More on Generalized Heisenberg Ferromagnet Models

We generalize the integrable Heisenberg ferromagnet model according to each Hermitian symmetric spaces and address various new aspects of the generalized model. Using the first order formalism of generalized spins which are defined on the coadjoint orbits of arbitrary groups, we construct a Lagrangian of the generalized model from which we obtain the Hamiltonian structure explicitly in the case of CP (N − 1) orbit. The gauge equivalence between the generalized Heisenberg ferromagnet and the nonlinear Schrödinger models is given. Using the equivalence, we find infinitely many conserved integrals of both models. 1 E-mail address; [email protected] 2 E-mail address; [email protected] In the past decades, there have been extensive investigations on the structure of the continuous Heisenberg ferromagnet (HM) model [1]-[4]. The dynamical variable of the conventional HM model is given by a spin variable Q which is defined on the coadjoint orbit S of SU(2), i.e. Q(x, t) = Q(x, t)T , ∑3 a=1Q Q = k, where T ’s are generators of the SU(2) algebra and k is a constant. This SU(2) spin HM model was later extended to the SU(N) case[5]. More generally, it was shown that there exists an extension of the HM model to each Hermitian symmetric spaces[6]. However, the extension in [6] is made only implicitly in connection with the nonlinear Schrödinger(NS) model. In particular, the integrability structure was not clear and the Lax pair formalism was lacking. The purpose of this Letter is to provide a systematic understanding of the generalized HM model. We formulate the model in terms of a Lagrangian using the first order formalism of generalized spins which are defined on the coadjoint orbits of arbitrary groups. The gauge equivalence of the generalized HM and the generalized NS model is demonstrated. Especially, starting from the associated linear equation of the HM model, we obtain a closed form of the generalized NS equation(Eq.(17)). We also find zero curvature expressions of both the HM and the NS equations in terms of which we obtain infinitely many conserved integrals. These conserved integrals are constructed systematically by making use of the properties of Hermitian symmetric space and they are given in a multicomponent form thus giving more than “one series” of integrals. As an explicit example of the Lagrangian description, we perform a reduction to CP (N − 1) orbit in detail and explain the resulting Hamiltonian structure of the HM model. We begin with a brief introduction on the Hermitian symmetric space. A symmetric space is a coset space G/K for Lie groups G ⊃ K whose associated Lie algebras g and k, with the decomposition g = k ⊕ m, satisfy the commutation relations, [k, k] ⊂ k, [k, m] ⊂ m, [m, m] ⊂ k. (1) A Hermitian symmetric space is a symmetric space equipped with a complex structure. For our purpose, we need only the following properties of Hermitian symmetric spaces [6][7] ; for each Hermitian symmetric space, there exists an element T in the Cartan subalgebra of g whose centralizer in g is k, i.e. k = {V ∈ g : [V, T ] = 0}. Also, up to a scaling, J = adT = [T, ∗] is a linear map J : m → m satisfying the complex structure condition 2 J = α for a constant α, or [T, [T, M ]] = αM, for M ∈ m. Now, we introduce an action for the HM model, A = ∫ dtdx Tr [2Tgġ + ∂x(gTg )∂x(gTg )− 2BgTg] (2) where g is a map g : R → G and B = B(t) is an arbitrary element in g describing an external magnetic field. The equation of motion can be written in terms of a generalized spin Q ≡ gTg, Q̇+ ∂x[Q, ∂Q] + [Q, B] = 0. (3) The integrability of the HM equation (3) arises from the existence of its associated linear equations: (∂̄ − B − λ[Q, ∂Q] + αλQ)ΨHM = 0, (∂ + λQ)ΨHM = 0, (4) where ∂ = ∂/∂x, ∂̄ = ∂/∂t and λ is an arbitrary complex constant whereas α is a constant to be fixed later. These linear equations are overdetermined systems whose consistency requires the integrability condition: 0 = [∂̄ − B − λ[Q, ∂Q] + αλQ, ∂ + λQ] = λ(∂̄Q+ ∂[Q, ∂Q] + [Q, B]) + λ(−α∂Q + [Q, [Q, ∂Q]]). (5) The λ-order term in the second line of Eq.(5) becomes precisely the HM equation since the λ-order term vanishes identically due to the complex structure property of T , [Q, [Q, ∂Q]] = g[T, [T, g(∂Q)g]]g = g[T, [T, [g∂g, T ]]]g = αg[g∂g, T ]g = α∂Q, (6) or, whenever T satisfies a stronger condition: T 2 = βT + γI, α = β + 4γ. (7) Without loss of generality, we may set β to zero by shifting T by T → T−β/2. This stronger condition holds at least for SU(p+ q)/S(U(p)×U(q)), SO(2n)/U(n), Sp(n)/U(n) compact Hermitian symmetric spaces and their noncompact counterparts. Since K is a subgroup of G commuting with T , this shows that Q in the case of Eq.(7) is in fact defined on the 3 coadjoint orbit G/K characterized by a number Q = γI. Note that the external magnetic field B(t) can be made disappear by taking the gauge transformation of g and ΨHM such that g → b(t)g, ΨHM → b (t)ΨHM where b(t) satisfies ∂̄bb −1 = B(t). From now on, we assume that B = 0. Having found the associated linear equations of the HM equation, we demonstrate the integrability of the HM model itself by deriving infinitely many conserved currents from the linear equations. In order to do so, we first derive the generalized NS equation from the generalized HM equation thereby proving the gauge equivalence of both models. Define ΨNS ≡ g ΨHM and rewrite the linear equation (4) in an equivalent form: (∂̄ + g∂̄g − λg∂g − λT )ΨNS = 0, (∂ + g ∂g + λT )ΨNS = 0, (8) where we have taken α = −1 without loss of generality. Since Q is invariant under g → gk for k ∈ K, we choose k such that g∂g is valued in m. The integrability of Eq.(8) becomes the zero curvature condition: [∂̄ + g∂̄g − λg∂g − λT, ∂ + g∂g + λT ] = 0. (9) Equivalently, we may require [T, g∂̄g]− ∂(g∂g) = 0 (10) and the identity [∂̄ + g∂̄g, ∂ + g∂g] = 0. (11) g∂̄g may be expressed in terms of g∂g by solving Eqs.(10) and (11) as follows; introduce a decomposition g∂̄g = (g∂̄g)m + (g ∂̄g)k where subscript m and k refer to the components of g∂̄g in those vector subspaces. Then, due to the algebraic properties of Eq.(1), Eq.(10) becomes [T, (g∂̄g)m]− ∂(g ∂g) = 0 (12) which can be solved for (g∂̄g)m by applying the adjoint action of T , [T, [T, (g∂̄g)m] = −(g ∂̄g)m = [T, ∂(g ∂g)]. (13)