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From mesoscopic magnet to quantum wire. The Bethe Ansatz application
Materials Science and Engineering C 25 (2005) 809 – 812 www.elsevier.com/locate/msec
From mesoscopic magnet to quantum wire. The Bethe Ansatz application A. Wal Institute of Physics, University of Rzeszow, Rejtana 16 a, 35 – 310 Rzeszow, Poland Available online 1 August 2005
Abstract Molecular nanomagnets are representatives of quantum systems in which significant magnetic size effects are observed. In most cases they are formed of a few or a dozen or so magnetic ions arranged into a regular ring (e.g. Fe8). Such small chains can be treated as quantum wires. In literature magnetic properties of these systems were studied experimentally and these results confirm fully their extraordinary quantum features. By far results were interpreted basing on solutions obtained numerically. In present paper we give the exact solutions of the system, obtained basing on the Bethe theory. The application of this theory for such small systems is quite complicated. We develop mathematical method for inspection of the solutions evolution starting from macroscopic via mesoscopic till nanoscopic size of the Heisenberg chain. Furthermore, we study the properties of the energy band structures (as noncrossing, rarefied effect, etc.) and their changes as a function of magnetic quantum dots size. D 2005 Elsevier B.V. All rights reserved. Keywords: Bethe Ansatz; Heisenberg chain; Energy spectrum; Nanoscopic system
1. Introduction The one dimensional Heisenberg magnet with the spin s = 1 / 2 has been studied more then 70 years. The analytical solution of such a system in the thermodynamical limit and with the periodic boundary conditions was given by Bethe [1] and Hulthen [2]. The method is known as the Bethe Ansatz. It allows to solve exactly the quantum system. The anti-ferromagnetic finite Heisenberg model as well as the ferromagnetic one were studied also numerically [3– 7]. The knowledge of physical properties for finite system became more important recently due to the advances in inorganic and organic chemistry which allow to synthesise well formed nanomagnets, e.g. perfect rings with metal ions [8,9]. These systems are good realisation of an antiferromagnetic Heisenberg ring with weak magnetic anisotropy. Among others properties the energy spectrum plays a crucial role for a possible application in spintronic. For a finite number N (range of dozen) it was studied by K. Fabricius et al. [6]. The differences between even and odd number of nodes were outlined by them. We restrict our investigation to the even case. The whole energy spectrum can be found by a routine numerical procedure or by solving Bethe equations [1,2].
The second method provides a complete knowledge of eigenstates and eigenvalues of the system. The method has its own disadvantages, e.g. a hard way to solve transcendental equations leading to eigenfunctions. However there is an effective method [10], which with the help of classification of the quantum states for Heisenberg ring, given the first time by Bethe [1] and then by Orbach [15], allows to find solutions, for the systems with different number of nodes N. The key is to start from asymptotic solution (N Y V) and then follow the solutions through the chain with thousands, hundreds and few nodes. In the Section 2 we discuss the solutions for mesoscopic rings (N is order of hundreds) whereas in the Sections 3 the solutions for smaller chains are presented. In the Section 4 the pecularities of the energy spectrum are addressed.
2. Infinity and mesoscopic magnetic chain The dynamic of the system is provided by the Heisenberg Hamiltonian N
Hˆ ¼ J ~ 4S n S nþ1 1 E-mail address: [email protected]. 0928-4931/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.msec.2005.06.034
n¼1
ð1Þ
810
A. Wal / Materials Science and Engineering C 25 (2005) 809 – 812 0,5 0,3 0,0
Ω=6
E/|J|
-0,3 -0,5 -0,8 -1,0 -1,3 -1,5
20
30
40 50 60708090100
200
300 400500600700
2 0 -2 -4 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24
N
-3,1416 -2,0944 -1,0472 0,0000 1,0472 2,0944 3,1416
Fig. 1. The energy states for two deviations (r = 2), clasified by k 1 = 3, k 2 = 3(X = 6), as a function of N. n
where S denotes the spin on the position n along the chain and the boundary condition S N+1 = S 1 is applied. The coefficient J is positive for the antiferromagnetic coupling and negative for the ferromagnetic one. The solutions for the system can be found using the Bethe Ansatz [1]. The inversions of the spin projections can be treated as pseudoparticles with the pseudomomenta k j . When the two particles k j and k l scatter, they exchange phase / j, l . The basic equations in the Bethe Ansatz approach provide the relation between these two quantities k j and / j, l 2cot
/l;j kj kl ¼ cot cot ; 2 2 2
N kl ¼ 2kkl þ ~ /l;j ;
E/|J|
/l;j ¼ /j;l
l ¼ 1; 2; . . . ; r:
ð2Þ ð3Þ
jl
The integer number k i should satisfy the condition N / 2 k 1 k 2. . . N / 2 and it plays the key role in the classification of the Bethe – Hulthen solutions [10]. The solutions for the asymptotic antiferromagnetic case were
k Fig. 3. The energy spectrum for finite magnet (N = 8 and r = 4) and antiferromagnetic coupling.
