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- Ground state properties of antiferromagnetic Heisenberg spin rings
- Ground state properties of one-dimensional Bose-Fermi mixtures
- Dynamic Spin Correlation Function of the Alternating Spin-12 Antiferromagnetic Chain with C
- Ground State Entropy of the Potts Antiferromagnet with Next-Nearest-Neighbor Spin-Spin Coup
- Spin lattices with two-body Hamiltonians for which the ground state encodes a cluster state
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- Ground state properties of antiferromagnetic Heisenberg spin rings
- Thermal conductivity of quasi-one-dimensional antiferromagnetic spin-chain materials
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- Ground State Properties of One Dimensional S=12 Heisenberg Model with Dimerization and Quad

Ground State Properties of one-dimensional Antiferromagnetic Spin-1 Chain with Single-ion Anisotropy

Ekrem Ayd?ner? and Cenk Aky¨ uz

Department of Physics, Dokuz Eyl¨ ul University Faculty of Arts and Sciences Tr-35160 Izmir, Turkey (Dated: February 2, 2008) In this study, we have investigated ground state properties of one-dimensional antiferromagnetic spin-1 chain with single-ion anisotropy at very low temperatures using the Transfer Matrix method. Magnetic plateaus, phase diagram, speci?c heat, susceptibility of the spin chain have been evaluated numerically from the free energy. Results are in good agreement with the experimental data for the spin-1 compounds [Ni2 (Medpt)2 (?-ox)(H2 O)2 ](ClO4 )2 2H2 O, [Ni2 (Medpt)2 (?-ox)(?N3 )](ClO4 )0.5H2 O, Ni(C2 H8 N2 )Ni(CN)4 and Ni(C10 H8 N2 )2 Ni(CN)4 .H2 O. However, spin-Peierls transition have not been observed in the temperature dependence of speci?c heat and magnetic susceptibility.

PACS numbers: 75.10.Hk; 75.10.Pq

arXiv:cond-mat/0501111v2 [cond-mat.stat-mech] 11 Jan 2005

The physics of low-dimensional i.e., one-dimensional (1D) or quasi-one-dimensional (Q1D) spin-S (S≥ 1) chains has attracted a considerable amount of attention after the prediction by Haldane1 that a 1D Heisenberg antiferromagnet should have an energy gap between the singlet ground state and the ?rst excited triplet states in the case of an integer spin quantum number, while the energy levels are gapless in the case of a half integer spin quantum number. The most fascinating characteristic of these low-dimensional systems is that they show magnetic plateaus i.e., quantization of magnetization at low temperatures near the ground state. This phenomenon has been observed not only in Haldane spin systems but also in other spin gapped systems for instance; spinPeierls and spin ladders systems. A general condition of quantization of the magnetization was derived from the Lieb-Schultz-Mattis theorem2 for low-dimensional magnetic systems. The fact that, Oshikawa, Yamanaka and A?eck (OYA)3 discussed this plateau problem and derived a condition p(S ? m) = integer, necessary for the appearance of the plateau in the magnetization curve of one-dimensional spin system, where S is the magnitude of spin, m is the magnetization per site and p is the spatial period of the ground state, respectively. The 2S + 1 magnetization plateaus (contained the saturated magnetization m = S ) can appear when the magnetic ?eld increases from zero to its saturation value hs . Theoretical studies have suggested the realization of quantization of magnetization in various systems,3,4,5,6,7,8,9,10,11,12,13,14 and it has been observed in experimental studies.15,16,17,18,19 Recent years, antiferromagnetic (AF) spin-1 systems among the other low-dimensional gapped spin-S systems have drawn attention from both theorists and experimentalists in literature. So far, many Q1D gapped AF spin-1 systems, which are called Haldane systems, spin-Peierls and ladder compounds, were synthesized and to understand the ground state behaviors of the spin-1 AF Heisenberg chain have been extensively studied. The ?rst Q1D spin-1 Haldane gap compound was Ni(C2 H8 N2 )2 NO2 (ClO4 ) (abbreviated NENP)

