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A generalised Landau-Lifshitz equation for isotropic SU(3) magnet

In the paper we obtain equations for large-scale fluctuations of the mean field (the field of magnetization and quadrupole moments) in a magnetic system realized by a square (cubic) lattice of atoms with spin s>= 1 at each site. We use the generalized Heisenberg Hamiltonian with biquadratic exchange as a quantum model. A quantum thermodynamical averaging gives classical effective models, which are interpreted as Hamiltonian systems on coadjoint orbits of Lie group SU(3).


Introduction
Being a multiparticle quantum system, a magnet can be considered on different levels of hierarchy: a quantum (microscopic) level and a classical (macroscopic) one. The quantum level is described by means of quantum electrodynamics, or by simpler models like the Hubbard model or the Heisenberg one. The most common model for the classical level is the mean field model. Dynamics of a mean field is described by the equations of Landau-Lifshitz type.
Each model is suitable to describe certain phenomena. For example, the problems of formation of large-scale structures (domain walls, topological solitons, nonlinear magnetization waves and so on) are naturally investigated from a classical point of view. More tenuous problems, like renormalization of the order parameter according to a temperature or an effective interaction constant, require a quantum point of view [1].
Here we start from the quantum level described by the Heisenberg model. In addition to the usual Heisenberg bilinear interaction −J(Ŝ n ,Ŝ m ), we consider the biquadratic one −K(Ŝ n ,Ŝ m ) 2 . By many theoretical and experimental researches it was shown that the biquadratic interactions have significant effects on magnetic properties. For example, a new ordered state (a nematic state, with zero magnetization) occurs as a separate phase transition [2]. Note, that the biquadratic interaction can be taken into account only if a magnetic system has the spin s 1.
In this paper we propose a classical generalization of the isotropic Landau-Lifshitz equation corresponding to the Heisenberg model with biquadratic exchange interaction. A transition from the quantum level to the classical one is performed by the mean field approximation. The classical model can be interpreted as a Hamiltonian system on a coadjoint orbit of the unitary group SU (3). Therefore, we acquire an additional mathematical apparatus, which gives a significant advantage.
The mean field approximation gives a qualitative analysis of ordered states [3,4], but has no answer about their stability. Moreover, in this approximation the temperature dependencies of order parameters considerably differ from the observed dependencies. That proves a necessity to take into account fluctuations of the mean field. The proposed effective classical models describe large-scale (or slow) fluctuations of mean field. One can come to slow fluctuations by an averaging over high frequencies [1]. However, remaining in the context of theory of magnetism, we choose the models associated with the equations of Landau-Lifshitz type.
The paper is organized as follows. Section 2 is devoted to the quantum model based on the spin Hamiltonian with biquadratic exchange interactions. We consider the SU(3)-invariant case. In section 3 we construct two effective models that describe largescale fluctuations of mean field (the field of magnetization and quadrupole moments). We obtain one of them by an averaging of the quantum Hamiltonian over coherent states. The other effective model is a result of an averaging over mixed states. These classical models appear to be Hamiltonian systems on coadjoint orbits of the group SU(3), that follows from SU(3)-invariance of the original quantum model. Each coadjoint orbit is determined by constrains, which are observed quantities becoming rigid after averaging. In section 4 we summarize results and give some ideas how to extend the proposed scheme to magnetic systems with higher spins.

