The Schwinger-boson mean-field theory (SBMFT) and the linearized tensor renormalization group (LTRG) methods are complementarily applied to explore the thermodynamics of the quantum ferromagnetic mixed spin (S, σ) chains. It is found that the system has double excitations, i.e. a gapless and a gapped excitation; the low-lying spectrum can be approximated by ω k S σ 2 ( S + σ ) J k 2 with J the ferromagnetic coupling; and the gap between the two branches is estimated to be △ ∼ J. The Bose-Einstein condensation indicates a ferromagnetic ground state with magnetization m tot z = N ( S + σ ) . At low temperature, the spin correlation length is inversely proportional to temperature (T), the susceptibility behaviors as χ = a 1 1 T 2 + a 2 1 T , and the specific heat has the form of C = c 1 T c 2 T + c 3 T 3 2 , with ai (i = 1, 2) and ci (i = 1, 2, 3) the temperature independent constants. The SBMFT results are shown to be in qualitatively agreement with those by the LTRG numerical calculations for S = 1 and σ = 1/2. A comparison of the LTRG results with the experimental data of the model material MnIINiII(NO2)4(en)2(en = ethylenediamine), is made, in which the coupling parameters of the compound are obtained. This study provides useful information for deeply understanding the physical properties of quantum ferromagnetic mixed spin chain materials.

Low-dimensional quantum spin systems have been attracting great interest in condensed matter physics both theoretically and experimentally in past decades. Owing to strong quantum fluctuations and competitions between various interactions in model compounds, a number of exotic and fascinating quantum emergent phenomena occur in ground states as well as in excited states. Among others, the spin-1 and spin-1/2 Heisenberg chains and their variants are typical examples. It is known that the spin-1/2 Heisenberg antiferromagnetic chains exhibit gapless excitations with a power-law decay of correlation functions, whereas the spin-1 Heisenberg antiferromagnetic chains bear exponentially decaying correlation function and gapped excitations in terms of Haldane conjecture.1 To gain insight into the interplay of the two kinds of quantum chains with spin integer and half-integer, people turn to inspect the antiferromagnetic quantum mixed-spin chains.2–9 Such systems usually have ferrimagnetic ground states due to uncompensated spins. The spin wave theory exposes the existence of two kinds of excitations, say, the ferromagnetic gapless excitations and antiferromagnetic gapped excitations. The two magnon branches also manifest themselves in thermodynamic properties: the specific heat shows a ferromagnetic behavior C υ T for T → 0 and an antiferromagnetic behavior at intermediate temperature range.6 The magnon gap of the antiferromagnetic mixed-spin (1,1/2) chain was numerically found to be △ = 1.759J.3,4

On the other hand, for the ferromagnetic spin-1/2 uniform chains, the modified spin wave method10 and Schwinger-boson mean field theory11 were usually exploited to probe the ground state and thermodynamic properties including the specific heat, magnetic susceptibility, and correlation functions. The asymptotic behavior of the so-obtained thermodynamic properties at low temperature agrees with the exact results based on the thermodynamic Bethe ansatz for S = 1/2.12 However, the studies on the thermodynamic properties of the Heisenberg quantum ferromagnetic mixed-spin chains are still sparse. Since the ferromagnetic mixed-spin chains might contain interesting features different from the corresponding antiferromagnetic counterpart, more elaborate analyses on this kind of quantum spin chain are quite necessary.

From the experimental point of view, many magnetic materials are model compounds corresponding to the one-dimensional (1D) Heisenberg quantum magnets. For instance, NENP,13 NDOAP,14 and NTENP15 can be well described by the spin S=1 bond alternating Heisenberg antiferromagnetic chain, and Na3Cu2SbO6 is one of the spin-1/2 alternating Heisenberg antiferromagnetic chains.16 Kahn et al. succeeded in synthesizing the ferrimagnetic chains with spin 1 2 and 1.17 Other ferrimagnetic chains with ( 1 2 , 5 2 ) alternating spins were also fabricated.18,19 In addition, some chain compounds with Heisenberg ferromagnetic couplings were designed as well although such magnetic materials are less common than those with antiferromagnetic couplings. An example is MnIINiII(NO2)4(en)2 with en = ethylenediamine.20–22 This compound exhibits intra-chain Heisenberg ferromagnetic interactions between Mn (S = 5/2) and Ni (σ = 1) ions. The magnetization, susceptibility, heat capacity, and other physical quantities were measured. The susceptibility reveals the existence of easy axis anisotropic energy (D term) because of the low local symmetry around Mn ions. Although the experimental studies on this compound gain obvious advances, some further studies are still called for.

Motivated by these findings, in this paper, we study systematically the low-lying excitations and thermodynamic properties of a Heisenberg ferromagnetic mixed spin (S, σ) chain by means of the Schwinger-boson mean field theory (SBMFT)11,23 and the linearized tensor renormalization group (LTRG) method.24 Analytical expressions of the low-lying spectra, the susceptibility and the specific heat are obtained by the SBMFT, and verified numerically by the LTRG method for the case with S=1 and σ = 1/2, showing the qualitative validity of these expressions. In addition, the LTRG method is applied to fit the experimental data on the compound MnIINiII(NO2)4(en)2 to obtain the coupling parameters of the material.

This paper is organized as follows. In Sec. II, the model Hamiltonian and the SBMFT theory is established. In Sec. III, the ferromagnetic ground state is identified by Bose condensation of Schwinger bosons, and the excitation properties of the spectra are also discussed. The thermodynamic properties are investigated by both the SBMFT and LTRG methods in Sec. IV. The experimental data of a ferromagnetic mixed spin chain compound are fitted to gain the relevant material parameters in Sec. V. Finally, a conclusion and discussion is presented.