which was synthesized by Meyer et al.20 in 1981, just before the Haldane’s prediction appeared. Renard et al.21 showed that NENP had the energy spectrum predicted by Haldane; the magnetic susceptibility decreased steeply at low temperatures for all of the crystal axes suggesting a single ground state and neutron inelastic scattering showed that the magnetic excitation had a gap at the AF zone center. Other Q1D spin-1 compounds Ni(C2 H8 N2 )2 Ni(CN)4 (abbreviated NENC),22,23,24,25 Ni(C11 H10 N2 O)2 Ni(CN)4 (abbreviated NDPK),22,23 Ni(C10 H8 N2 )2 Ni(CN)4 .H2 O (abbreviated NBYC),22,23,24 Ni(C5 H14 N2 )2 N3 (PF6 ) (abbreviated NDMAP),26,27,28 and Ni(C5 H14 N2 )2 N3 (ClO4 ) (abbreviated NDMAZ)29 have also been identi?ed as e?ective Heisenberg chains. This class of compounds exhibits a non-degenerate ground state which can be separated from the lowest excitation. As predicted by Haldane, it is experimentally shown that these systems have an energy gap between the single ground state and ?rst excited triplet.21,22,23,24,25,26,27,28,29 In addition, exception Haldane systems, several Q1D AF spin-1 gapped compounds such as AF dimer compound [Ni2 (Medpt)2 (?-ox)(H2 O)2 ](ClO4 )2 2H2 O,15 and AF alternating chain compound [Ni2 (Medpt)2 (?-ox)(?N3 )](ClO4 )0.5H2 O,16 where (Medpt=methyl-bis(3aminopropyl)amine), the ladder system 3,3’,5,5’-tetrakis (N-tert-butylaminxyl)biphenyl (BIP-TENO),17,30,31 the spin-Peierls systems32,33,34,35 (Li,Na)V(Si,Ge)2 O6 have been synthesized and magnetic properties have been studied. The magnetic quantization behavior has been experimentally observed in only several gapped spin-1 AF materials as predicted by OYA.3 For example, Narumi et al.15,16 observed a magnetic plateaus in the magnetization curve for both [Ni2 (Medpt)2 (?ox)(H2 O)2 ](ClO4 )2 2H2 O and [Ni2 (Medpt)2 (?-ox)(?N3 )](ClO4 )0.5H2 O. Goto et al.17 reported the existence of the magnetization a plateau at 0.25 in spin-1 3,3’,5,5’tetrakis (N-tert-butylaminxyl)biphenyl (BIP-TENO). On the other hand, in theoretical studies, Chen et

2 al.4 employed the classical Monte Carlo method to investigate the magnetization plateaus of 1D classic spin-1 AF Ising chain with a single-ion anisotropy under the external ?eld at low temperatures, and they showed that the system has 2S + 1 magnetic plateaus. Tonegawa et al.9 observed the plateau of at m =0.0, 0.5, and 1.0 in the ground state of spin-1 AF Heisenberg chain with bond alternating and uniaxial single-ion anisotropy. Sato and Kindo obtained a magnetic plateau at m =0.0, 0.5, and 1.0 for spin-1 bond alternating Heisenberg chain using renormalization group method.14 Other thermodynamical features of low-dimensional AF spin-1 systems have been extensively studied in theoretical concepts as well as in experimental studies. For example, White and Huse36 calculated the ground state energy and Haldane gap of a spin-1 AF Heisenberg chain using the density matrix renormalization group techniques. Yamamoto and Miyashita37 studied the speci?c heat and magnetic susceptibility of a spin-1 AF chain by Monte Carlo simulation. The spin-wave theory was used by Rezende38 to study the temperature dependence of excitation spectra, and it has been found that the excitation energy or Haldane gap increases with increasing temperature in the region of low temperature. Campana et al.39 discussed the speci?c heat and magnetic susceptibility of a ?nite chain by a numerical calculation method. Bao40 employed a second-order Green function method to discuss the variations of internal energy and speci?c heat with temperature for a spin-1 Heisenberg AF chain. Li and Zhu,41 using a self-consistent mean-?eld approximation, studied the thermodynamic properties of a spin-1 AF Heisenberg chain with an anisotropy in the presence of an external magnetic ?eld at a ?nite temperature, they found that the internal energy and Haldane gap increase with increasing temperature. Batchelor et al.42 investigated the ground state and thermodynamic properties of spin-1 via the integrable su(3) model. Lou et al.43 calculated the magnetic susceptibility and speci?c heat of one-dimensional spin-1 bilinear-biquadratic Heisenberg model using transfer matrix renormalization group. On the other hand, a ?eld-theoretic approach to the Heisenberg spin-1 chain with single-ion anisotropy was suggested by Tsvelik.44 Now, it understood from experimental and theoretical studies that quantization of magnetization appears due to gap mechanism originating from dimerization, frustration, single-ion anisotropy, periodic ?eld so on. However, it notes that low-dimensional gapped spin systems have still open problems and noteworthy physics. Therefore, in this study, we interested in ground state properties of 1D AF spin-1 chain. In general, the most of the theoretical studies about low-dimensional spin-S systems have been based on Heisenberg Hamiltonian because of non-trivial quantum e?ects. However, in several theoretical studies have been shown that it is possible to used classical spin systems to obtain magnetic plateaus and other thermodynamical behaviors.4,11,13 Considering this fact, we have constructed herein the Hamiltonian of the 1D spin-1 system in a quasi-classical manner which allows to use the Ising type variable instead of the quantum spin operators. Such a Hamiltonian with single-ion anisotropy for 1D spin-1 systems is described by