Description of the model
The magnetic system in question is realized by a homogeneous lattice of atoms with the spin s 1 at each site. The lattice can be one-, two-, or three-dimensional, and has the distance l between the nearest-neighbor sites. We assign three spin operators (Ŝ 1 n ,Ŝ 2 n ,Ŝ 3 n ) to each site n; they obey the standard commutation relations: [Ŝ α n ,Ŝ β m ] = iε αβγŜγ n δ nm , where α, β, γ run over the set {1, 2, 3}, and δ nm denotes the Kronecker symbol.
We use the localized spin model for the magnetic system. In many cases this model adequately describes a magnetic system by the Heisenberg Hamiltonian, which includes only the bilinear exchange interaction. Nevertheless, there are a lot of magnets that require taking into account higher powers of exchange interaction. Our model is applicable to magnets with the spin s 1.
In the present paper we consider the Hamiltonian with biquadratic exchange and call it bilinear-biquadratic: whereŜ n = (Ŝ 1 n ,Ŝ 2 n ,Ŝ 3 n ) is a vector of spin operators at site n, and δ runs over the nearest-neighbor sites. This Hamiltonian was discussed, for example, in [2][3][4][5]. The constants J and K serve as exchange integrals. We suppose that J and K are positive. It means that we consider a ferromagnetic interaction in preference.
The operators {Ŝ α n } (here n is fixed) are defined over the (2s + 1)-dimensional space of irreducible representation of the group SU(2). They generate an associative matrix algebra over this space. The complete matrix algebra can be represented as a direct sum of irreducible sets of tensor operators with respect to the action adŜ α . In the case of s = 1, we have: Mat 3×3 ≃ [9] = [1] + [3] + [5]. Evidently, the operators {Ŝ α n } form a basis in the 3-dimensional irreducible set. One can construct a basis in the 5-dimensional irreducible set from the tensor operators of weight 2. These are the quadrupole operators {Q 12 n ,Q 13 n ,Q 23 n ,Q [2,2] n ,Q [2,0] n } defined by the formulas: . The spin and quadrupole operators are normalized by the following relation: Tr(P ) 2 = 1 3 s(s + 1)(2s + 1).
As s = 1 we have Tr(P ) 2 = 2 . The chosen normalization is matched to the relation (Ŝ 1 n ) 2 + (Ŝ 2 n ) 2 + (Ŝ 3 n ) 2 = s(s + 1). Now, fix the canonical basis {|+1 , |−1 , |0 } in the space of representation. Then one obtains the following matrix representation for the spin and quadrupole operators: We denote all spin and quadrupole operators: {Ŝ 1 n ,Ŝ 2 n ,Ŝ 3 n ,Q 12 n ,Q 13 n ,Q 23 n ,Q [2,2] n ,Q [2,0] n } by {P a n } 8 a=1 . The operators {P a n } obey the following commutation relations: [P a n ,P b m ] = iC abcP c n δ nm , where C abc are structure constants; nonzero components are C 123 = C 145 = C 167 = C 264 = C 257 = C 356 = 1, The Hamiltonian (1) becomes bilinear in the terms of {P a n }: where N denotes the total number of sites. Obviously, the Hamiltonian is SU(2)invariant, and one can transform the operators {Ŝ α n } and {Q a n } by the formulas of adjoint representation whereD αβ (Û ) andD ab (Û ) are matrices of the real irreducible 3-and 5-dimensional representations of the group SU(2) respectively, andÛ = exp{ α ϕ αŜ α n }, where {ϕ α } are group parameters. As K = J the SU(2)-symmetry is extended to the SU(3)-one, and the Hamiltonian (2) gets the form

Mean field approach and ordered states
Instead of interactions between the spin and quadrupole operators {P a n } according to the Hamiltonian (2), we consider effective interactions of the operators {P a n } with a classical mean field. We suppose that components of the mean field at site n are proportional to averages (quasiaverages) of the quantum operators {P a n }. In the mean field approximation the Hamiltonian (2) has the form where z is a number of the nearest-neighbor sites. We have to give a warning about averages of {P α n }. If one calculates the averages by means of the density matrix ρ(T ) = exp{− H kT }, one obtains zeros. This follows from the SU(2)-symmetry of the Hamiltonian (2). Nonzero values of the averages appear if the symmetry is broken. Symmetry breaking can be stimulated by an external magnetic field that vanishes after specifying an order in the magnetic system. Such averages are called quasiaverages [6].
Suppose that the magnetic system in question has nonzero quasiaverages { P α n }. They form a classical 8-component vector field {µ a (x n )} 8 a=1 , which we call a mean field. Suppose that the mean field is constant over the whole magnetic system. This happens in the case of thermodynamic equilibrium and an infinite lattice. Then under an action of the group SU(2) the Hamiltonian (4) can be reduced to a diagonal form, namely: where the components µ 3 = Ŝ 3 and µ 8 = Q [2,0] do not depend on the spatial point x n . These components are suitable to be order parameters. Evidently, µ 3 describes a normalized magnetization (a ratio of z-projection of magnetic moment to a saturation magnetization), µ 8 is similarly connected to a quadrupole moment. Now we briefly show that the proposed quantum model admits ordered states. In the mean field approximation a partition function is calculated by the formula where h MF denotes the one-site Hamiltonian The mentioned mean field exists if self-consistent relations are held, in other words, if the system has a solution. After calculation of the mean field averages one obtains the self-consistent relations in the form Solutions of the system correspond to ordered states of the magnetic system in question.
An evident solution is the paramagnetic state (µ 3 = 0, µ 8 = 0). All other solutions depend on a temperature T , and the exchange integrals J and K. Note, that we consider the ferromagnetic interaction in preference: J > 0. Nontrivial solutions appear at temperatures low than the critical one T crit = 2 3k (J − 1 2 K). As K < 0 there exists a ferromagnetic state with the values (µ 3 = 1, µ 8 = 1 √ 3 ) at zero temperature, and a nematic state with the values (µ 3 = 0, µ 8 = 1 √ 3 ) at zero temperature. As K > 0 there exist four nontrivial solutions: two ferromagnetic states with the values (µ 3 = 1, 3 ) at zero temperature, and two nematic states with the values 3 ) at zero temperature. The same states are declared in [3,4]. The states (µ 3 = 1, 3 ) are stable. The problem of transient processes in the mean field approach is discussed, for example, in [4]. The analysis of solutions of the self-consistent relations proves that ordered states in the proposed model exist.
In the sequel we deal with the case J = K, which corresponds to the boundary between the ferromagnetic and the nematic regions (see the phase diagram of the bilinear-biquadratic s = 1 model in [5]). In this case, the Hamiltonian (2) and its mean field approximation are SU(3)-invariant. The latter gets the form n aP a n P a n − 4 3 JN z = − 1 2 Jz n aP a n µ a − 4 3 JN z. (5)