The Hamiltonian for the 1D Heisenberg ferromagnetic mixed spin (S, σ) chain reads

H = J j N S j σ j 1 2 + S j σ j + 1 2 ,
(1)

where J(>0) is the ferromagnetic coupling constant, N is the number of primitive cell, Sj and σ j ± 1 2 are the spin operators at j and j + 1 2 sites, respectively. The lattice spacing is set to unity. The spin operators can be represented by the Schwinger bosons

S ˆ j + = a ˆ j b ˆ j , S ˆ j = b ˆ j a ˆ j , S ˆ j z = 1 2 ( a ˆ j a ˆ j b ˆ j b ˆ j ) ,
(2)
σ ˆ j + 1 2 + = A ˆ j + 1 2 B ˆ j + 1 2 , σ ˆ j + 1 2 = B ˆ j + 1 2 A ˆ j + 1 2 , σ ˆ j + 1 2 z = 1 2 ( A ˆ j + 1 2 A ˆ j + 1 2 B ˆ j + 1 2 B ˆ j + 1 2 ) ,
(3)

where a, b, A and B are Bose operators.The spin S and σ define the physical subspaces of Schwinger bosons on each site:

2 S = a ˆ j a ˆ j + b ˆ j b ˆ j ,
2 σ = A ˆ j + 1 2 A ˆ j + 1 2 + B ˆ j + 1 2 B ˆ j + 1 2 .
(4)

Then the Hamiltonian Eq. (1) can be written as

H = J 2 j N F ˆ j , j + 1 2 F ˆ j , j + 1 2 + F ˆ j , j 1 2 F ˆ j , j 1 2 4 S σ ,
F ˆ j , j + 1 2 = a ˆ j A ˆ j + 1 2 + b ˆ j B ˆ j + 1 2 .
(5)

Because of the constraints [Eqs. (4)], we introduce two Lagrangian multipliers λj and λ j + 1 2 and decouple the Hamiltonian by introducing the auxiliary field Q j , j ± 1 2 via the Hubbard-Stratonovich transformation. By assuming 〈λj,S〉 = λS, λ j + 1 2 σ = λ σ and Q j , j + 1 2 = Q j , j 1 2 = Q in the SBMFT, the mean-field Hamiltonian is obtained

H MF = i N λ S a ˆ j a ˆ j + b ˆ j b ˆ j + λ σ A ˆ j + 1 2 A ˆ j + 1 2 + B ˆ j + 1 2 B ˆ j + 1 2 i N Q F ˆ j , j + 1 2 + F ˆ j , j + 1 2 + Q F ˆ j , j 1 2 + F ˆ j , j 1 2 + z 2 N J | Q | 2 2 N ( σ λ σ + S λ S ) ,
(6)

where z is the number of nearest neighbors. By making the Fourier transform and defining the form factor γ k = cos k 2 , we express the Hamiltonian in a more compact form

H MF = H aA MF + H bB MF + z 2 N J | Q | 2 2 N ( σ λ σ + S λ S ) ,
(7a)
H aA MF = k [ λ S a ˆ k a ˆ k + λ σ A ˆ k A ˆ k z Q γ k ( a ˆ k A ˆ k + a ˆ k A ˆ k ) ] ,
(7b)
H bB MF = k [ λ S b ˆ k b ˆ k + λ σ B ˆ k B ˆ k z Q γ k ( b ˆ k B ˆ k + b ˆ k B ˆ k ) ] .
(7c)

It is clear that H aA MF and H bB MF can be obtained from each other by exchanging the subscripts a, A to b, B and vice versa, indicating that the system has two degenerate excitations. By introducing the Bogoliubov transformation,

P ˆ k , + = u k a ˆ k v k A ˆ k , Q ˆ k , + = v k a ˆ k + u k A ˆ k ,
(8a)
P ˆ k , = u k b ˆ k v k B ˆ k , Q ˆ k , = v k b ˆ k + u k B ˆ k ,
(8b)

where the coefficients u k 2 = 1 2 + 1 2 ( λ S λ σ ) 2 ( λ S λ σ ) 2 + 4 ( z Q γ k ) 2 , v k 2 = 1 2 1 2 ( λ S λ σ ) 2 ( λ S λ σ ) 2 + 4 ( z Q γ k ) 2 , u k v k = z Q γ k λ S λ σ ( λ S λ σ ) 2 ( λ S λ σ ) 2 + 4 ( z Q γ k ) 2 , we can get the diagonalized Hamiltonian HMF and the free energy FMF

H MF = k , m ω k , P P ˆ k , m P ˆ k , m + ω k , Q Q ˆ k , m Q ˆ k , m + z 2 N J | Q | 2 2 N ( σ λ σ + S λ S ) ,
(9a)
F MF 2 N = m 2 N β k ln 1 e β ω k , P + ln 1 e β ω k , Q + z J | Q | 2 ( σ λ σ + S λ S ) ,
(9b)

where β = 1 k B T . The two branches of excitation spectra are given by

ω k , P = 1 2 ( λ S + λ σ ) + 1 2 ( λ S λ σ ) 1 + 4 ( z Q γ k ) 2 ( λ S λ σ ) 2 ,
(10a)
ω k , Q = 1 2 ( λ S + λ σ ) 1 2 ( λ S λ σ ) 1 + 4 ( z Q γ k ) 2 ( λ S λ σ ) 2 .
(10b)

It is convenient to introduce the scaled parameters Λ±, c, and κ by

Λ ± = 1 2 ( λ S ± λ σ ) , c = z Q , κ 2 = ( z Q ) 2 + Λ 2 ( z Q ) 2 .
(11)

The excitation spectra can then be expressed as ω k , P = Λ + c κ 2 + ( γ k 2 1 ) , ω k , Q = Λ + + c κ 2 + ( γ k 2 1 ) . The spectra are depicted in Fig. 1, where we assume Λ < 0. The parameters λS, λσ and Q are determined by the saddle point equations δFMF/δλS = 0, δFMF/δλσ = 0, and δFMF/δQ = 0. After a straightforward algebra, we have

π π d k 2 π 1 e β ω k , P 1 + 1 e β ω k , Q 1 = S + σ ,
(12a)
π π d k 2 π 1 e β ω k , P 1 1 e β ω k , Q 1 Λ Λ 2 + c 2 γ k 2 = σ S ,
(12b)
π π d k 2 π 1 e β ω k , P 1 1 e β ω k , Q 1 z γ k 2 Λ 2 + c 2 γ k 2 = 2 J .
(12c)
FIG. 1.