N N z z Si Si+1 i=1 N z 2 (Si ) i=1

H=J

+D

+h

i=1

z Si

(1)

where J denotes the exchange coupling of antiferromagnetic type (J > 0), D describe the single-ion anisotropy, z and h is the external ?eld. Also, Si refers to spin of magnitude 1, which takes on 0, and ±1 values. We used Transfer Matrix method to obtain ground state properties of 1D AF spin-1 system which was de?ned by Eq. (1). Using this method, it is possible to get the analytical and comparable results for 1D in?nite spin system, though it is very simple approach. For the Eq. (1) transfer matrix V is given as

? exp (?βJ ? βh ? βD) exp(?βh/2 ? βD/2) exp(βJ ? βD) 1 exp(βh/2 ? βD/2) ? V = ? exp(?βh/2 ? βD/2) exp(βJ ? βD) exp(βh/2 ? βD/2) exp(?βJ + βh ? βD)

?

(2)

where β = 1/kT . Using relation (2), partition function of the 1D spin-1 AF systems can be represented in terms of transfer matrix V as Z = T r VN . Hence free energy per spin of this system is given by f (h, T ) = ?kT lim 1 ln Z . N →∞ N (4) (3)

We are interested in the three thermodynamical expressions of (i) ferromagnetic order m, (ii) the speci?c heat C , and (iii) the magnetic susceptibility χ, which respectively are m=? ?f (h, T ) ?h ? 2 f (h, t) ?T 2 (5a)

C = ?T

(5b)

3 the beginning point of the ?eld for the appearance of the plateau m = 0.5 which corresponds to critical ?eld hc values, and its ending point is signed by the square-dot line which corresponds to the saturated ?eld hs values. In addition, the distance between both the lines for the same value of D is the width of the plateau m = 0.5. For D ≥ 1.0, the value of saturated ?eld hs increases with the increase of D. The initial ?eld hc decreases with the increase of D in the interval 0.0 < D < 1.0 and increases for around D > 1.0. Almost same magnetization phase diagram for spin-1 has been carried out using Monte Carlo method with ?nite spin in Ref.4. Figure 3 (a) and (b) show the speci?c heat C plotted as a function of temperature T , for ?xed value of single-ion anisotropy ?eld D = 0.2 at various values of external ?eld h = 0.3, 0.5, 0.7, and for ?xed value of h = 0.1 at various values of D = 0.1, 0.3, 0.5, respectively. These curves show a board maximum as Schottky-like round hump about between T = 8 and 13. But, it is no means that phase transition occurs in one-dimensional system. Schottky-like round hump in the speci?c heat probably re?ects energy of the system. For several higher external ?elds and single-ion anisotropy ?elds, at ?xed value of D and h, respectively, the peaks get smaller and move towards zero temperature. It seems clear that the peaks are related to the magnetization phase diagram of the ground state of AF Ising chain with single-ion anisotropy under ?nite magnetic ?eld. Furthermore, the speci?c heat curves of the spin chain which have obtained as a function of temperature are in good agreement with the experimental data for the compounds Ni(C2 H8 N2 )Ni(CN)4 ,22,23,24,25 Ni(C10 H8 N2 )2 Ni(CN)4 H2 O,22,23,24 and the theoretical studies4,27,28 in literature, respectively. An other way is to understand the ground state behavior of the system is investigation ?eld dependence of the susceptibility. In this reason, the susceptibility were plotted in Fig. 4 as a function of the magnetic ?eld. It is seen that there are two peaks in the susceptibility for D = 0.5 at T = 0.01. First peak occur at initial ?eld value, and second peak occur saturated ?eld value for ?xed value of D. Both of the peaks indicate the critical ?elds where magnetization plateaus appear. Similar two peaks have been observed in experimental study for compound [Ni2 (Medpt)2 (?-ox)(H2 O)2 ](ClO4 )2 2H2 O.15 On the other hand, for di?erent values of the magnetic ?eld, the susceptibility was plotted as a function of the temperature T for ?xed value of D = 0.5 in Fig. 5. The susceptibility curve for h = 1.0, marked by square-dot line, has relatively a sharp peak around T = 5. However, it has a broad maximum for h = 0.5, and h = 0.1 which are denoted by circulardot and triangular-dot line, respectively. We roughly say that susceptibility curves decay exponentially with increasing temperature by the following relation χ (T ) ≈ exp (?Eg /kB T ). This characteristic behavior can be interpreted that AF spin-1 chain with the singleion anisotropy has the Haldane gap like Heisenberg

