Semiclassical Modeling of Isotropic Non-Heisenberg Magnets for Spin 𝑆=1 and Linear Quadrupole Excitation Dynamics

Abstract

Equations describing one-dimensional non-Heisenberg model are studied by use of generalized coherent states in real parameterization, and then dissipative spin wave equation for dipole and quadrupole branches is obtained if there is a small linear excitation from the ground state. Finally, it is shown that for such exchange-isotropy Hamiltonians, optical branch of spin wave is nondissipative.

1. Introduction

Many condensed matter systems are fully described by use of effective continuum field models. Topologically nontrivial field configurations have an important role in modeling systems with reduced spatial dimensionality [1]. Magnetic systems are usually modeled with the help of the Heisenberg exchange interaction [2–4].

However, for spin 𝑆>1/2, the general isotropic exchange goes beyond the purely Heisenberg interaction bilinear in spin operators ^→𝑆_𝑖 and includes higher order terms of the type (^→𝑆_𝑖^→𝑆_𝑗)^𝑛 with 𝑛 up to 2𝑆 [5]. Due to the spin states, the 2𝑆+1 complex parameters are necessary to describe each of them, and this corresponds with the 4𝑆+2 degrees of freedom. Two degrees of freedom are omitted, one because of normalization condition and the other for arbitrary phase decrease, hence 4S parameters are required to completely modeled the remainder 4𝑆 degrees of freedom of spin states [6].

Particularly, in case 𝑆=1 with the isotropic nearest neighbor exchange on a lattice is derived by use of the Hamiltonian 𝐻=−_𝑖𝐽^→𝑆_𝑖^→𝑆_𝑖+1+𝐾^→𝑆_𝑖^→𝑆_𝑖+1². (1)

Here 𝑆^𝑥_𝑖,𝑆^𝑦_𝑖,𝑆^𝑧_𝑖 are the spin operators acting at a site 𝑖, 𝐽 and 𝐾 are, respectively, the bilinear (Heisenberg) and biquadratic exchange integrals. The model (1) has been discussed recently in connection with 𝑆=1 bosonic gases in optical lattices [7] and in the context of the deconfined quantum criticality [8, 9]. Hamiltonian (1) is a special form presented in [10] and because of importance of quadrupole excitation in ferromagnetic Materials, it is considered here. This paper does not consider the antiferromagnetic and nematic states.

Considering the effects of both dipole and quadrupole branches gives a nonlinear approximation. If higher order multipole effects are considered, the approximation is more accurate but at the same time, deriving the equations is too complicated. In this paper, only the effect of quadrupole branch for Hamiltonians described by (1) is considered. Study of isotropic and anisotropic spin Hamiltonian with non-Heisenberg terms is complicated due to quadrupole excitation dynamics [5, 11, 12]. Antiferromagnetic property of this excitation in states near the ground proves the existence of it, and Dzyaloshinskii calculated the effect of this excitation [13]. Also, numerical calculations more accurately justify laboratory results if the effect of quadrupole excitation in nanoparticles Fe₈ and Mn₁₂ is considered [14, 15]. In addition, this method may be promising for description of the multispin configuration of the Fe in the different ligand coordinations [16].

To calculate effect of quadrupole excitation, at first it is necessary to obtain classical equivalent of Hamiltonian (1), and then to find out the solution of spin wave, it is necessary to analyze resulted equations in case there is small linear excitation from the ground states. Therefore, the stages of process are the following.(1)Obtaining coherent states for spin 𝑠=1 which are coherent states of SU(3) group.(2)Calculation of average values of spin operator.(3)Classical spin Hamiltonian equation is obtained using previously calculated values.(4)Calculating Lagrangian equation by use of Feynman path integral over coherent states and then computing classical equations of motion.(5)It is necessary to substitute resulted Hamiltonian in classical equations of motion to obtain nonlinear equations of magnets. Solutions of these nonlinear equations result in soliton description of magnet that is not interested here.(6)Now, ground states of magnets calculated and then nonlinear equations are linearized above the ground states for small linear excitation.(7)Finally, spin wave equations and dispersion equations must be calculated.