given by Bethe and they describe properties of the ground state. From the other hand the exact solutions were studied also for the small numbers of spins [4]. They were found by direct diagonalisation of the Hamiltonian. In last time the short Heisenberg chains were investigated extensively in frame of Bethe Ansatz [11 – 13]. These two methods were not applied so far for the chains with hundreds of spins due to the difficulties in solving such large set of equations. The method proposed in Ref. [10,16] allows to find solutions for wide range of chain lengths N. The method bases on the assumption that the solutions have quasi-continous form for a wide range of number N. The starting point for the calculations are the asymptotic solutions, which can be found separately for each type of solutions classified by the relations between the winding numbers [14]. Then, using the calculation procedure one can follow the solutions by
-14 2,5
kl k1 k2
2,0
-16
{λ 1,λ 2} = {-1,1}
1,5 -18
E/|J|
1,0 0,5
-20
0,0 -0,5
-22
-1,0 -1,5
-24
-2,0
-4
-2
0
2
4
k
-2,5 4
10
100
1000
N Fig. 2. The quasimomenta ki for two deviations and N q (6; 100). The solutions are obtained for k 1 = 1, k 2 = 1.
Fig. 4. The lower energy levels for antiferromagnetic coupling for chain with N = 8 spins (s = 1/2). The solid line denotes the fitting for the function E / |J| = A|sin(Bk)| + C with the following values: A = 6.077, B = 1.02, and C = 21.45.
A. Wal / Materials Science and Engineering C 25 (2005) 809 – 812
3. Nanomagnets The monotonous form N-dependence of the Bethe solutions is not preserved for the finite magnet with small number of nodes. These properties follow from a size effect. Some of solutions labelled by winding numbers {k i } change their character from bounded to scattered states, other solutions disappear for the special length of the spin chain. These peculiarities can be characterised by special points: critical, limiting and transition points [10]. Fig. 2 presents the solutions of Bethe equations for selected set of winding numbers. The number of nodes N changes from 100 up to 6.
4. Properties of energy spectrum The energy spectrum E(k) for the antiferromagnet coupling for the chain with N = 8 is presented in Fig. 3. The ground state is located in the centre of the Brillouine zone k = 0 [6]. The first excited state has spin s = 1 and the shape of the lowest laying states can be described by the function E(k) = A|sin(Bk)| + C [17]. These lowest states are presented in Fig. 4. They fit this formula well and are comparable with results presented in Ref. [17]. The distribution of the states in the Brillouin zone is not monotonous. The number of states corresponding to quasimomentum k is not the same for each k—the so-called Frarefied band_ effect is observed [18]. The distribution of the states can be described with the help of number theory [18]. The winding numbers k i provide the key to classification of the energy states for the Heisenberg chain. However if we consider the energy as a function of N the more useful labels should be constructed, which assure the classification to be valid for all chain lengths. For the states of two deviations
2 0 Λ=0
-2 -4
0,0 -0,1
-6
E/|J|
decreasing the number of nodes N and fixing the set of winding numbers {k i }. The form of the solutions does not change drastically in lowering N. For the mesoscopic case, i.e. for the chains with a few hundreds of spins, the pseudomomenta for given and fixed set {k i } change slightly in this region [16]. The energy of corresponding state given by the textbook equation E ðk Þ ¼ 4~ri¼1 1 cos kj does not change considerably for this region as well (see Fig. 1). It should be remarked that by fixing the set of winding numbers in the procedure we restrict ourselves to special solutions for a given N. To obtain a full energy spectrum for fixed values of N, we should repeat calculation procedure for all possible values of winding numbers {k i }. However in most cases we are interested in the properties of the ground and first excited states, e.g. for antiferromagnets. Because the properties of winding numbers and total quasimomentum k for such states can be predicted [5,6], the method can be applied with success.
