Semiclassical Description of Anisotropic Magnets for Spin 𝑆=1

Abstract

Nonlinear equations describing one-dimensional non-Heisenberg ferromagnetic model are studied by the use of generalized coherent states in a real parameterization. Also, dissipative spin wave equation for dipole and quadruple branches is obtained if there is a small linear excitation from the ground state.

1. Introduction

In the past decades, magnets with spin value 𝑠=1/2 have been studied completely. There are dipoles, quadruples and higher-order branches that affect the behavior of magnet crystal. However, the only necessary tool for describing the behavior of this kind of magnets is dipole branch effect, and the order branches are not necessary. This results in linear approximation for describing the magnet behavior.

Indeed, only dipole branch effect has been used for describing magnets with spin value 𝑠≥1, and the effect of quadrupole and higher-order branches has been ignored. Recently, however, due to the new developments in mathematics and technology and also due to the great potential of quadrupole branch in description of nanoparticles, its important role cannot be ignored [1, 2].

Using the effects of both dipole and quadrupole branches results in a nonlinear approximation. The use of higher-order multipole effects yields more accurate approximations which demand more complicated equations. In this paper, only the effect of quadruple branch for Hamiltonians described by (1) is considered. Study of isotropic and anisotropic spin Hamiltonian with non-Heisenberg terms are complicated due to quadruple excitation dynamics [3–5]. Antiferromagnetic property of this excitation in states near the ground proves the existence of it. The effect of this calculation has been studied by Dzyaloshinskii [6]. The results obtained through the quadrupole excitation in nanoparticles Fe₈ and Mn₁₂ are more in line with numerical calculations and laboratory results [7, 8].

In classical physics term, the number of parameters required for a full macroscopic description of the magnet behavior is equal to 4𝑠, where 𝑠 is the spin value. Also real-parameterized coherent states based on related group is used to obtain classical equation of motion and to describe multipole dynamics [9, 10]. Here, Heisenberg ferromagnets with anisotropic term as described by (1) are considered:𝐻=−𝐽_𝑖^⇀𝑆𝑆_𝑖^⇀_𝑖+1+𝛿𝑆^𝑧_𝑖𝑆^𝑧_𝑖. (1)

Here, 𝑆^𝑥_𝑖,𝑆^𝑦_𝑖, and 𝑆^𝑧_𝑖 are the spin operators acting at a site 𝑖, and 𝛿 is the anisotropy coefficient. This Hamiltonian is related to a one-dimensional ferromagnetic spin chains, and the coefficient 𝐽 is positive.

In order to calculate the effect of quadrupole excitation, first, the classical equivalent of Hamiltonian (1) is obtained and then, by analyzing such equation for small linear excitation from the ground states, the spin wave solution is found. This process requires the following steps.(1)Obtaining coherent states for spin 𝑠=1 which are coherent states of SU(3) group.(2)Calculating the average values of spin operator.(3)Obtaining classical spin Hamiltonian equation using previously calculated values.(4)Computing Lagrangian equation by the use of Feynman path integral over coherent states and then computing classical equations of motion.(5)For finding nonlinear equations of magnet behavior, it is necessary to substitute resulted Hamiltonian in classical equations of motion. Solutions of these nonlinear equations result in soliton description of magnet that is not needed here.(6)Calculating ground states of magnet and then linearizing the nonlinear equations around the ground states for small excitation.(7)At the end, calculating spin wave equation and dispersion equation.

In what follows, the mathematical descriptions of the above steps are presented.

2. Theory and Calculation

In quantum mechanics, coherent states are special kind of quantum states that their dynamics are very similar to their corresponding classical system. These states are obtained by act of Weil-Heisenberg group operator on vacuum state. Vacuum state of SU(3) group is (1,0,0)^𝑇, and coherent state is introduced as [11]𝜓=𝐷^1/2𝜃,𝜙𝑒^−𝑖𝛾𝑆^𝑧𝑒^2𝑖𝑔𝑄^𝑥𝑦|0⟩=𝐶₀|0⟩+𝐶₁|1⟩+𝐶₂|2⟩, (2) where 𝐷^1/2(𝜃,𝜙) is Wigner function and 𝑄^𝑥𝑦 is quadruple moment which is written in the following form:𝑄^𝑥𝑦=𝑖2.001000−100 (3)

Coefficients 𝐶₀ to 𝐶₂ are computed from these equations𝐶₀=𝑒^𝑖𝜙𝑒^−𝑖𝛾sin²𝜃2cos𝑔+𝑒^𝑖𝛾cos²𝜃2,𝐶sin𝑔₁=sin𝜃√2𝑒^−𝑖𝛾cos𝑔−𝑒^𝑖𝛾,𝐶sin𝑔₂=𝑒^−𝑖𝜙𝑒^−𝑖𝛾cos²𝜃2cos𝑔+𝑒^𝑖𝛾sin²𝜃2sin𝑔. (4)

Two angles, 𝜃 and 𝜙, determine the direction of classical spin vector in spherical coordinate system. The angle 𝛾 determines the direction of quadruple moment around the spin vector, and parameter 𝑔 shows change of the magnitude of spin vector.

In order to obtain the classical equivalent of Hamiltonian (1), the classical equivalent of spin vector and its corresponding products should be computed. So, consider^⇀𝑆=𝜓^⇀𝑆𝜓 (5) as classical spin vector, and also consider𝑄^𝑖𝑗=𝜓𝑆^𝑖𝑆^𝑗𝜓 (6) as components of quadruple moment. Spin operators can be commute in different lattices; so𝜓𝑆^𝑖_𝑛𝑆^𝑗_𝑛+1𝜓=𝜓𝑆^𝑖_𝑛𝜓𝜓𝑆^𝑗_𝑛+1𝜓, (7) where |𝜓⟩=|𝜓⟩_𝑛|𝜓⟩_𝑛+1.