In this paper, we write coherent states in real parameters, because each parameter in this representation related to one degree of freedom. In complex parameterization, each parameter related to two or more degrees of freedom. Then in physical problems, the first representation is very helpful.

In next sections, we develop mathematical descriptions of the pervious stages and analyze these descriptions.

2. Theory and Calculations

In quantum mechanics, coherent states are special kinds of quantum states that their dynamics are very close to their corresponding classical system. Type of coherent state is used in problem and depends on the operators symmetry in Hamiltonian. Due to the operators symmetry in Hamiltonian (1), for more detailed description and considering all multipolar excitation of coherent states we used coherent states in SU(3) group. Vacuum state of this group is (1,0,0)^𝑇 and coherent state for a single site in this group is introduced as [11]: 𝜓=𝐷¹𝜃,𝜙𝑒^−𝑖𝛾𝑆^𝑧𝑒^2𝑖𝑔𝑄^𝑥𝑦|0⟩, (2) where 𝐷¹(𝜃,𝜙) is wigner function and 𝑄^𝑥𝑦 is quadrupole moment. Two angles, 𝜃 and 𝜙, the Euler angles, determine the direction of classical spin vector in spherical coordinate system. The angle 𝛾 determines the direction of quadrupole moment around the spin vector and parameter 𝑔 shows change of magnitude of spin vector and average value of quadrupole moment. Two angles 𝜙 and 𝛾 change between 0 to 𝜋 and angle 𝜃 changes between −𝜋 to 𝜋.

In order to derive Lagrangian from path integral, we drive the path integral from the following transition amplitude: 𝑃𝜓₁,𝑡₁;𝜓,𝑡=𝜓₁−𝑖expℎ𝐻𝑡₁−𝑡𝜓. (3) Using completeness relation and doing some mathematical work, the following equation for transition amplitude obtained: 𝑃𝜓₁,𝑡₁;𝜓,𝑡=lim_{𝑛→∞𝑗}𝐷𝜇_𝑗𝜓exp−𝑖ℎ^𝑡1𝑡𝐿_𝑗𝜃,𝜙,𝑔,𝛾𝑑𝜏. (4)

In the pervious equation 𝐿 is Lagrangian and has the following form [11]: 𝐿=ℎcos2𝑔cos𝜃𝜙_𝑡+𝛾_𝑡−𝐻𝜃,𝜙,𝑔,𝛾. (5)

𝑥_𝑡=(𝜕𝑥/𝜕𝑡)(𝑥=𝜃,𝜙) and 𝐻(𝜃,𝜙,𝑔,𝛾) is classical Hamiltonian. By use of (4) and (5) and the action stationary principle, the classical equations of motion are obtained.

When Lagrangian is obtained from path integral, another two terms appear, one is kinetic term and has “Berry phase” properties that are important in spin tunneling phenomena and the other depends on boundary condition values. In this paper, we do not consider these two terms.

Now classical equivalent of spin vector and their products must be computed so that the classical equivalent of Hamiltonian (1) is obtained. So consider the following: ^→𝑆=𝜓^→𝑆𝜓 (6) as classical spin vector, and also consider the following: 𝑄^𝑖𝑗=12𝑆_𝑖𝑆_𝑗+𝑆_𝑗𝑆_𝑖−43𝛿_𝑖𝑗𝐼 (7) components of quadrupole moment. Because we can write any coherent state as product of single site coherent states, namely, 𝜓=_𝑖𝜓_𝑖. (8)

Then spin operators in ground state of nonsingle ions Hamiltonian can be commuted in different lattices [11], so 𝜓𝑆^𝑖_𝑛𝑆^𝑗_𝑛+1𝜓=𝜓𝑆^𝑖_𝑛𝜓𝜓𝑆^𝑖_𝑛+1𝜓, (9) where |𝜓⟩=|𝜓⟩_𝑛|𝜓⟩_𝑛+1.

