# Effective action for ferromagnetism and ferroelectricity

In Three Lectures On Topological Phases Of Matter section 2.1 mentioned, that: $$I^\prime = \int dt d^3x \; \left(\vec{a}\vec{E}+\vec{b}\vec{B}\right)$$ correspond to ferromagnetism and ferroelectricity. And that $$I^{\prime\prime} = \int dt d^3x \; \left(a_{ij}E^iE^j+b_{ij}B^iB^j\right)$$ correspondence to electric and magnetic susceptibility.

Could somebody clarify, why? I will be very appreciate for answers!

The energy of an electric dipole moment $$\bf{p}$$/magnetic dipole moment $$\bf{m}$$ in the external field is proportional to it, $$W = -\bf{p\cdot E}$$ or $$W = -\bf{m\cdot B}$$. In a ferromagnetic sample the local magnetic dipole moment is propotional to element of volume $$d^3x$$. This is just the same as you have in the expression for $$I'$$.
Then, if you consider a dielectric or diamagnetic sample, it gets polarized in the external electric or magnetic field. The susceptibility is a tensor that relates, for example, the dipole moment of unit volume $$\bf{P}$$ (polarization) and the external field $$\bf{E}$$: $$P^i=a^i_jE^j.$$ Since $$\bf{P}$$ is again the dipole moment, you get the expression $$I''$$ for action.
• Okay, thank you. But I trying to understand, how somebody can to using this action? What is the next step after writing such action? As I understand, varying by $A_\mu$ we can obtain current, as reflection of small change in external field. But can we find something more interesting? Commented Mar 15, 2020 at 21:50