Motion equations for large-scale fluctuations of mean field
Return to the quantum SU(3)-invariant spin model with the Hamiltonian (3). The Heisenberg equation for an evolution ofP a n has the form i dP a n dt = [P a n ,Ĥ].
We suppose that the magnetic system is ordered, then we take an average of equation (6) over the Heisenberg (time independent) coherent states , Alternatively, one can take an average by means of the density matrix. In the both cases we neglect correlations between fluctuations of the quantum fields {P a n } 8 a=1 at distinct sites, that is P a nP b m ≈ P a An averaging of equation (6) results in the following equation for µ a (x n ): which is a Hamiltonian one with respect to the Lie-Poisson bracket.
In order to investigate large-scale fluctuations of the mean field {µ a (x n )} 8 a=1 , we consider a continuum space instead of the discrete lattice. It can be achieved by the well-known limiting process. In the case of SU(2)-magnetic system (only bilinear intereations are taken into account), this limiting process underlies the macroscopic phenomenological theory of magnetism [7]. The limiting process replaces quantum operators by densities of their averages, which serve as dynamical variables. In our case, we deal with the densities M a of averages of the spin and quadrupole moments: where V 0 denotes a physically infinitesimal region of the lattice, and U (x) is the infinitesimal neighborhood of x. The Lie-Poisson bracket for {M a (x)} is defined by where δ(x) is the Dirac function. Since dimensionless quantities are more suitable, we introduce µ a (x) = V 0 M a (x) instead of M a (x). Then equation (8) gets the form where l is the lattice distance, and d is the lattice dimension. Note, that in the 2-dimensional case we obtain a scale-invariant Hamiltonian. Evidently, (9) is a generalization of the well-known Landau-Lifshitz equation to the case of 8-component vector field {µ a }. In the same way one can obtain the standard Landau-Lifshitz equation, if considers a spin system with s = 1 2 over the 2-dimensional space of representation of SU(2).
We rewrite (9) in the matrix form Hereμ is a Hermitian 3×3 matrix, [·, ·] denotes the matrix commutator, ∆ is the Laplas operator. Being SU(3)-invariant equation (10) as well as (9) preserves the quantities h 0 = 1 2 Trμ 2 and f 0 = 1 2 Trμ 3 , which we call invariants. They serve as constrains for the Hamiltonian system and define the manifold where the vector field {µ a } lives. At the same time, this manifold is an orbit of coadjoint representation of the group SU(3).

Classical Hamiltonian systems on coadjoint orbits of SU(3)
In the 1-dimensional case the Hamiltonian system (9) appears to be integrable, what is shown below by means of the orbital approach.