The schematic plot of the excited spectrum for the Heisenberg ferromagnetic mixed spin (S,σ) chain. In presence of a weak magnetic field, the spectrum is four-fold. An excited state | ψ 1 = c 1 P ˆ k , P ˆ 0 + | ψ 0 + c 2 Q ˆ k , P ˆ 0 + | ψ 0 can be obtained by acting m ˆ k to the ground state, where the processes of Q ˆ k , P ˆ 0 + and P ˆ k , P ˆ 0 + are indicated by “(I)” and “(II)”, respectively.

FIG. 1.

The schematic plot of the excited spectrum for the Heisenberg ferromagnetic mixed spin (S,σ) chain. In presence of a weak magnetic field, the spectrum is four-fold. An excited state | ψ 1 = c 1 P ˆ k , P ˆ 0 + | ψ 0 + c 2 Q ˆ k , P ˆ 0 + | ψ 0 can be obtained by acting m ˆ k to the ground state, where the processes of Q ˆ k , P ˆ 0 + and P ˆ k , P ˆ 0 + are indicated by “(I)” and “(II)”, respectively.

Close modal

Let us discuss the connection between the ferromagnetic mixed spin chain and the uniform ferromagnetic spin chain. When Sσ, then λSλσ ∼ Λ+λ, Λ ∼ 0, giving rise to ωk,Pλk, and ωk,Qλ + k. Fig. 2 presents these two excitation branches, where each branch is doubly degenerate, because there exist two flavors of the Schwinger bosons. At finite temperature, the ferromagnetic mixed spin chain and the uniform ferromagnetic spin chain both sustain gaps, i.e. Λ+ and λc at k = 0, respectively. In contrast to the uniform ferromagnetic spin chain with only one branch of excitation in the entire Brillouin zone (BZ), we get two branches of excitations shown in Fig. 2. The two branches touch each other at the boundary of the BZ at energy λ. The reason behind this fact is that we have two spins in a unit cell and the length of the unit vector is two times larger. As a consequence, the BZ is thus reduced two times, and the spectrum has to be folded at the boundary of the BZ, forming two branches. When Sσ, the self-consistent equations become

π π d k 2 π 1 e β ω k , P 1 + 1 e β ω k , Q 1 = 2 S ,
(13a)
π π d k 2 π 1 e β ω k , P 1 1 e β ω k , Q 1 γ k = c J .
(13b)
FIG. 2.

The SBMFT excitation spectrum for the Heisenberg ferromagnetic mixed spin (S,σ) chain in the limit Sσ, where the first Brillouin zone is from −π to π. When T ∼ 0, ω k , P J S k 2 + O ( k 4 ) .

FIG. 2.

The SBMFT excitation spectrum for the Heisenberg ferromagnetic mixed spin (S,σ) chain in the limit Sσ, where the first Brillouin zone is from −π to π. When T ∼ 0, ω k , P J S k 2 + O ( k 4 ) .

Close modal

At low temperature, the upper branch ωkQ can be ignored in Eq. (13) and the integral is dominated by the region of small energies and momenta of branch ωkP. We therefore obtain the same expression of the uniform ferromagnetic spin chain at TJS: c 2 = J S + O ( T / J S ) , lim T 0 , k 0 ω k , P J S k 2 + O ( k 4 ) .

The ferromagnetic ground state in the mixed spin chain can be identified by the Bose condensation of Schwinger bosons.25 Suppose that an infinitesimal magnetic field h is coupled to the mixed spin chain, and the Hamiltonian is H = J j N ( S j σ j 1 2 + S j σ j + 1 2 ) h j N ( S j z + σ j + 1 2 z ) . The energy spectra ωk,P and ωk,Q split due to the magnetic field, giving rise to

ω k , P ± = Λ + c κ 2 + ( γ k 2 1 ) h 2 ,
ω k , Q ± = Λ + + c κ 2 + ( γ k 2 1 ) h 2 .
(14)

Among the four branches, the spectrum ωk,P+ has a minimum at k = 0. As T → 0K, the Schwinger bosons have to condense at ωk=0,P+ satisfying the constraints. The first saddle point equation of Eqs. (12) is modified as

lim T 0 1 2 π π d k 2 π 1 e β ω k , P + 1 + 1 e β ω k , Q + 1 + 1 e β ω k , P 1 + 1 e β ω k , Q 1 = 1 2 N n k = 0 , + P = S + σ .
(15)

The ground state magnetization is obtained by

m tot z = lim T 0 j S ˆ j z + σ ˆ j z
= lim T 0 1 2 k n k , + P + n k , + Q n k , P n k , Q
= 1 2 n k = 0 , + P = N ( S + σ ) .
(16)

The ground state |ψ0〉 therefore possesses ferromagnetic long-range order with m tot z = N ( S + σ ) which corresponds to that the total 2N(S + σ) particles condense at k = 0 of ωk,P+ branch. By acting m ˆ k = j ( S ˆ j e i k r j + σ ˆ j + 1 2 e i k ( r j + 1 2 ) ) on the ground state |ψ0〉, we get the excited state | ψ 1 = c 1 P ˆ k , P ˆ 0 , + | ψ 0 + c 2 Q ˆ k , P ˆ 0 , + | ψ 0 , where c1 = uku0 + vkv0 and c2 = u0vkukv0. The state |ψ1〉 is an eigenstate of m ˆ tot z with eigenvalue N(S + σ) − 1. Obviously, m ˆ k + | ψ 0 = 0 . The ferrimagnetic chain8,28,29 also have two branches, i.e. the gapless ferromagnetic branch and the gapped antiferromagnetic branch. The numerical and mean-field studies confirm that the lowest gapless excitation is corresponding to the state with eigenvalue N(sσ) − 1, and the state with eigenvalue N(sσ) + 1 is gapped, where the N(sσ) is the eigenvalue of m ˆ tot z for the ground state. Different from the ferrimagnetic chain, if a particle is annihilated at ωk=0,P+ of the ferromagnetic mixed spin chain, the state would be in the gapless excitation of ωk≠0,P+ or gapped excitation of ωk,Q+. The system is still in the state characterized by m tot z = N ( S + σ ) . If the particle is excited to ωk,P or ωk,Q, the system is in the state with m tot z = N ( S + σ ) 1 . Both the gapless ωk,P and gapped ωk,Q branches contribute to the state with m tot z = N ( S + σ ) 1 .