The average spin values in SU(3) group for coherent states (2) are defined as [17] 𝑆⁺=𝑒^𝑖𝜙𝑆cos2𝑔sin𝜃,₋=𝑒^−𝑖𝜙𝑆cos2𝑔sin𝜃,_𝑧=cos2𝑔cos𝜃. (10) And also 𝑆²=cos²,𝑞2𝑔₂=sin²,𝑆2𝑔₂+𝑞²=1. (11)

In the pervious relation, 𝑆² is related to dipole moment and 𝑞² is related to quadrupole moment. 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 is 𝐻=−𝑑𝑥𝑎₀𝐽cos²2𝑔+𝐾cos⁴−𝑎2𝑔₂₀24𝑔²_𝑥sin²2𝑔𝐽+2𝐾cos²2𝑔+cos²2𝑔𝐽+2𝐾cos²×𝜃2𝑔_2𝑥+𝜙²_𝑥sin²𝜃, (12) where 𝑎₀ is length of crystal sites. The pervious classical Hamiltonian is substituted in equation of motion that obtained from the Lagrangian, and the result is classical equations of motion: 1𝜔₀𝜃_𝑡=−𝑎²₀cos2𝑔𝐽+𝐾+𝐾cos4𝑔𝜙_𝑥𝑥1sin𝜃,𝜔₀𝜙_𝑡=𝑎²₀cos2𝑔𝐽+𝐾+𝐾cos4𝑔𝜙²_𝑥cos𝜃+𝜃_𝑥𝑥csc𝜃,1𝜔₀𝑔_𝑡1=0,𝜔₀𝛾_𝑡+=4cos2𝑔𝐽+𝐾+𝐾cos4𝑔𝐾cos³2𝑔16𝑔²_𝑥−8𝜃²_𝑥−5𝜙²_𝑥+3𝜙²_𝑥cos(2𝜃)−2𝜃_𝑥𝑥×+cos2𝑔8𝑔²_𝑥(𝐽−𝐾)−2𝐽𝜃²_𝑥+8𝑔²_𝑥+1𝐾cos4𝑔2𝐽𝜙²_𝑥(−3+cos(2𝜃))−𝐽𝜃_𝑥𝑥cot𝜃+4𝑔_𝑥𝑥𝐽sin2𝑔+2𝑔_𝑥𝑥𝐾sin2𝑔+sin6𝑔𝑎²₀. (13)

In these equations, 𝜔₀ is ℎ𝑎₀. These equations describe nonlinear dynamics of non-Heisenberg ferromagnetic chain completely. If we omitted quadrupole excitation (𝑔=0) in the pervious equations, these equations reduced to Landau-Lifshitz equations. Then in comparison with Landau-Lifshitz equations, these equations are more complete and contain more degrees of freedom. Note that solutions of these equations are different forms of magnetic solitons.

In this paper, only the linearized form of (13) for small excitation above the ground states is considered. To this end,at first, classical ground states must be calculated, so in above Hamiltonian only nonderivative part is considered: 𝐻₀=−𝑑𝑥𝑎₀𝐽cos²2𝑔+𝐾cos⁴2𝑔. (14)

To find the smallest value of the 𝐻₀ we vary it respect to all the parameters, with the ground state is obtained at; 𝜋𝑔=0,𝑔=2. (15)

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 (15), are defined: 2𝑔→𝜋+𝑔. (16)

In this situation, the linearized classical equations motion are 1𝜔₀𝜃_𝑡=−𝑎²₀(𝐽+2𝐾)𝜙_𝑥𝑥,1𝜔₀𝜙_𝑡=𝑎²₀(𝐽+2𝐾)𝜃_𝑥𝑥,1𝜔₀𝑔_𝑡1=0,𝜔₀𝛾_𝑡=−4(𝐽+2𝐾). (17)

Consider functions 𝜃 and 𝜙 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)}. (18)

Substitution of these equations in (17) results in dispersion equation of spin wave near the ground states. 𝜔²₁=(𝐽+2𝐾)²𝑎²₀𝑘⁴𝜔²₀,𝜔₂=−4(𝐽+2𝐾)𝜔₀. (19)

It is evident from (19) that in addition to the dispersion acoustic branch, there exist nondispersion optical branches which is related to the dipole and quadrupole excitations.

3. Conclusion

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

In addition, it is shown that for isotropic ferromagnets, the magnitude of average quadrupole moment is constant (𝑔_𝑡=0) and its dynamic, is rotational dynamics around the classical spin vector (𝛾_𝑡≠0).