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Landau–Lifshitz model

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(Redirected from Landau–Lifshitz equation (continuous spin field)) For another Landau–Lifshitz equation describing magnetism, see Landau–Lifshitz–Gilbert equation.

In solid-state physics, the Landau–Lifshitz equation (LLE), named for Lev Landau and Evgeny Lifshitz, is a partial differential equation describing time evolution of magnetism in solids, depending on 1 time variable and 1, 2, or 3 space variables.

Landau–Lifshitz equation

The LLE describes an anisotropic magnet. The equation is described in (Faddeev & Takhtajan 2007, chapter 8) as follows: it is an equation for a vector field S, in other words a function on R taking values in R. The equation depends on a fixed symmetric 3-by-3 matrix J, usually assumed to be diagonal; that is, J = diag ⁡ ( J 1 , J 2 , J 3 ) {\displaystyle J=\operatorname {diag} (J_{1},J_{2},J_{3})} . The LLE is then given by Hamilton's equation of motion for the Hamiltonian

H = 1 2 ∫ [ ∑ i ( ∂ S ∂ x i ) 2 − J ( S ) ] d x ( 1 ) {\displaystyle H={\frac {1}{2}}\int \left\,dx\qquad (1)}

(where J(S) is the quadratic form of J applied to the vector S) which is

∂ S ∂ t = S ∧ ∑ i ∂ 2 S ∂ x i 2 + S ∧ J S . ( 2 ) {\displaystyle {\frac {\partial \mathbf {S} }{\partial t}}=\mathbf {S} \wedge \sum _{i}{\frac {\partial ^{2}\mathbf {S} }{\partial x_{i}^{2}}}+\mathbf {S} \wedge J\mathbf {S} .\qquad (2)}

In 1+1 dimensions, this equation is

∂ S ∂ t = S ∧ ∂ 2 S ∂ x 2 + S ∧ J S . ( 3 ) {\displaystyle {\frac {\partial \mathbf {S} }{\partial t}}=\mathbf {S} \wedge {\frac {\partial ^{2}\mathbf {S} }{\partial x^{2}}}+\mathbf {S} \wedge J\mathbf {S} .\qquad (3)}

In 2+1 dimensions, this equation takes the form

∂ S ∂ t = S ∧ ( ∂ 2 S ∂ x 2 + ∂ 2 S ∂ y 2 ) + S ∧ J S ( 4 ) {\displaystyle {\frac {\partial \mathbf {S} }{\partial t}}=\mathbf {S} \wedge \left({\frac {\partial ^{2}\mathbf {S} }{\partial x^{2}}}+{\frac {\partial ^{2}\mathbf {S} }{\partial y^{2}}}\right)+\mathbf {S} \wedge J\mathbf {S} \qquad (4)}

which is the (2+1)-dimensional LLE. For the (3+1)-dimensional case, the LLE looks like

∂ S ∂ t = S ∧ ( ∂ 2 S ∂ x 2 + ∂ 2 S ∂ y 2 + ∂ 2 S ∂ z 2 ) + S ∧ J S . ( 5 ) {\displaystyle {\frac {\partial \mathbf {S} }{\partial t}}=\mathbf {S} \wedge \left({\frac {\partial ^{2}\mathbf {S} }{\partial x^{2}}}+{\frac {\partial ^{2}\mathbf {S} }{\partial y^{2}}}+{\frac {\partial ^{2}\mathbf {S} }{\partial z^{2}}}\right)+\mathbf {S} \wedge J\mathbf {S} .\qquad (5)}

Integrable reductions

In the general case LLE (2) is nonintegrable, but it admits two integrable reductions:

a) in 1+1 dimensions, that is Eq. (3), it is integrable
b) when J = 0 {\displaystyle J=0} . In this case the (1+1)-dimensional LLE (3) turns into the continuous classical Heisenberg ferromagnet equation (see e.g. Heisenberg model (classical)) which is already integrable.

See also

References

  • Faddeev, Ludwig D.; Takhtajan, Leon A. (2007), Hamiltonian methods in the theory of solitons, Classics in Mathematics, Berlin: Springer, pp. x+592, doi:10.1007/978-3-540-69969-9, ISBN 978-3-540-69843-2, MR 2348643
  • Guo, Boling; Ding, Shijin (2008), Landau-Lifshitz Equations, Frontiers of Research With the Chinese Academy of Sciences, World Scientific Publishing Company, ISBN 978-981-277-875-8
  • Kosevich A.M., Ivanov B.A., Kovalev A.S. Nonlinear magnetization waves. Dynamical and topological solitons. – Kiev: Naukova Dumka, 1988. – 192 p.
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