χ=?

? 2 f (h, T ) . ?h2

(5c)

A conventional way to calculate free energy Eq. (4) which is required to obtain expression in Eq. (5) is that partition function is expressed in terms of eigenvalues of transfer matrix (2) as

N N Z = T rVN = λN 1 + λ2 + λ3 .

(6)

The eigenvalues of the Transfer Matrix (2) have been carried out with the aid of Mathematica 4.0 package. However, we have not listed herein since they occupied many pages. After we tested all eigenvalues numerically, we have sorted and so-called them from the biggest to smaller as λ1 , λ2 , λ3 respectively. Using standard assumptions of Transfer Matrix method, Eq. (6) was de?ned as, Z = T rVN = λN 1 [1 + ( λ2 N λ3 ) + ( )N ]. λ1 λ1 (7)

It is clearly seen that on the RHS of Eq. (7) the second and the third terms goes to zero as N → ∞ since both λ3 λ2 | < 1 and | λ | < 1. Consequently, terms are being | λ 1 1 the free energy Eq. (4) is reduced to f (h, T ) = ?kT ln λ1 (8)

then, some of the thermodynamical quantities of the system, we interested in Eq. (5), can be calculated numerically using Eq. (8) which is depend on the biggest eigenvalue λ1 of the matrix (2). Obtained numerical results are followed: In Fig. 1(a) and (b), the magnetization m is plotted at T = 0.01 (units by J ) as a function of external ?eld h for D = 0.5, and D = 1.0, respectively. It is seen that 2S + 1 = 3 plateaus appear (i.e. m =0, 0.5, and 1) for all values of positive single-ion anisotropy D at the low temperature near the ground state under external ?eld h. No magnetization appears in the low ?eld region. By increasing the ?eld there is a plateau of which starting point is called initial critical ?eld hc and then with increasing the ?eld, a plateau of which starting point is called saturated ?eld hs occurs again. These results clearly indicated that numbers of plateaus of classical 1D spin-1 AF system obey to OYA criterion as well as Monte Carlo prediction,4 and compatible with the experimental data for compound [Ni2 (Medpt)2 (?-ox)(H2 O)2 ](ClO4 )2 2H2 O.15 It suggested that single-ion anisotropy plays a significant role for magnetic plateau.4 In order to examine the e?ect of single-ion anisotropy on the magnetization plateaus, the magnetization m was calculated at ?nite h for a series of values of D (0.1 ≤ D ≤ 1.5), and the data were plotted in Fig. 2. For D > 0.0, three plateau lines placed at m = 0.0, m = 0.5, and m = 1.0 are divided by the initial ?eld and saturated ?eld lines as shown in Fig. 2. The longitudinal coordinate of the circular-dot line is

