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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!

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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.

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  • $\begingroup$ Thank you for answer! But in this functionals we consider fields as non-dinamical external parameters, so what information one can extract from such actions? $\endgroup$
    – Nikita
    Commented Mar 15, 2020 at 16:43
  • $\begingroup$ @Nikita, Well, since this is a cond-mat action, in general you are mostly interested in the properties of your sample. External field is considered only as a condition, such as pressure or temperature, for instance. You can study the position of Fermi surface or density of states in your system.. $\endgroup$
    – CuteKitty
    Commented Mar 15, 2020 at 16:49
  • $\begingroup$ What kind of information one can extract from action? Is such object useful for something? $\endgroup$
    – Nikita
    Commented Mar 15, 2020 at 18:38
  • $\begingroup$ @Nikita, a classical problem that one can solve given the susceptibility is to determine the magnetic field inside a sample that is placed in some other magnetic field. I am not an expert in magnetism but, of course, this quantity is of big practical use. An example that I can imagine is the determination of magnetic field inside a detector. If you are measuring tracks of charged particles moving in ext. mag. field, you surely need to know the field inside. Then using a kind of expression for Lorenz force one can deduce particle mass/charge. $\endgroup$
    – CuteKitty
    Commented Mar 15, 2020 at 21:29
  • $\begingroup$ 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? $\endgroup$
    – Nikita
    Commented Mar 15, 2020 at 21:50

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