Phase space for SU(3)-symmetric generalization of Landau-Lifshitz equation
In this section we briefly construct the orbital interpretation of a finite-zone phase space for the SU(3)-symmetric generalization of the Landau-Lifshitz equation.
Consider an algebra of polynomials in λ with coefficients from the Lie algebra su(3). Denote by g + the algebra su(3) ⊗ P(λ), where P(λ) is a ring of polynomials in λ with the standard multiplication. Let A, B ∈ g + have the form: The operation (11) turns g + into a graded Lie algebra. LetP a,n = λ nP a , where a runs from 1 to 8. The set {P a,n } serves as a basis in g + . Recall that [P a ,P b ] = iC abcP c ; the nonzero components C abc have the following values: C 123 = C 145 = C 167 = C 264 = C 257 = C 356 = 1, Introduce a bilinear ad-invariant form on g + by The basis {P a,n } is orthonormal with respect to the bilinear form. Let M = g * + be a dual space to the algebra g + with respect to (12). Orthonormality of {P a,n } implies that {P a,n } also form a basis in M. Consider the following elements of M: The functionsμ(λ) form a closed ad-invariant subset of M, we denote it by M N +1 . One can compute the coordinate µ n a ofμ(λ) by the formula µ n a = μ(λ),P a,−n+N +1 . Define a Lie-Poisson bracket in C(M N +1 ) as with the Poisson tensor field W mn ab = μ(λ), [P a,−m+N +1 ,P b,−n+N +1 ] . Introduce also two ad-invariant functions I 2 (λ) and I 3 (λ) by the formulas where d abc = √ 3 4 √ 5 Tr(P aP bP c +P bP aP c ), and µ a (λ) denotes the polynomial The invariant functions are also polynomials in λ: It is easy to prove that the coefficients {h 0 , . . ., h N +1 , f 0 , . . ., f N +1 } are annihilators with respect to the bracket (13). We fix these coefficients and obtain the system of algebraic equations h n = const, f n = const, n = 0, . . . , N + 1, It becomes possible if one restricts the system to the orbit O N +1 ⊂ M N +1 .

Classification of orbits of SU(3)
It is evident, that the orbit O N +1 defined by (14) is a vector bundle over a coadjoint orbit of the group SU(3). That is why we need to classify orbits of SU(3). The group SU(3) is simple [8], hence its algebra g ≃ su(3) coincides with the dual space g * . Consequently, the coordinates {µ a } in g * can be regarded as coordinates in su(3) as well as in su * (3). A generic elementμ ∈ su * (3) has the form Let h be the maximal commutative subalgebra (also called the Cartan subalgebra) of g.
The dual space h * to the Cartan subalgebra h coincides with h. By definition the set Oμ in = {g −1μ in g, ∀g ∈ SU(3)} is the coadjoint orbit of SU(3) through an initial pointμ in ∈ su * (3). All elements g ′ ∈ SU(3) such that g ′−1μ in g ′ =μ 0 form the stationary subgroup Sμ in atμ in . The orbit Oμ in is a homogeneous space, which is diffeomorphic to the coset space SU (3) of dimension 4. It is proven by R. Bott that each orbit of coadjoint action of a semisimple group G intersects h * precisely in an orbit of the Weyl group W(G).
The full Weyl group of SU(3) consists of six elements {e, σ 1 , σ 2 , σ 1 σ 2 , σ 2 σ 1 , σ 1 σ 2 σ 1 }, where σ 1 , σ 2 are reflections across the hyperplanes orthogonal to the simple roots α 1 , α 2 (see figure 1). The open domain C = {μ ∈ h * , μ, α > 0, ∀α ∈ ∆ + } is called a positive Weyl chamber. Here ∆ + denotes the set of positive roots. We call the set Γ α = {μ ∈ h * , μ, α = 0} a wall of the Weyl chamber. An orbit of the Weyl group W(G) is obtained by the action of W(G) on a point of C.  Each orbit of the Weyl group W(G), and consequently, each coadjoint orbit of G intersects the positive Weyl chamber in an only point. That is why we can classify coadjoint orbits of G by points of the positive Weyl chamber.
In the case of group SU(3), there exist two types of orbits of the Weyl group. A generic orbit contains six elements and passes through the interior of the positive Weyl chamber. A degenerate orbit contains three elements and passes through a wall of the positive Weyl chamber. According to this, we call an orbit of SU(3) a generic one ifμ in lies in the interior of the positive Weyl chamber, and a degenerate one ifμ in belongs to a wall of the positive Weyl chamber.
In our case,μ in has the following diagonal form , where m and q denote initial values of the variables µ 3 and µ 8 respectively, or boundary values (at zero temperature) of the corresponding components of the mean field. As m > 0, q > 0 the coadjoint action of SU(3) gives a generic orbit. If m = 0 or m = √ 3 q, we obtain a degenerate orbit. In the sequel we consider degenerate orbits with m = 0.