In this section, we discuss the low-temperature behaviors of the Heisenberg ferromagnetic mixed spin (S, σ) chain through the SBMFT. Generally, the Schwinger boson mean-field theory usually gives qualitatively correct results at low temperature. In order to benchmark the validity of the mean-field theory, we calculate the thermodynamic properties numerically by using recently proposed LTRG method24 that may give quite accurate results at low temperature. When the LTRG algorithm is applied to the 1D quantum lattice model, the partition function of the system is transformed to a two-dimensional tensor network by means of the Trotter-Suzuki decomposition. Therefore, the partition function of the Heisenberg ferromagnetic mixed spin (S, σ) model can be represented as

Z N T r [ e β H 1 / K e β H 2 / K ] K = s j i , σ j + 1 / 2 i i = 1 K S j 2 i 1 σ j + 1 / 2 2 i 1 | e β H 1 / K | S j 2 i σ j + 1 / 2 2 i × S j 2 i σ j + 1 / 2 2 i | e β H 2 / K | S j 2 i + 1 σ j + 1 / 2 2 i + 1 ,
(17)

where H 1 = j S j σ j + 1 2 , H 2 = j S j σ j 1 2 , and the periodic boundary condition along the Trotter and spacial direction is assumed. Defining a fourth-order tensor μ s j 2 i 1 σ j + 1 / 2 2 i 1 , s j 2 i σ j + 1 / 2 2 i = S j 2 i 1 σ j + 1 / 2 2 i 1 | e β ( S j σ j + 1 2 ) / K | S j 2 i σ j + 1 / 2 2 i and υ σ j + 1 / 2 2 i s j + 1 2 i , σ j + 1 / 2 2 i + 1 s j + 1 2 i + 1 = σ j + 1 / 2 2 i S j + 1 2 i | e β ( σ j + 1 2 S j + 1 ) / K | σ j + 1 / 2 2 i + 1 S j + 1 2 i + 1 , Eq. (17) can be further decomposed into

Z N s j i , σ j + 1 / 2 i i = 1 K j μ s j 2 i 1 σ j + 1 / 2 2 i 1 , s j 2 i σ j + 1 / 2 2 i ν σ j + 1 / 2 2 i s j + 1 2 i , σ j + 1 / 2 2 i + 1 s j + 1 2 i + 1 .
(18)

Thus, the partition function is transformed into a classical transfer matrix tensor network, as demonstrated in Fig. 3(a), and can be calculated by summing over all the intermediate states s j i , σ j + 1 / 2 i in the tensor network. In order to contract this tensor network, the matrix singular value decomposition (SVD) is applied to the fourth order tensor μ s j 2 i 1 σ j + 1 / 2 2 i 1 , s j 2 i σ j + 1 / 2 2 i , ν σ j + 1 / 2 2 i s j + 1 2 i , σ j + 1 / 2 2 i + 1 s j + 1 2 i + 1 , which transforms them to third order tensors T a μ , T b μ and T a ν , T b ν :

μ s j 2 i 1 σ j + 1 / 2 2 i 1 , s j 2 i σ j + 1 / 2 2 i = x U s j 2 i 1 s j 2 i , x μ λ x μ ( V T ) x , σ j + 1 / 2 2 i 1 σ j + 1 / 2 2 i μ = x ( T a μ ) s j 2 i 1 , s j 2 i , x ( T b μ ) x , σ j + 1 / 2 2 i 1 , σ j + 1 / 2 2 i ,
(19a)
ν σ j + 1 / 2 2 i s j + 1 2 i , σ j + 1 / 2 2 i + 1 s j + 1 2 i + 1 = y U σ j + 1 / 2 2 i σ j + 1 / 2 2 i + 1 , y ν λ y ν ( V T ) y , s j + 1 2 i s j + 1 2 i + 1 ν = y ( T a ν ) σ j + 1 / 2 2 i , σ j + 1 / 2 2 i + 1 , y ( T b ν ) y , s j + 1 2 i , s j + 1 2 i + 1 ,
(19b)

where ( T a μ ) s j 2 i 1 , s j 2 i , x = U s j 2 i 1 s j 2 i , x μ λ x μ , ( T b μ ) x , σ j + 1 / 2 2 i 1 , σ j + 1 / 2 2 i = λ x μ ( V T ) x , σ j + 1 / 2 2 i 1 σ j + 1 / 2 2 i μ , and the same is applied to ( T a ν ) σ j + 1 / 2 2 i , σ j + 1 / 2 2 i + 1 , y , ( T b ν ) y , s j + 1 2 i , s j + 1 2 i + 1 . This procedure is shown in Fig. 3(b). After this, Fig. 3(c) shows that the square tensor network becomes a hexagonal one with four third-order tensors T a μ , T b μ and T a ν , T b ν , and every two rows of the third-order tensor form a transfer matrix, as represented in the dash rectangle of Fig. 3(c). There are K layers of this transfer matrix in total. Then, the s and σ bonds encircled by the dash oval line are contracted, leading to the two fourth-order tensors

( M a ) udlr = p ( V T ) l p , u ν U d p , r μ ,
(20a)
( M b ) udlr = q ( V T ) l , d q μ U q u , r ν ,
(20b)

which form a matrix product operator (MPO) lying in the bottom line of the whole tensor network. Each bond between (Ma)udlr and (Mb)udlr is assigned with a diagonal matrix λμ,ν, as depicted in Fig. 3(d). Next, one needs to project K − 1 times the transfer matrix on the MPO. During the projection, the horizontal bond dimension of the tensor (Ma)udlr and (Mb)udlr (or the dimension of the diagonal matrix λμ and λν) will increase exponentially with the projection time. So, we use the infinite time evolution block decimation algorithm26,27 to cut the dimension to an acceptable value. After the projection, we obtain the final MPO made of the fourth-order (Ma)′ and (Mb)′ tensors, each of which have two physical indices u and d, as illustrated in Fig. 3(e). By tracing out the physical index due to the periodic boundary condition and absorbing the λμ,ν matrix to the corresponding bond, we arrive at M a 1 = u , d δ u , d ( M a ) udlr ( λ ν ) l ( λ μ ) r and M b 1 = u , d δ u , d ( M b ) udlr ( λ μ ) l ( λ ν ) r , as indicated in Fig. 3(f). It gives a matrix product in the spatial direction. As we have totally 2N states, the partition function is obtained by shrinking the neighboring matrix N times and tracing out the final matrix. The lattice size increases exponentially with shrinking time. In our calculations, we have no size effect or boundary effect, as this scheme is implemented in the thermodynamic limit. The free energy per site f at inverse temperature β = is obtained by f = 1 2 N β ln Z N . During the LTRG calculations, we always keep the Trotter step τ = 0.1, the cut dimension Dc = 800 and the coupling constant J = 1.