811
Ω=2 Ω=4 Ω=6
-0,2
-8
-0,3 -0,4
-10
-0,5 -0,6
-12
100
-14 -16 10
100
N Fig. 5. The energy levels classified by numbers K = k 1 + k 2 and X = k 2 k 1 for system with N spins and r = 2 deviations.
{k 1k 2} the new parameters K and X should be introduced in the following way [19]: K ¼ k1 þ k2 N ; X ¼ k 2 k1 :
ð4Þ
The first parameter corresponds to total quantum number k of the system whereas the second labels the energy levels for the given k. The change of the chain length causes the shift of energy levels, in such a way, that their ordering provided by the number X is conserved. In Fig. 5 these noncrossing levels are presented for the centre of the Brillouine zone k = 0 (K = 0).
5. Conclusion Bethe Ansatz provides a full description of the quantum states of the finite magnet. The energies as well the eigenfunctions can be obtained using this approach. The method of calculation presented in this paper can be effectively applied for a wide range of chain length N. The energy levels are classified by total quantum number k and for the same k the additional labels are introduced related to winding numbers. The levels classified in this way reveal the noncrossing properties, i.e. they do not cross when the number of nodes N is changed. The energy spectrum for small systems manifests rarefied phenomena. References [1] [2] [3] [4] [5] [6]
H.A. Bethe, Z. Phys. 71 (1931) 205. L. Hulthe´n, Arkiv Mat. Astron. Fys. 26A (1938) 1. L.F. Mettheiss, Phys. Rev. 123 (1958) 1209. J.C. Bonner, M.E. Fisher, Phys. Rev. 135 (1964) A640. R.B. Griffiths, Phys. Rev. 133 (1964) A768. K. Fabricius, U. Lo¨w, K.-H. Mu¨tter, J. Ueberholz, Phys. Rev., B 44 (1991) 7476. [7] K. Ba¨rwinkel, P. Hage, H.-J. Schmidt, J. Schnack, Phys. Rev., B 68 (2003) 054422-1. [8] K.L. Taft, Ch.D. Delfs, G.C. Papaefthymiou, S. Foner, D. Gatteschi, S.J. Lippard, J. Am. Chem. Soc. 116 (1994) 823.
812
A. Wal / Materials Science and Engineering C 25 (2005) 809 – 812
[9] R.W. Saalfrank, I. Bernt, E. Uller, F. Hampel, Angew. Chem. Int. Ed. 36 (1997) 2482. [10] W.J. Caspers, M. Labuz, A. Wal, M. Kuzma, T. Lulek, J. Phys. A, Math. Gen. 36 (2003) 5369. [11] B. Lulek, T. Lulek, Rep. Math. Phys. 38 (1996) 267. [12] T. Lulek, in: T. Lulek, B. Lulek, A. Wal (Eds.), Symmetry and Structural Properties of Condensed Matter, World Scientific, Singapore, 1999, p. 52. [13] B. Lulek, T. Lulek, A. Wal, P. Jakubczyk, Physica B337 (2003) 375.
[14] W.J. Caspers, in: T. Lulek, B. Lulek, A. Wal (Eds.), Symmetry and Structural Properties of Condensed Matter, World Scientific, Singapore, 2003, p. 224. [15] R. Orbach, Phys. Rev. 112 (1958) 309316. [16] M. Labuz, M. Kuzma, A. Wal, Mater. Sci. Eng., C 23 (2003) 945. [17] J. des Cloizeaux, J.J. Pearson, Phys. Rev. 128 (1962) 2131. [18] B. Lulek, Seminaire Lotharingien de Combinatoire B26e (1991). [19] W.J. Caspers, M. Kuzma, Mol. Phys. Rep. 23 (1999) 142.
