How to derive the effective Langrangian of matter field, say, fermi field and quasiparticle, say, spin wave?

Question

In the Fermi-Hubbard model, it may lead to spin wave due to SO(3) symmetry breaking. I know how to derive the effective Lagrangian for the spin wave by integrating out the Fermi fields. However, how to derive a Lagrangian for the Fermi fields term, spin wave dynamic term and their interaction ? An example,

This post imported from StackExchange Physics at 2015-02-04 06:01 (UTC), posted by SE-user Xinloong Han

Comments

What do you mean, "derive" the Lagrangian? The Lagrangian is assumed, it is the starting point of any (well, most) QFTs, it is not derived, though one can often tell which terms may not occur through symmetry considerations.

This post imported from StackExchange Physics at 2015-02-04 06:01 (UTC), posted by SE-user ACuriousMind

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I mean the effective theory. For example, starting from a Fermi Hubbard model, it may be obtain the effective model about the spin wave when one consider the spin fluctuation and integrate out the fermi field.

This post imported from StackExchange Physics at 2015-02-04 06:01 (UTC), posted by SE-user Xinloong Han