FIG. 3.

(a) A transfer-matrix tensor network, where each bond denotes the s or σ index in Eq. (17) and Eq. (18). (b) A local transformation of a fourth-order tensor into two third-order tensors through a matrix singular value decomposition, i.e., Eq. (19a) and Eq. (19b). (c) Transform the tensor network to a hexagonal network, and the tensor network in dash rectangle can be seen as a transfer matrix. (d) By contracting the dash oval bond in (c), one can obtain a matrix product operator (MPO) lying in the bottom of the tensor network. There are K − 1 layers of transfer matrix on top of this MPO. (e) After projecting the transfer matrix K − 1 times, the final MPO is obtained. (f) A matrix product along the spatial direction. At every contraction, by multiplying the neighboring matrix to a new one, one can reduce the number of the matrix by half.

FIG. 3.

(a) A transfer-matrix tensor network, where each bond denotes the s or σ index in Eq. (17) and Eq. (18). (b) A local transformation of a fourth-order tensor into two third-order tensors through a matrix singular value decomposition, i.e., Eq. (19a) and Eq. (19b). (c) Transform the tensor network to a hexagonal network, and the tensor network in dash rectangle can be seen as a transfer matrix. (d) By contracting the dash oval bond in (c), one can obtain a matrix product operator (MPO) lying in the bottom of the tensor network. There are K − 1 layers of transfer matrix on top of this MPO. (e) After projecting the transfer matrix K − 1 times, the final MPO is obtained. (f) A matrix product along the spatial direction. At every contraction, by multiplying the neighboring matrix to a new one, one can reduce the number of the matrix by half.

Close modal

The thermodynamic properties of the system can be first analyzed qualitatively in terms of the self-consistent equations in the SBMFT. For sufficient low temperature ( T O ( J ) ) , the integral in Eqs. (12) is dominated by the contributions of low energies. We thus mainly consider the branch ωk,P near the center zone of k [around (k = 0)]. Eqs. (12) become

π π d k 2 π 1 e β ω k , P 1 = S + σ ,
(21)
π π d k 2 π 1 e β ω k , P 1 1 F ( k ) = S σ ,
(22)
π π d k 2 π 1 e β ω k , P 1 F ( k ) 1 F ( k ) = 2 z Q 2 J Λ ,
(23)

where F ( k ) = 1 + 4 ( z Q γ k ) 2 ( λ S λ σ ) 2 . The energy gap is Λ+ at k = 0, which implies at T = 0 that the system condenses. ωk,P can be approximated by expanding to the second-order in k

ω k , P Λ + c κ + c 8 κ k 2 .
(24)

At low temperature, the dispersion relation ωk,P leads to the nontrivial solutions of the mean-field equations (Appendix A 1)

Λ J σ S , Λ + J σ + S , Q 2 J 2 S σ .
(25)

Meanwhile, the dispersion ωk,P at T → 0 in the long wavelength limit is of the form

ω k , P 1 S σ ( S + σ ) T 2 2 J + S σ 2 ( S + σ ) J k 2 .
(26)

This suggests that the mixed spin system possesses gapless excitation at T → 0 and the quadratic form of the low-lying spectrum near k = 0. The energy gap between the upper branch ωk,Q and the lower branch ωk,P can be estimated to be Δ = − 2ΛJ.

The spin-spin correlation function can be calculated through the following equations:

S ˆ i S ˆ j = 3 2 f 1 ( R i , j ) 2 ,
f 1 ( R i , j ) = 1 N k e i k ( R j R i ) u k 2 n k P + v k 2 n k Q ,
σ ˆ i + 1 2 σ ˆ j + 1 2 = 3 2 f 2 ( R i , j ) 2 ,
f 2 ( R i , j ) = 1 N k e i k ( R j R i ) u k 2 n k Q + v k 2 n k P ,
S i ˆ σ ˆ j + 1 2 = 3 2 f 3 ( R i , j ) 2 ,
f 3 ( R i , j ) = 1 N k e i k ( R j R i ) u k v k n k Q n k P .

At low temperatures and in the long wavelength limit (k → 0), we get the asymptotic expression of the correlation length (Appendix A 2)

ξ 2 S σ 2 J T .
(27)

The divergence of the correlation length when lowering the temperature down to zero indicates that the T = 0 case is critical. At finite temperature, because of the absence of the Bose condensation, the ferromagnetic long-range order is destroyed, which is in agreement with the Mermin-Wagner theorem.30 

In light of the linear response theory, the physical responses △Sz(k, ω) and △σz(k, ω) of magnetization Sz and σz are given by

S z ( k , ω ) σ z ( k , ω ) = g S S R ( k , ω ) , g S σ R ( k , ω ) g σ S R ( k , ω ) , g σ σ R ( k , ω ) h S ( k , ω ) h σ ( k , ω ) ,
(28)

where hS(k, ω) and hσ(k, ω) are external magnetic fields applied to the spin S and σ, respectively. gR(k, ω)’s are retarded Green’s functions, which can be solved by means of the standard equation of motion, say, ω〈〈A|B〉〉ω = 〈[A, B]〉 + 〈〈[A, H]|B〉〉ω with gAB = 〈〈A|B〉〉ω the double-time Green’s function:

