behaviour of Hamiltonian (3). The results for the second derivative of ground state energy per site obtained from

No. 2 Communications in Theoretical Physics 269 DMRG show that a second order quantum phase transition happens at a critical point α c. The results of the spin gap drawn from CCM and DMRG both indicate that the spin gap of Hamiltonian (3) opens up when the parameter α exceeds α c. By analyzing the scaling behavior of the spin gap, we find that the precise value of α c is. Thus, when α c >, there is an m = 0 plateau state in the alternating Heisenberg chain. Due to its special spatial structure, the alternating Heisenberg chain also possesses an m = /4 quantum plateau state that appears between two critical fields. Similar to the m = 0 plateau state, the m = /4 plateau state also appears when α > and its width grows with the increase of α. By calculating the expectation values Si z using DMRG, we investigated the mechanism for the m = /4 plateau state. The behavior of the magnetization m near the critical fields h c2 and h c3 exhibits a conventional critical behavior. Acknowledgment It is a pleasure to thank Dr. Damain Farnell for his careful reading of the manuscript and his valuable suggestions. References [] F.D.M. Haldane, Phys. Rev. Lett. 50 (983) 53. [2] I. Affleck, T. Kennedy, E.H. Lieb, and H. Tasaki, Phys. Rev. Lett. 59 (987) 799. [3] M. den Nijs and K. Rommelse, Phys. Rev. B 40 (989) 4709. [4] T. Kennedy and H. Tasaki, Phys. Rev. B 45 (992) 304. [5] S.R. White, Phys. Rev. Lett. 69 (992) 2863. [6] M. Yamashita, T. Ishii, and H. Matsuzaka, Coord. Chem. Rev. 98 (2000) 347. [7] M. Oshikawa, M. Yamanaka, and I. Affleck, Phys. Rev. Lett. 78 (997) 984. [8] T. Sakai and S. Yamamoto, Phys. Rev. B 60 (999) 4053. [9] J.J. Jiang, Y.J. Liu, F. Tang, and C.H. Yang, Physica B 406 (20) 78. [0] W. Chen, K. Hida, and H. Nakano, J. Phys. Soc. Jpn. 68 (999) 625. [] H.T. Lu, Y.H. Su, L.Q. Sun, J. Chang, C.S. Liu, H.G. Luo, and T. Xiang, Phys. Rev. B 7 (2005) 44426. [2] B. Gu, G. Su, and S. Gao, J. Phys.: Condens. Matter 7 (2005) 608. [3] S.S. Gong and G. Su, Phys. Rev. B 78 (2008) 0446. [4] S.S. Gong, S. Gao, and G. Su, Phys. Rev. B 80 (2009) 0443. [5] S. Mahdavifar and J. Abouie, J. Phys.: Condens. Matter 23 (20) 246002. [6] M.S. Naseri, G.I. Japaridze, S. Mahdavifar, and S.F. Shayesteh, J. Phys.: Condens. Matter 24 (202) 6002. [7] D.V. Dmitriev, V.Y. Krivnov, and A.A. Ovchinnikov, Phys. Rev. B 65 (2002) 72409. [8] R.F. Bishop, J.B. Parkinson, and Y. Xian, Phys. Rev. B 44 (99) 9425. [9] R.F. Bishop, D.J.J. Farnell, and J.B. Parkinson, Phys. Rev. B 58 (998) 6394. [20] R.F. Bishop, D.J.J. Farnell, S.E. Krüger, J.B. Parkinson, J. Richter, and C. Zeng, J. Phys.: Condens. Matter 2 (2000) 6887. [2] D.J.J. Farnell, R.F. Bishop, and K.A. Gernoth, J. Stat. Phys. 08 (2002) 40. [22] D.J.J. Farnell, J. Schulenburg, J. Richter, and K.A. Gernoth, Phys. Rev. B 72 (2005) 72408. [23] R. Darradi, J. Richter, K.A. Gernoth, and D.J.J. Farnell, Phys. Rev. B 72 (2005) 04425. [24] D. Schmalfuß, R. Darradi, J. Richter, J. Schulenburg, and D. Ihle, Phys. Rev. Lett. 97 (2006) 5720. [25] R.F. Bishop, P.H.Y. Li, D.J.J. Farnell, and C.E. Campbell, Phys. Rev. B 79 (2009) 74405. [26] D.J.J. Farnell, R. Zinke, J. Schulenburg, and J. Richter, J. Phys.: Condens. Matter 2 (2009) 406002. [27] J. Richter, R. Darradi, J. Schulenburg, D.J.J. Farnell, and H. Rosner, Phys. Rev. B 8 (200) 74429. [28] D.J.J. Farnell, R. Darradi, R. Schmidt, and J. Richter, Phys. Rev. B 84 (20) 04406. [29] O. Goetze, D.J.J. Farnell, R.F. Bishop, P.H.Y. Li, and J. Richter, Phys. Rev. B 84 (20) 224428. [30] P.H.Y. Li, R.F. Bishop, D.J.J. Farnell, J. Richter, and C.E. Campbell, Phys. Rev. B 85 (202) 0855. [3] T. Papenbrock, T. Barnes, D.J. Dean, M.V. Stoitsov, and M.R. Strayer, Phys. Rev. B 68 (2003) 02446. [32] K. Okamoto, T. Tonegawa, and M. Kaburagi, J. Phys.: Condens. Matter 5 (2003) 5979. [33] Y.C. Tzeng, H.H. Hung, Y.C. Chen, and M.F. Yang, Phys. Rev. A 77 (2008) 06232. [34] T. Barnes, E. Dagotto, J. Riera, and E.S. Swanson, Phys. Rev. B 47 (993) 396. [35] G. De. Chiara, M. Lewenstein, and A. Sanpera, Phys. Rev. B 84 (20) 05445. [36] J.J. Jiang and Y.J. Liu, Physica B 403 (2008) 3498. [37] N. Okazaki, J. Miyoshi, and T. Sakai, J. Phys. Soc. Jpn. 69 (2000) 37. [38] T. Sakai and M. Takahashi, Phys. Rev. B 57 (998) R809. [39] T. Sakai and H. Nakano, Phys. Rev. B 83 (20) 00405(R).