The average spin values in SU(3) group are defined as [12]𝑆⁺=𝑒^𝑖𝜙𝑆cos2𝑔sin𝜃,₋=𝑒^−𝑖𝜙𝑆cos2𝑔sin𝜃,_𝑧𝑆=cos2𝑔cos𝜃,₂=cos²2𝑔. (8) Classical Hamiltonian can be obtained from the average calculation of Hamiltonian (1) over coherent states. The classical continuous limit of Hamiltonian in SU(3) group is𝐻_cl=−𝐽𝑑𝑥𝑎₀cos²+𝛿2𝑔2cos²𝜃+sin2𝑔cos2𝛾sin²𝜃−𝑎²₀2𝜃²_𝑥+𝜙²_𝑥sin²𝜃cos²2𝑔+4𝑔²_𝑥sin²2𝑔. (9)

The above classical Hamiltonian is substituted in equation of motion that was obtained from the Lagrangian, and the result is classical equations of motion:1𝜔₀𝜙_𝑡=𝛿cos𝜃sec2𝑔−cos2𝛾tan2𝑔+𝑎²₀cos2𝑔𝜃_𝑥𝑥csc𝜃+𝜙²_𝑥,1cos𝜃𝜔₀𝜃_𝑡=𝛿2sin(2𝜃)sin2𝛾tan2𝑔−𝑎²₀𝜙_𝑥𝑥1cos2𝑔sin𝜃,𝜔₀𝑔_𝑡𝛿=−2sin2𝛾sin²1𝜃,𝜔₀𝛾_𝑡=4cos2𝑔−𝛿cos2𝛾cot4𝑔−cos(2𝜃)csc4𝑔+cos²𝜃sec+2𝑔cos2𝑔8𝑔²_𝑥−2𝜃²_𝑥+12𝜙²_𝑥(−3+cos(2𝜃))−𝜃_𝑥𝑥cot𝜃+4𝑔_𝑥𝑥sin2𝑔𝑎²₀. (10) These equations describe nonlinear dynamics of non-Heisenberg ferromagnetic chain completely. Solutions of these equations are magnetic solitons that are not studied in this paper.

In this paper, only the linearized form of (10) for small excitation from the ground states is considered. To this end, first, classical ground states must be calculated. therefore in the above Hamiltonian, only nonderivative part is taken into account:𝐻₀=−𝐽𝑑𝑥𝑎₀cos²+𝛿2𝑔2cos²𝜃+sin2𝑔cos2𝛾sin²𝜃. (11) It is necessary to calculate derivative of (11) with respect to all variables to find out minimum of 𝐻₀. As a result, if 𝛿<0, ground states are at these points𝜋𝜃=2𝜋,𝛾=2,sin2𝑔₀=|𝛿|4,|𝛿|<4. (12)

In this paper, only dispersion of spin wave in neighborhood of the ground states is studied. For this purpose, small linear excitations from the ground states, as shown in (13), are defined:𝜋𝜃⟶2−𝜃,2𝛾⟶𝜋+𝛾,2𝑔⟶𝑔₀+𝑔. (13)

In this situation, the linearized classical equations of motion are1𝜔₀𝜙_𝑡=𝛿sec𝑔₀+tan𝑔₀𝜃+𝑎²₀cos𝑔₀𝜃_𝑥𝑥,1𝜔₀𝜃_𝑡=−𝑎²₀𝜙_𝑥𝑥cos𝑔₀,1𝜔₀𝑔_𝑡𝛿=−21𝛾,𝜔₀𝛾_𝑡=−22sin𝑔₀+𝛿cos𝑔₀𝑔+4𝑎²₀𝑔_𝑥𝑥sin𝑔₀. (14)

Consider functions 𝜃, 𝜙, 𝛾, and 𝑔 as plane waves to obtain dispersion equation:𝜙=𝜙₀𝑒^{𝑖(𝜔𝑡−𝑘𝑥)}+𝜙₀𝑒^{−𝑖(𝜔𝑡−𝑘𝑥)},𝜃=𝜃₀𝑒^{𝑖(𝜔𝑡−𝑘𝑥)}+𝜃₀𝑒^{−𝑖(𝜔𝑡−𝑘𝑥)},𝑔=𝑔₀𝑒^{𝑖(𝜔𝑡−𝑘𝑥)}+𝑔₀𝑒^{−𝑖(𝜔𝑡−𝑘𝑥)},𝛾=𝛾₀𝑒^{𝑖(𝜔𝑡−𝑘𝑥)}+𝛾₀𝑒^{−𝑖(𝜔𝑡−𝑘𝑥)}. (15)

Substitute of these equations in (14), then𝜔²₁=𝜔²₀𝑘²𝑎²₀𝛿1+sin𝑔₀+𝑘²𝑎²₀cos²𝑔₀,𝜔²₂=𝜔²₀2sin𝑔₀𝑘²𝑎²₀+𝛿4𝛿sin²𝑔₀−2sin𝑔₀. (16)

These equations are dispersion equations of spin wave near the ground states in SU(3) group.

3. Conclusions

In this paper, describing equations of one-dimensional anisotropic non-Heisenberg Hamiltonians are obtained using real-parameter coherent states. It was indicated that both dipole and quadruple excitations have different dispersion if there is small linear excitation from the ground state.

In addition, it was indicated, that for anisotropic ferromagnets, the magnitude of average quadruple moment is not constant and its dynamics consists of two parts. One part is rotational dynamics around the classical spin vector (𝛾_𝑡≠0) and the other related to the change of the magnitude of quadruple moment (𝑔_𝑡≠0).