g S S R ( k , ω ) = S ˆ k z S ˆ k z R ( ω ) = q 1 2 u q 2 u k + q 2 n q P n k + q P ω ( ω k + q , P ω q , P ) + i 0 + + 1 2 u k + q 2 v q 2 n q Q n k + q P ω ( ω k + q , P ω q , Q ) + i 0 + + 1 2 u q 2 v k + q 2 n q P n k + q Q ω ( ω k + q , Q ω q , P ) + i 0 + + 1 2 v q 2 v k + q 2 n q Q n k + q Q ω ( ω k + q , Q ω q , Q ) + i 0 + ,
(29a)
g S σ R ( k , ω ) = S ˆ k z σ ˆ k z R ( ω ) = q 1 2 u q v q u k + q v k + q n q P n k + q P ω ( ω k + q , P ω q , P ) + i 0 + + 1 2 u q v q u k + q v k + q n k + q P n q Q ω ( ω k + q , P ω q , Q ) + i 0 + + 1 2 u q v q u k + q v k + q n k + q Q n q P ω ( ω k + q , Q ω q , P ) + i 0 + + 1 2 u q v q u k + q v k + q n q Q n k + q Q ω ( ω k + q , Q ω q , Q ) + i 0 + .
(29b)

g σ σ R ( k , ω ) and g σ S R ( k , ω ) can be obtained by exchanging the superscript PQ in Eqs. (29a) and (29b), respectively. When hS(k, ω) = hσ(k, ω) = h(k, ω), the uniform susceptibility is given by

χ = lim k 0 lim ω 0 S z ( k , ω ) + σ z ( k , ω ) 2 h ( k , ω )
= β 4 N k n k P ( n k P + 1 ) + n k Q ( n k Q + 1 ) .
(30)

Again, in the limit of Sσ, λSλσλ, and T → 0, we only need to consider ωk,P branch, and the magnetic susceptibility in this case reduces to that of the uniform ferromagnetic spin chain, χ u n i = β 4 N k [ n k P ( n k P + 1 ) ] .

At low temperature, the magnetic susceptibility behaviors (Appendix A 3)

χ S σ ( S + σ ) 2 4 J T 2 + S + σ 4 T .
(31)

The magnetic susceptibility diverges at T = 0. For S = 1 and σ = 1/2, Eq. (31) gives χ 0 . 28 T J 2 + 0 . 375 1 T . We also utilize the LTRG method to validate the SBMFT results for this case. Both LTRG and SBMFT results are shown in Fig. 4. We use a different form χ = lim h 0 m ( h ) m ( 0 ) h to calculate the susceptibility. It is found that the magnetization per unit cell is lim h 0 , T 0 m ( h ) = 1 2 ( 1 + 0 . 5 ) , revealing that our LTRG result about the susceptibility is reliable. The fitting equation χ ( T ) = a 1 T 2 + a 2 T indicated by Eq. (31) is employed to fit the LTRG results with the fitting parameters a1 ≈ 0.143, a2 ≈ 1.347. At high temperature, the Curie’s law says χ O 1 T and the O 1 T 2 term is negligible. Compared with the LTRG calculations, one may see that the SBMFT gives a qualitatively correct result at low temperature.

FIG. 4.

Temperature dependence of the magnetic susceptibility of the Heisenberg ferromagnetic mixed spin (1, 1/2) chain calculated by SBMFT and LTRG methods.

FIG. 4.

Temperature dependence of the magnetic susceptibility of the Heisenberg ferromagnetic mixed spin (1, 1/2) chain calculated by SBMFT and LTRG methods.

Close modal

Next, we would like to discuss the low-temperature behavior of the specific heat. The free energy per spin can be obtained by the SBMFT as (Appendix A 4)

1 T F MF 2 N E 0 = ζ ( 3 2 ) 2 π 2 ( S + σ ) S σ T J 1 2 1 S σ T J + ζ ( 1 2 ) 4 S σ 2 π S σ ( S + σ ) T J 3 2 ,
(32)

where ζ(α) is the Riemann’s zeta function and E0 = − 2SσJ is the ground state energy per spin. From Eq. (32), we obtain the low temperature specific heat C ( T ) 1 . 35 ( T / J ) 1 2 + 4 ( T / J ) + 3 . 78 ( T / J ) 3 2 for S = 1 and σ = 1/2. The temperature dependence of the specific heat is also calculated numerically by the LTRG method down to very low temperature, as shown in Fig. 5. It is seen that the specific heat C(T) goes to zero when temperature is lowered to zero or raised to infinite temperature.

FIG. 5.

Temperature dependence of the specific heat of the Heisenberg ferromagnetic mixed spin (1, 1/2) chain. The inset shows both LTRG and SBMFT results at low temperature. The red dash line is the fitting curve given by Eq. (33).

FIG. 5.

Temperature dependence of the specific heat of the Heisenberg ferromagnetic mixed spin (1, 1/2) chain. The inset shows both LTRG and SBMFT results at low temperature. The red dash line is the fitting curve given by Eq. (33).

Close modal

At low temperature, the LTRG calculation (Inset of Fig. 5) gives

C ( T ) = 0 . 669 T J 1 2 0 . 896 T J + 0 . 715 T J 3 2 ,
(33)

which shows, again, that the SBMFT result is qualitatively correct at low temperature.

The compound MnIINiII(NO2)4(en)2 exhibits ferromagnetic interactions between the Mn (S = 5/2) and Ni (σ = 1) ions within the chains and weaker antiferromagnetic interactions between neighbouring chains.20 Experimental measurements on susceptibility disclose the existence of easy axis anisotropy due to the low local symmetry being only around Mn ions, that is, the single-ion anisotropic energy D ( S i z ) 2 should not be negligible. So, this compound can be described by the following Hamiltonian:

H = J j N S j σ j 1 2 + S j σ j + 1 2 + D j N ( S j z ) 2 h z j N μ B g S S j z + g σ σ j + 1 2 z h x j N μ B g S S j x + g σ σ j + 1 2 x .
(34)

Here, hz and hx are external longitudinal and transverse magnetic fields, gS = 2 and gσ = 2.24 are the g-factors of the Ni and Mn ions, respectively and μB is the Bohr magneton.

