Landau–Lifshitz model

From Wikipedia, the free encyclopedia

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 equations describing time evolution of magnetism in solids, depending on 1 time variable and 1, 2, or 3 space variables.

Contents

[edit] Landau-Lifshitz equation

The LLE (a partial differential equation in 1 time and n space variables (n is usually 1,2,3) describes an anisotropic magnet. The LLE 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 R1+n taking values in R3. The equation depends on a fixed symmetric 3 by 3 matrix J, usually assumed to be diagonal; that is, J=\operatorname{diag}(J_{1}, J_{2}, J_{3}). It is given by Hamilton's equation of motion for the Hamiltonian

H=\frac{1}{2}\int \left[\sum_i\left(\frac{\partial \mathbf{S}}{\partial x_i}\right)^{2}-J(\mathbf{S})\right]\, dx\qquad (1)

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

\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

\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

\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 LLE looks like

\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)

[edit] Integrable reductions

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

a) in the 1+1 dimensions that is Eq. (3), it is integrable. The Lax representation for this case reads as
???\qquad (6a)
???\qquad (6b)
b) if J = 0 then the (1+1)-dimensional LLE (3) turn to the continuous classical Heisenberg ferromagnet equation (see e.g. Heisenberg model (classical)) which is as well known integrable.

[edit] See also

[edit] References

  • Faddeev, Ludwig D. & Takhtajan, Leon A. (2007), Hamiltonian methods in the theory of solitons, Classics in Mathematics, Berlin: Springer, pp. x+592, MR2348643, ISBN 978-3-540-69843-2 
  • Guo, Boling & Ding, Shijin (2008), Landau-Lifshitz Equations, Frontiers of Research With the Chinese Academy of Sciences, World Scientific Publishing Company, ISBN 978-9812778758 
  • Kosevich A.M., Ivanov B.A., Kovalev A.S. Nonlinear magnetization waves. Dynamical and topological solitons.- Kiev: Naukova Dumka, 1988. - 192 p.