From mesoscopic magnet to quantum wire. The Bethe Ansatz application A. Wal Institute of Physics, University of Rzeszow, Rejtana 16 a, 35 – 310 Rzeszow, Poland Available online 1 August 2005
Abstract Molecular nanomagnets are representatives of quantum systems in which significant magnetic size effects are observed. In most cases they are formed of a few or a dozen or so magnetic ions arranged into a regular ring (e.g. Fe8). Such small chains can be treated as quantum wires. In literature magnetic properties of these systems were studied experimentally and these results confirm fully their extraordinary quantum features. By far results were interpreted basing on solutions obtained numerically. In present paper we give the exact solutions of the system, obtained basing on the Bethe theory. The application of this theory for such small systems is quite complicated. We develop mathematical method for inspection of the solutions evolution starting from macroscopic via mesoscopic till nanoscopic size of the Heisenberg chain. Furthermore, we study the properties of the energy band structures (as noncrossing, rarefied effect, etc.) and their changes as a function of magnetic quantum dots size. D 2005 Elsevier B.V. All rights reserved. Keywords: Bethe Ansatz; Heisenberg chain; Energy spectrum; Nanoscopic system
1. Introduction The one dimensional Heisenberg magnet with the spin s = 1 / 2 has been studied more then 70 years. The analytical solution of such a system in the thermodynamical limit and with the periodic boundary conditions was given by Bethe [1] and Hulthen [2]. The method is known as the Bethe Ansatz. It allows to solve exactly the quantum system. The anti-ferromagnetic finite Heisenberg model as well as the ferromagnetic one were studied also numerically [3– 7]. The knowledge of physical properties for finite system became more important recently due to the advances in inorganic and organic chemistry which allow to synthesise well formed nanomagnets, e.g. perfect rings with metal ions [8,9]. These systems are good realisation of an antiferromagnetic Heisenberg ring with weak magnetic anisotropy. Among others properties the energy spectrum plays a crucial role for a possible application in spintronic. For a finite number N (range of dozen) it was studied by K. Fabricius et al. [6]. The differences between even and odd number of nodes were outlined by them. We restrict our investigation to the even case. The whole energy spectrum can be found by a routine numerical procedure or by solving Bethe equations [1,2].
The second method provides a complete knowledge of eigenstates and eigenvalues of the system. The method has its own disadvantages, e.g. a hard way to solve transcendental equations leading to eigenfunctions. However there is an effective method [10], which with the help of classification of the quantum states for Heisenberg ring, given the first time by Bethe [1] and then by Orbach [15], allows to find solutions, for the systems with different number of nodes N. The key is to start from asymptotic solution (N Y V) and then follow the solutions through the chain with thousands, hundreds and few nodes. In the Section 2 we discuss the solutions for mesoscopic rings (N is order of hundreds) whereas in the Sections 3 the solutions for smaller chains are presented. In the Section 4 the pecularities of the energy spectrum are addressed.
2. Infinity and mesoscopic magnetic chain The dynamic of the system is provided by the Heisenberg Hamiltonian N
Hˆ ¼ J ~ 4S n S nþ1 1 E-mail address: [email protected]. 0928-4931/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.msec.2005.06.034
n¼1
ð1Þ
810
A. Wal / Materials Science and Engineering C 25 (2005) 809 – 812 0,5 0,3 0,0
Ω=6
E/|J|
-0,3 -0,5 -0,8 -1,0 -1,3 -1,5
20
30
40 50 60708090100
200
300 400500600700
2 0 -2 -4 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24
N
-3,1416 -2,0944 -1,0472 0,0000 1,0472 2,0944 3,1416
Fig. 1. The energy states for two deviations (r = 2), clasified by k 1 = 3, k 2 = 3(X = 6), as a function of N. n
where S denotes the spin on the position n along the chain and the boundary condition S N+1 = S 1 is applied. The coefficient J is positive for the antiferromagnetic coupling and negative for the ferromagnetic one. The solutions for the system can be found using the Bethe Ansatz [1]. The inversions of the spin projections can be treated as pseudoparticles with the pseudomomenta k j . When the two particles k j and k l scatter, they exchange phase / j, l . The basic equations in the Bethe Ansatz approach provide the relation between these two quantities k j and / j, l 2cot
/l;j kj kl ¼ cot cot ; 2 2 2
N kl ¼ 2kkl þ ~ /l;j ;
E/|J|
/l;j ¼ /j;l
l ¼ 1; 2; . . . ; r:
ð2Þ ð3Þ
jl
The integer number k i should satisfy the condition N / 2 k 1 k 2. . . N / 2 and it plays the key role in the classification of the Bethe – Hulthen solutions [10]. The solutions for the asymptotic antiferromagnetic case were
k Fig. 3. The energy spectrum for finite magnet (N = 8 and r = 4) and antiferromagnetic coupling.
