How to derive ferromagnet Lagrangian from the Heisenberg model: $H=-J_{ij}S_i S_j$ ? I understand how to obtain potential energy term, but it is not clear how to get "kinetic energy". \begin{equation} L=\int d^2 x\frac{\rho}{2} \left[ \frac{(\partial_t \theta)^2}{v^2}-(\nabla \theta)^2\right] \end{equation}
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$\begingroup$ What is $theta$, $v$ and $\rho$ ? $\endgroup$– Frederic ThomasCommented Apr 4, 2020 at 13:07
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You only get an equation with a second-order time derivative for an antiferromagnet. If you have feromagnet there is only a first time derivative and you get a Landau-Lifshitz equation rather than a conventional wave equation. Deriving the antiferromagnetic equation is tricky. See F. D. M. Haldane, Phys. Lett. 93A, 464 (1983).
answered Apr 4, 2020 at 19:12
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$\begingroup$ Yes, thank you! I have already realized that the spectrum is linear so it is certainly not a Lagrangian for a ferromagnet. For those who are interested in deriving ferromagnet Lagrangian from path integral formulation see Altland and Simons. $\endgroup$ Commented Apr 5, 2020 at 8:27