In Ref. 20, the coupling constant J is determined by a semi-classical method where Mn ions were treated as classical spins and the anisotropy term is absent. The parameter D was estimated by the polycrystalline data. In order to determine more reliably the parameters J and D of this compound, we calculate the magnetic susceptibility and magnetization curve by LTRG and fit our results with experimental data. The susceptibility measurements on MnIINiII(NO2)4(en)2 shows a peak at TN ∼ 2.45K and then drop rapidly with temperature going down, where the sharp peak is associated with the transition to three-dimensional long-range antiferromagnetic ordering because of the existence of the interchain antiferromagnetic coupling. The magnetic curve shows a phase transition at about 2.5K, and the antiferromagnetic transition is shifted continuously to lower temperatures with increasing of the applied magnetic field. The specific heat measurement also marks the antiferromagnetic transition by the λ-type peak at TN ∼ 2.4K. Because the existence of the phase transition, only the temperature range T ≥ 4.5K is used to fit the parameters J and D. Fig. 6 gives the temperature dependence of susceptibilities under both longitudinal and transverse magnetic field hz = hx = 200Oe. The LTRG results are nicely fitted to the experimental data, giving rise to a set of material parameters: J ∼ 1.8K, D ∼ − 0.36K. It is the anisotropic term that makes the longitudinal susceptibility larger than that of the transverse one. The experimental data of the magnetization curves are also fitted with the LTRG results with temperature from 1.8K to 4.5K for different fields in Fig. 7. Here, the same set of parameters J ∼ 1.8K, D ∼ − 0.36K are used. It is noted that, because of the existence of antiferromagnetic interchain coupling, the magnetic curve drops at about 2.5K and the LTRG results are much larger than the experimental data. At about T ≥ 4K, the LTRG results approach the experimental measurement. In Ref. 21, the high-temperature series expansion method gives the parameter J ∼ 2.4K, D ∼ − 0.36K which are slightly different from ours. The difference may arise from the finite-size effects, because in Ref. 21, the finite-size effects are more pronounced in the ferromagnetic case. Our LTRG method gives directly the thermodynamics limit results that should be more reasonable.

FIG. 6.

Temperature dependence of the magnetic susceptibility of MnIINiII(NO2)4(en)2 measured experimentally is well fitted to the LTRG results for both longitudinal and transverse magnetic fields. The critical temperature is TN ∼ 2.45K. The experimental data are taken from Ref. 20.

FIG. 6.

Temperature dependence of the magnetic susceptibility of MnIINiII(NO2)4(en)2 measured experimentally is well fitted to the LTRG results for both longitudinal and transverse magnetic fields. The critical temperature is TN ∼ 2.45K. The experimental data are taken from Ref. 20.

Close modal
FIG. 7.

Temperature dependence of the magnetization curve of MnIINiII(NO2)4(en)2. Because of the existence of weak interchain antiferromagnetic coupling, the LTRG results are larger than experiment data with temperature going down. The experimental data are taken from Ref. 20.

FIG. 7.

Temperature dependence of the magnetization curve of MnIINiII(NO2)4(en)2. Because of the existence of weak interchain antiferromagnetic coupling, the LTRG results are larger than experiment data with temperature going down. The experimental data are taken from Ref. 20.

Close modal

We have applied the SBMFT to investigate the ground states, excitation spectra and thermodynamic properties of the Heisenberg ferromagnetic mixed spin (S, σ) chains. The numerical LTRG method is also used to validate the result of the mean-field theory. We have found that compared with the uniform ferromagnetic spin chains, there are two degenerate excitation branches for the mixed spin chain, the gapless excitation ωk,P and the gapped excitation ωk,Q. Applying a small stagger magnetic field, the double degeneracy is lifted. At T = 0, the Schwinger bosons condense in the ωk=0,P+ state indicating a ferromagnetic long-range order of the ground state with m tot z = N ( S + σ ) . If the spin density operator m ˆ k acting on the ground state, a boson is annihilated at ωk=0,P+ and is excited to the state that is a combination of ωk,P and ωk,Q, which is characterized by m tot z = N ( S + σ ) 1 . The boson can be excited to ωk,P+ or ωk,Q+ which are the eigenstates characterized by m ˆ tot z = N ( S + σ ) . At low temperature, we obtained the spectrum ω k , P 1 S σ ( S + σ ) T 2 2 J + S σ 2 ( S + σ ) J k 2 . The gap between the two branches is estimated to be △ ∼ J by the SBMFT.

The low temperature thermodynamic properties are explored by the SBMFT and also examined by the numerical LTRG method for S = 1 and σ = 1/2. The SBMFT results are observed to be qualitatively in agreement with those calculated by the LTRG method. The correlation length diverges as J T . By fitting the LTRG numerical results, the susceptibility and specific heat of the Heisenberg ferromagnetic mixed spin chain at low temperature behavior as the form of χ = 0 . 143 1 T 2 + 1 . 347 1 T , and C = 0 . 669 T J 1 2 0 . 896 T J + 0 . 715 T J 3 2 . The experimental data of a ferromagnetic mixed spin chain compound MnIINiII(NO2)4(en)2 are compared with the LTRG results, giving rise to the coupling parameters of the material: J ∼ 1.8K, D ∼ − 0.36K, which are more reasonable than previous results.

We are indebted to Wei Li, Shi-Ju Ran, Tao Liu, Yue Wu and Qing-Rong Zheng for helpful discussions. This work is supported in part by the MOST of China (Grant No. 2012CB932900 and No. 2013CB933401), the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB07010100), and the NSFC (Grant No. 11474279).

In one dimension at low temperature T O ( J ) , the region of small momentum is dominate, and we can use the approximations made for the uniform ferromagnetic chain.31 So the upper limit of the integral in self-consistent equations can be sent to +∞ for convenience. Under these simple but reasonable approximations, we can get the solution of the self-consistent equations and the low temperature properties of thermodynamic quantities.