given by Bethe and they describe properties of the ground state. From the other hand the exact solutions were studied also for the small numbers of spins [4]. They were found by direct diagonalisation of the Hamiltonian. In last time the short Heisenberg chains were investigated extensively in frame of Bethe Ansatz [11 – 13]. These two methods were not applied so far for the chains with hundreds of spins due to the difficulties in solving such large set of equations. The method proposed in Ref. [10,16] allows to find solutions for wide range of chain lengths N. The method bases on the assumption that the solutions have quasi-continous form for a wide range of number N. The starting point for the calculations are the asymptotic solutions, which can be found separately for each type of solutions classified by the relations between the winding numbers [14]. Then, using the calculation procedure one can follow the solutions by
-14 2,5
kl k1 k2
2,0
-16
{λ 1,λ 2} = {-1,1}
1,5 -18
E/|J|
1,0 0,5
-20
0,0 -0,5
-22
-1,0 -1,5
-24
-2,0
-4
-2
0
2
4
k
-2,5 4
10
100
1000
N Fig. 2. The quasimomenta ki for two deviations and N q (6; 100). The solutions are obtained for k 1 = 1, k 2 = 1.
Fig. 4. The lower energy levels for antiferromagnetic coupling for chain with N = 8 spins (s = 1/2). The solid line denotes the fitting for the function E / |J| = A|sin(Bk)| + C with the following values: A = 6.077, B = 1.02, and C = 21.45.
A. Wal / Materials Science and Engineering C 25 (2005) 809 – 812
3. Nanomagnets The monotonous form N-dependence of the Bethe solutions is not preserved for the finite magnet with small number of nodes. These properties follow from a size effect. Some of solutions labelled by winding numbers {k i } change their character from bounded to scattered states, other solutions disappear for the special length of the spin chain. These peculiarities can be characterised by special points: critical, limiting and transition points [10]. Fig. 2 presents the solutions of Bethe equations for selected set of winding numbers. The number of nodes N changes from 100 up to 6.
4. Properties of energy spectrum The energy spectrum E(k) for the antiferromagnet coupling for the chain with N = 8 is presented in Fig. 3. The ground state is located in the centre of the Brillouine zone k = 0 [6]. The first excited state has spin s = 1 and the shape of the lowest laying states can be described by the function E(k) = A|sin(Bk)| + C [17]. These lowest states are presented in Fig. 4. They fit this formula well and are comparable with results presented in Ref. [17]. The distribution of the states in the Brillouin zone is not monotonous. The number of states corresponding to quasimomentum k is not the same for each k—the so-called Frarefied band_ effect is observed [18]. The distribution of the states can be described with the help of number theory [18]. The winding numbers k i provide the key to classification of the energy states for the Heisenberg chain. However if we consider the energy as a function of N the more useful labels should be constructed, which assure the classification to be valid for all chain lengths. For the states of two deviations
2 0 Λ=0
-2 -4
0,0 -0,1
-6
E/|J|
decreasing the number of nodes N and fixing the set of winding numbers {k i }. The form of the solutions does not change drastically in lowering N. For the mesoscopic case, i.e. for the chains with a few hundreds of spins, the pseudomomenta for given and fixed set {k i } change slightly in this region [16]. The energy of corresponding state given by the textbook equation E ðk Þ ¼ 4~ri¼1 1 cos kj does not change considerably for this region as well (see Fig. 1). It should be remarked that by fixing the set of winding numbers in the procedure we restrict ourselves to special solutions for a given N. To obtain a full energy spectrum for fixed values of N, we should repeat calculation procedure for all possible values of winding numbers {k i }. However in most cases we are interested in the properties of the ground and first excited states, e.g. for antiferromagnets. Because the properties of winding numbers and total quasimomentum k for such states can be predicted [5,6], the method can be applied with success.