Hamiltonian equations on orbits of SU(3)
Return to the system of equations (18), which is degenerate in M N +1 . However, it can be solved if one restricts the system to the orbit O N +1 ⊂ M N +1 . Each orbit is determined by the following equation [9]: where χ min (μ) is the minimal characteristic polynomial inμ ∈ O N +1 . Equation (20) serves as a constrain for the system (18), which has the form Now we solve (21) on orbits of the group SU(3). A degenerate orbit is determined by the equation Using this constrain, one obtains the following solution of (21): where we write µ a instead of µ 0 a and scale the flow parameter t by 16A. The dimensional constant A provides a correspondence between (22) as h 1 = 0 and (10) as d = 1. That is (22) describes large-scale fluctuations of the mean field {µ a }.
A generic orbit is determined by the characteristic equation . On this orbit we obtain the following solution of (21): The motion equation (17) on the generic orbit has the form ∂µ a ∂t = 2A As h 1 = 0, f 1 = 0 equation (23) also describes large-scale fluctuations of the mean field.
One can obtain (23) from (6) by averaging with a more complicate correlation rule.
Equations (22) and (23) imply the following Hamiltonians respectively: In addition to the 1-dimensional case one can consider the corresponding 2-or 3-dimensional Hamiltonian systems with the effective Hamiltonians where µ denotes {µ a } 8 a=1 . The exchange integral J = A/ gives the Hamiltonian the required physical dimension. By H we denote the Hamiltonian density One can use these effective Hamiltonians for describing the magnetic system considered in section 2. Note, that H deg is the same as the Hamiltonian of (9). The proposed Hamiltonians describe large-scale (slow) fluctuations of the mean field µ. After averaging over high frequencies some observed quantities become rigid (or invariant); these quantities are h 0 = δ ab µ a µ b , and f 0 = 5/3 d abc µ a µ b µ c . They serve as constrains for the Hamiltonian systems, and are equivalent to (20). The constrains determine the orbit where the system has to be considered.
In the case of SU(3)-invariant model we deal with the magnet whose ferromagnetic and nematic states are equiprobable. A generic orbit corresponds to a state with the ferromagnetic order at zero temperature, because of nonzero magnetization (m = 0). A degenerate orbit (m = 0) corresponds to a state with the nematic order at zero temperature. So equations (22) and (23) describe fluctuations of the mean field µ near a nematic and a ferromagnetic ordered states respectively. of the complex parameters the density becames independent on boundary values of µ. Moreover, the Hamiltonian density serve as a Riemannian metrics on the manifold.
In the case of SU(3)-invariant model we deal with the magnet whose ferromagnetic and nematic states at zero temperature are equiprobable. That is why we propose two effective Hamiltonians: H gen for states with the ferromagnetic order at zero temperature, and H deg for states with the nematic order (when magnetization is zero) at zero temperature. Also we produce equations (22) and (23) describing large-scale fluctuations of the mean field µ near a nematic and a ferromagnetic ordered states respectively.
The proposed classical models can be used to construct topological excitations [11], which are stationary solutions of the Landau-Lifshitz like equations. These excitations realize destruction of a long-range order in 2-dimensional spin systems at nonzero temperatures, according to the Mermin-Wagner theorem.
The considered scheme is easily extended to the case with higher powers of exchange interaction. For an arbitrary spin s the spin operators {Ŝ α n } are defined over the (2s+1)-dimensional space of representation of the group SU(2). The complete matrix algebra generated by the spin operators is Mat (2s+1)×(2s+1) . Then one can consider a spin Hamiltonian with powers of exchange interaction up to 2s. Such Hamiltonian admits a bilinear form, if one takes into account multipole moments. In the mean field approximation this quantum model corresponds to a Hamiltonian system on a coadjoint orbit of the group SU(2s+1). Each orbit has a Hamiltonian system, which serves as an effective classical model.