1. Solution of the self-consistent equations

By using the identity 1 + 4 ( z Q γ k ) 2 ( λ S λ σ ) 2 = ω k , P Λ + Λ and the simple approximations at low temperature, we find that the integrals of Eqs. (21)-(23) can be expressed by

π π d k 2 π 1 e β ω k , P 1 0 + d k π T Λ + c κ + c 8 κ k 2 + O ( ( T / J ) 2 ) 2 κ T 2 c κ Λ + 2 c 2 κ 2 + O ( ( T / J ) 2 ) S + σ .
(A1)
π π d k 2 π 1 e β ω k , P 1 1 1 + 4 ( z Q γ k ) 2 ( λ S λ σ ) 2 π π d k 2 π T ω k , P Λ ω k , P Λ + + O ( ( T / J ) 2 )
2 T Λ π c Λ + ln 2 2 κ π 2 2 κ + π 2 Λ Λ + κ T κ c Λ + c 2 κ 2
+ O ( ( T / J ) 2 ) S σ .
(A2)
Λ + Λ ( S + σ ) + ( S σ ) = 2 z Q 2 J Λ + 1 Λ π π d k 2 π ω k , P e β ω k , P 1 2 z Q 2 J Λ + 1 Λ π π d k 2 π T ω k , P ω k , P + O ( ( T / J ) 2 ) 2 z Q 2 J Λ + O ( T / Λ ) .
(A3)

From Eq. (A1) and Eq. (A2), we get

2 T Λ π c Λ + ln 2 2 κ π 2 2 κ + π Λ Λ + ( S + σ ) + O ( ( T / J ) 2 ) S σ .
(A4)

At low temperature, Eq. (A4) gives

Λ + Λ S + σ σ S .
(A5)

Combining Eq. (A5) with Eq. (A3), we have

2 z Q 2 J Λ 4 S σ σ S .
(A6)

Besides, Eq. (A1) gives

Λ + c κ T 2 ( S + σ ) 2 2 κ c .
(A7)

Substituting Eqs. (A5) and (A6) into Eq. (A7), we get

Λ J σ S , Λ + J σ + S , Q 2 J 2 S σ .
(A8)

2. Correlation function and correlation length

Here, we take S i S j = 3 2 f 1 ( R i , j ) 2 as an example to calculate the asymptotic behavior of the correlation function at low temperature. f1(Ri,j) can be written as

f 1 ( R i , j ) = π π d k 2 π e i k ( R j R i ) ( u k 2 n k , P + v k 2 n k , Q ) π π d k 2 π e i k ( R j R i ) 1 2 1 e β ω k , P 1 1 2 Λ Λ + ω k , P 1 e β ω k , P 1 π π d k 2 π e i k ( R j R i ) 1 2 T Λ + c κ + c 4 κ k 2 + Λ c 4 κ c κ T Λ + c κ + c 4 κ k 2 d k 2 π e i k ( R j R i ) 1 2 T Λ + c κ + c 4 κ k 2 + + d k 2 π Λ c 4 κ c κ T Λ + c κ + c 4 κ k 2 .
(A9)

The residues are used to calculate the two parts in Eq. (A9) and the integral loop is along the real axis and on the upper half plane. All residues are single-poles. For the first term, the residues are on the imaginary axis, and for the second term, the residues are on both real and imaginary axises. The integral gives

f 1 ( R i , j ) κ T 2 c κ Λ + c 2 κ 2 exp 2 κ Λ + c κ 2 R j R i .
(A10)

The correlation function and correlation length bear the form of, respectively

S i S j 3 8 T 2 c Λ + κ c 2 exp 4 κ Λ + c κ 2 R j R i ,
(A11)
ξ 1 4 κ Λ + c κ 2 2 S σ 2 J T .
(A12)

3. Susceptibility

At low temperature, the susceptibility is also dominated by the small-momentum part of the n k P in the integral Eq. (30). Thus we have

χ 1 4 T π π d k 2 π n k P ( n k P + 1 ) 1 4 T π π d k 2 π n k P + 1 4 T π π d k 2 π n k P n k P S + σ 4 T + T 4 8 κ c 2 0 d k π 1 k 2 + 8 κ ( Λ + c κ ) / c 2 S + σ 4 T + T 4 8 κ c 2 1 4 8 κ ( Λ + c κ ) / c 3 .
(A13)

From Eq. (A1), we have Λ + c κ T 2 2 J 1 S σ ( S + σ ) and κc ≈ (S + σ) J. The susceptibility behaviors at low temperature as

χ S + σ 4 T + S σ ( S + σ ) 2 4 J T 2 .
(A14)

4. Low-temperature behavior of free energy

At low temperature, only small momentum and energy of branch ωk,P dominate the free energy. Then, we have

F MF 2 N 1 N β k ln ( 1 e β ω k , P ) + z J | Q | 2 S λ S σ λ σ .
(A15)

Introducing new parameters f, v, t−1:

f = 1 N β k ln ( 1 e β ω k , P ) , v = Λ + c κ T , t 1 = c 8 κ T ,
(A16)

we get

F MF 2 N z J | Q | 2 S λ S σ λ σ = f .
(A17)

f can be obtained by

f T π π d k 2 π ln 1 e β ω k , P
= 0 π d k π ln 1 e β ω k , P
0 d k π ln 1 exp Λ + c κ T c 8 κ T k 2
= 0 d k π ln 1 exp v t 1 k 2
= 0 d x 2 π t 1 2 x 1 2 ln 1 exp v x ,
(A18)

where x = t−1k2. Considering the formula10 

0 u α 1 ln 1 e u v d u = Γ ( α ) F ( α + 1 , v ) ,
(A19)

where Γ(α) is the gamma function and F(α, v) is the Bose-Einstein integral function. At low temperature v = Λ + c κ T T 2 J 1 S σ ( S + σ ) 0 and t = 8 κ c T 2 ( S + σ ) S σ T J 0 , we have

f T = t 1 2 2 π Γ 1 2 F 3 2 , v t 1 2 2 π Γ 1 2 Γ 1 2 v 1 2 ζ 3 2 + v ζ 1 2 = t 1 2 2 π π ζ 3 2 t 1 2 v 1 2 + t 1 2 2 π π ζ 1 2 v = ζ ( 3 2 ) 2 π 2 ( S + σ ) S σ T J 1 2 1 S σ T J + ζ ( 1 2 ) 4 S σ 2 π S σ ( S + σ ) T J 3 2 .
(A20)

From Eq. (A8), the constant term in Eq. (A15) gives

E 0 z J | Q | 2 S λ S σ λ σ = 2 J Q 2 S ( Λ + + Λ ) σ ( Λ + Λ ) 2 S σ J .
(A21)
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