811
Ω=2 Ω=4 Ω=6
-0,2
-8
-0,3 -0,4
-10
-0,5 -0,6
-12
100
-14 -16 10
100
N Fig. 5. The energy levels classified by numbers K = k 1 + k 2 and X = k 2 k 1 for system with N spins and r = 2 deviations.
{k 1k 2} the new parameters K and X should be introduced in the following way [19]: K ¼ k1 þ k2 N ; X ¼ k 2 k1 :
ð4Þ
The first parameter corresponds to total quantum number k of the system whereas the second labels the energy levels for the given k. The change of the chain length causes the shift of energy levels, in such a way, that their ordering provided by the number X is conserved. In Fig. 5 these noncrossing levels are presented for the centre of the Brillouine zone k = 0 (K = 0).
5. Conclusion Bethe Ansatz provides a full description of the quantum states of the finite magnet. The energies as well the eigenfunctions can be obtained using this approach. The method of calculation presented in this paper can be effectively applied for a wide range of chain length N. The energy levels are classified by total quantum number k and for the same k the additional labels are introduced related to winding numbers. The levels classified in this way reveal the noncrossing properties, i.e. they do not cross when the number of nodes N is changed. The energy spectrum for small systems manifests rarefied phenomena. References [1] [2] [3] [4] [5] [6]
H.A. Bethe, Z. Phys. 71 (1931) 205. L. Hulthe´n, Arkiv Mat. Astron. Fys. 26A (1938) 1. L.F. Mettheiss, Phys. Rev. 123 (1958) 1209. J.C. Bonner, M.E. Fisher, Phys. Rev. 135 (1964) A640. R.B. Griffiths, Phys. Rev. 133 (1964) A768. K. Fabricius, U. Lo¨w, K.-H. Mu¨tter, J. Ueberholz, Phys. Rev., B 44 (1991) 7476. [7] K. Ba¨rwinkel, P. Hage, H.-J. Schmidt, J. Schnack, Phys. Rev., B 68 (2003) 054422-1. [8] K.L. Taft, Ch.D. Delfs, G.C. Papaefthymiou, S. Foner, D. Gatteschi, S.J. Lippard, J. Am. Chem. Soc. 116 (1994) 823.
812
A. Wal / Materials Science and Engineering C 25 (2005) 809 – 812
[9] R.W. Saalfrank, I. Bernt, E. Uller, F. Hampel, Angew. Chem. Int. Ed. 36 (1997) 2482. [10] W.J. Caspers, M. Labuz, A. Wal, M. Kuzma, T. Lulek, J. Phys. A, Math. Gen. 36 (2003) 5369. [11] B. Lulek, T. Lulek, Rep. Math. Phys. 38 (1996) 267. [12] T. Lulek, in: T. Lulek, B. Lulek, A. Wal (Eds.), Symmetry and Structural Properties of Condensed Matter, World Scientific, Singapore, 1999, p. 52. [13] B. Lulek, T. Lulek, A. Wal, P. Jakubczyk, Physica B337 (2003) 375.
[14] W.J. Caspers, in: T. Lulek, B. Lulek, A. Wal (Eds.), Symmetry and Structural Properties of Condensed Matter, World Scientific, Singapore, 2003, p. 224. [15] R. Orbach, Phys. Rev. 112 (1958) 309316. [16] M. Labuz, M. Kuzma, A. Wal, Mater. Sci. Eng., C 23 (2003) 945. [17] J. des Cloizeaux, J.J. Pearson, Phys. Rev. 128 (1962) 2131. [18] B. Lulek, Seminaire Lotharingien de Combinatoire B26e (1991). [19] W.J. Caspers, M. Kuzma, Mol. Phys. Rep. 23 (1999) 142.