4 AF systems. Our results clearly compatible with the experimental results for compounds [Ni2 (Medpt)2 (?ox)(H2 O)2 ](ClO4 )2 2H2 O,15 [Ni2 (Medpt)2 (?-ox)(?N3 )](ClO4 )0.5H2 O,16 Ni(C5 H14 N2 )2 N3 (PF6 ),27 and 3,3’,5,5’-tetrakis (N-tert-butylaminxyl) biphenyl,17 and theoretical results27,28 in literature. However, in the present study, the speci?c heat and magnetic susceptibility curves which obtained as a function of temperature have not a spin-Peierls transition. Because in the compounds synthesized experimentally, a spin-Peierls transition in C and χ at low temperatures are due to a small amount of magnetic impurities and derivations from stoichiometry.34 In the majority of plateau mechanism which have been proposed up to now the purely quantum phenomena play a curial role. The concepts of magnetic quasi-particles and the strong quantum ?uctuations are regarded to be ?rst important for understanding of these processes. Particularly, for a number of systems it was shown that the plateaus are caused by the presence of the spin gap in the spectrum of magnetic excitations in the external ?eld. However, we have used quasi-classical Hamiltonian (1) in this study, and magnetic plateaus, magnetic phase diagram, speci?c heat, magnetic susceptibility of 1D AF spin-1 chain have been obtained. Obtaining results clearly consistent with experimental and theoretical results which ensure that quasi-classical approach can be used to examine the ground state properties of 1D AF spin-S gapped systems as spin-1. Finally, we conclude that the single-ion anisotropy in the Ising model has a signi?cant e?ect on the magnetic properties under the external ?eld. Therefore, we studied this phenomenon in the one-dimensional spin-1 AF Ising chain with single-ion anisotropy. When the external ?eld varied from the zero to the saturated ?eld, the 2S+1 steplike plateaus occurred owing to the co-existence of AF inz z teraction of (Si Si+1 ) and positive single-ion anisotropy z 2 (Si ) . Hence, the energy levels, separated by the crystal ?eld (i.e. single-ion anisotropy) in the system of two particles, correspond to the plateaus in the ground state. We acknowledge useful discussion with Hamza Polat.

? 1

2

3

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14 15

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17

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6

1,0

D=0.5

0,8

0,6

m

0,4 0,2 0,0

0

1

h

2

C

h

3

4

S

h

(a)

1,0

D=1.0

m

0,5

0,0 0 1

h

C

2

h

3

h

S

4

(b)

FIG. 1: The magnetization m as a function of magnetic ?eld h for (a) D = 0.5; and (b) D = 1.0. Where hc and hs are the initial and the saturated ?eld, respectively. (T = 0.01, unit by J )

7

3.6

m=1

3.2

2.8

2.4

m=1/2

2.0

h

1.6 1.2 0.8

m=0

0.4

0.2

0.4

0.6

0.8

1.0

1.2

1.4

D

FIG. 2: Magnetization phase diagram of the ground state of antiferromagnetic Ising chain with single-ion anisotropy under ?nite magnetic ?eld. The circle dot-line and the square-dot line represent the initial ?eld and the saturated ?eld, respectively.

8

4,5

4,0

h=0.7 h=0.5

3,5

h=0.3

3,0

2,5

C

2,0

1,5

1,0

0,5

0,0 0 5 10 15 20 25 30

T

(a)

5

D=0.5

4

D=0.3 D=0.1

3

C

2 1 0 0 5 10 15 20 25 30

T

(b)

FIG. 3: The speci?c heat C as a function of the temperature T (unit by J ): (a) h = 0.3, 0.5, 0.7 for ?xed value of D = 0.2 (b) D = 0.1, 0.3, 0.5 for ?xed value of h = 0.1, respectively.

9

8

7

D=0.5

6

5

4

c

3 2 1 0 1,0 1,5 2,0 2,5 3,0

h

FIG. 4: Susceptibility as a function of magnetic ?eld h for D = 0.5 and T = 0.01 (unit by J ).

0,35

h=1.0

0,30

h=0.5 h=0.1

0,25

0,20

c

0,15 0,10 0,05 0,00 0 10 20 30 40 50 60

T

FIG. 5: Susceptibility as a function of temperature for various h and D = 0.5 (unit by J ).