Magnetization and Lyapunov exponents on a kagome chain with multi-site exchange interaction
The Ising approximation of the Heisenberg model in a strong magnetic field, with two, three and six spin exchange interactions is studied on a kagome chain. The kagome chain can be considered as an approximation of the third layer of absorbed on the surface of graphite (kagome lattice). By using dynamical approach we have found one and multi-dimensional mappings (recursion relations) for the partition function. The magnetization diagrams are plotted and they show that the kagome chain is separating into four sublattices with different magnetizations. Magnetization curves of two sublattices exhibit plateaus at zero and 2/3 of the saturation field. The maximal Lyapunov exponent for multi-dimensional mapping is considered and it is shown that near the magnetization plateaus the maximal Lyapunov exponent also exhibits plateaus.
keywords:Multiple spin exchange, solid and fluid 3He, recursive lattice, dynamical systems, magnetization plateau, antiferromagnetic Ising model, kagome chain
Pacs:75.10.Pq, 05.50.+q, 75.50.Ee, 67.30.hj, 05.45.-a
The partition function of a statistical system can be calculated by means of several methods: transfer matrix approach books (), dynamic technique obzordynamic1 ()-dynamicheski2 () etc. The most useful for recursive lattices is the dynamical approach, which is a very powerful tool for research of a number of theoretical problems in statistical mechanics. By means of this approach one can obtain exact recursive relations for the partition function on recurrent lattices.
As the systems of almost localized identical fermions one can model solid and fluid films. Since the light mass spin-1/2 atoms are subject to a week attractive potential, the theoretical explanation of magnetism is based on the multiple-spin exchange mechanism Roger (). An important case is represented by solid and fluid films absorbed on the surface of graphite graffite (); graffite2 (); graffite3 () since it is a typical example of a two-dimensional frustrated quantum-spin system frustrated (). The first and second layers of the system form a triangular lattice, while the third one forms a system of quantum 1/2 spins on a kagome lattice kagome2 ().
In a strong external magnetic field Heisenberg model can be approximated by the Ising one. If the strong magnetic field is directed along the z-axis, a reduction of transverse fluctuations is expected to occur; it is supposed that in this case and -spin components are infinitely small infinitely small and may be neglected Roger (); platoes6 ().
The phenomenon of magnetization plateau has been studied during the past decade both experimentally and theoretically. The plateaus may be exhibited in the magnetization curves of quantum spin systems at very low temperatures in case of strong external fields. The plateaus of this kind were first observed by Hida hida () in trimerized S=1/2 spin chains and they were analytically explained by Okamoto okamato (). Magnetization plateaus appear in a wide range of models on chains, ladders, hierarchical lattices, theoretically analysed by dynamical, transfer matrix approaches and exact diagonalization in clusters (see Ref. platoes1 ()-platoes13 ()). The experimental evidence for the appearance of magnetization plateaus has also been found for two-dimensional systems experemental ()-experementa5 (). On triangular lattices a magnetization plateau was observed at for compounds like CEu CEu1 (), CsCuCl CsCuCl () (see also Ref. NH4CuCl32 ()-NH4CuCl36 ()); plateaus have also been observed on a kagome lattice (see Ref. kagomeplato ()-kagomeplato12 ()).
In the present paper the dynamical (recursive) approach was used to study magnetic properties of a kagome chain. The magnetization plateaus were found at low temperatures and moreover, the kagome chain was separated into four sublattices with different magnetizations. Two of these exhibited plateaus, whereas the others did not. The stability of the system is described by Lyapunov exponents lyapunov3 () by means of which the exponential rate is measured, at which the adjacent orbits converge or diverge lyapunov1 ()-lyapunov6 (). It is interesting to check whether the maximum Lyapunov exponent has plateaus that coincide with the magnetization one.
This paper is organized as follows: in Section 2 the two, three and six-site exchange interactions in a strong magnetic field are described, in Section 3 the exact recursion relation for the partition function is derived, the magnetization curves for different values of temperature and exchange parameters are displayed in Section 4. In Section 5 a multidimensional mapping for the partition function is obtained and the corresponding maximum Lyapunov exponent is investigated. Finally, Section 6 contains the concluding remarks.
2 Exchange Hamiltonian with two, three and six- site exchange
The Heisenberg Hamiltonian for kagome chain consists of two parts
where is the spin exchange Hamiltonian and is the Zeeman Hamiltonian which is responsible for magnetism, which has the following form
where is the gyromagnetic ratio for a nucleus, and is the magnetic field Roger ().
According to Roger () the multiple spin exchanges Hamiltonian can be written as
where is the pair transposition operator, () is the operator making a cyclic rearrangement in the triangle (hexagon). The explicit expression of pair transposition operator was been given by Dirac
where is the Pauli matrix, acting on the spin at the -th site. The expressions for , and in terms of Pauli matrixes were derived in platoes11 (). By using these expressions one can rewrite the Heisenberg Hamiltonian in terms of Pauli matrices. By taking into account that in a strong external magnetic field directed along the z-axis the contribution from x and y spin component will be negligible and the only relevant contribution will come from the z component (which can effectively take values ), we may consider an Ising model instead of Heisenberg one. By using expressions for , and and by substituting the Pauli matrix with we get the Ising approximation of the Heisenberg Hamiltonian:
where represents the six particle exchange term in each hexagon:
where the first sum goes over all pairs in the hexagon, and the second one goes over all quartets in the hexagon.
3 The recursion relation for the partition function
The partition function of the system is defined in the following way
where is the Hamiltonian of the system, is the Boltzmann constant, is the temperature and the sum goes over all configurations of the system. Below all the parameters will be rescaled using the Boltzmann constant (). To obtain recursion relations for the partition function one can separate the kagome ladder into two identical parts (branches) and firstly perform a summation over all spin configurations on each branch, and secondly sum over the central spin variable (see Fig. 1). The summation on each branch yields the same result and such a term only depends on the value of central spin:
where denotes the contribution of each branch. The expression for can be written (, see Fig. 1) in the form:
where and is the number of generations and the expression for has the following form:
Since does not depend on , equation (9) may be obtained as
where we denote by according to the sign of and is the sum of over the ten intermediate spins between and
The recursion relation is obtained by introducing a new variable :
4 Magnetization plateaus
The Thermodynamic functions of the system, such as magnetization, can be expressed in terms of . The local magnetization at site is expressed as
By using equation (8) one can write for the central vertex
Suppose it is required to find the local magnetization for the -th site (): in that case we must expand sums over , and
is denoted as depending on the sign of and sign. In the same way the expression for may be obtained. The difference is that each is replaced by :
For fixed values of temperature and exchange parameters the dependence of magnetization on the external magnetic field may be obtained by means of a simple iteration scheme induced by the recursion relations, starting from some initial value , The thermodynamic limit then corresponds to the asymptotic limit .
Because of the symmetry of the Hamiltonian, the values of magnetization for some spins are equal: we have in general (See Fig. 1),
In Figure 2 magnetization curves are plotted for different sites. As it can be seen from the figure, the magnetization functions have plateaus only for the sublattices B and C.
5 Two dimensional mapping and Lyapunov exponent
It is interesting to calculate the Lyapunov exponent near plateaus but one cannot use recurrent function (13) because it is related to site. To obtain the recursion relation for the partition function that is related to site one can separate the kagome chain into two identical parts as shown in Figure 3:
and the contribution of each branch is denoted by . The recursion relations for can be obtained by performing eleven (, see Fig. 3) summations in the kagome chain:
where is as follows
Since one can show that and, hance, the recurrence relations are two-dimensional. By introducing and the recursion relations may be written in the following form:
Taking into account that , we get after the expansion of sums:
Relations (5) permit the calculation of Lyapunov exponents near the plateaus: in the two-dimensional space they have the form
where is Jacobian matrix evaluated at the point
The only subtle point in the estimation of Lyapunov exponents Sprott (); skokos (); kuznecov () is in the rescaling procedure in each iteration to avoid the numerical overflow. The results of calculation are shown in Fig. 4 with the corresponding magnetization function: it is seen in the figures that the maximum Lyapunov exponent also exhibits plateaus. Moreover, for the maximum Lyapunov exponent the location of magnetization plateaus coincide with those of magnetization curves.
In the present paper the theory of dynamical systems has been used to study magnetization on the kagome chain. In a strong external magnetic field the Ising model has been considered instead of the Heisenberg one for the case of two, three and six-site exchange interactions. Exact recursion relations have been derived for the partition function. The magnetization curves for different temperatures and exchange parameters were obtained and the kagome chain was observed to split into four sublattices with different magnetizations. Two of them exhibit plateaus at zero and 2/3 of the saturation field value at low temperatures and certain exchange parameters. A multidimensional mapping was deduced to study the maximum Lyapunov exponent near the plateaus and it was shown that the location of plateaus on the maximal Lyapunov exponent and for magnetization was the same.
This work was supported by the ANSEF 1518-PS, 1981-PS and ECSP-09-08_sasp NFSAT research grants.
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