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In explaining/introducing second-order phase transition using Ising system as an example, it is shown via mean-field theory that there are two magnetized phases below the critical temperature. This derivation is done for zero external magnetic field B=0 and termed spontaneous symmetry breaking The magnetic field is then called the symmetry breaking field. But, if the symmetry breaking occurs "spontaneously" at zero external field why do we need to call the external magnetic field the symmetry breaking field? I am confused by the terminology.

$M=\sqrt{{\displaystyle \frac{4U}{3}}}\mathrm{\varphi <!--\; \varphi \; -->}$
where M is the ferromagnetic order parameter and $\mathrm{\varphi <!--\; \varphi \; -->}$ is the auxiliary field from the Hubbard-Stratonovich transformation. The book argues that because the above equation is correct, the mean field theory which is derived from the Hartree-Fock approach is equivalent to the the saddle point approximation formalism for H-S transformation auxiliary field Lagrangian. But I can not understand the equation.

I am currently studying magnetism as a part of AP Physics 2, and what confused me was that a Tesla was the unit for magnetic field strength(which decreases with distance) but it was also used to measure the strength of magnets. Why is this the case? Is there not another metric that could be used to measure the total strength of the magnet? Also, when the strength of a magnet is given in Teslas, at what point from the magnet is that number derived?

In Three Lectures On Topological Phases Of Matter section 2.1 mentioned, that:
$I^{\mathrm{\prime <!--\; \prime \; -->}}=\int <!--\; \int \; -->dt{d}^{3}x(\stackrel{\to <!--\; \to \; -->}{a}\stackrel{\to <!--\; \to \; -->}{E}+\stackrel{\to <!--\; \to \; -->}{b}\stackrel{\to <!--\; \to \; -->}{B})$
correspond to ferromagnetism and ferroelectricity. And that
$I^{\mathrm{\prime <!--\; \prime \; -->}\mathrm{\prime <!--\; \prime \; -->}}=\int <!--\; \int \; -->dt{d}^{3}x(a_{ij}{E}^i{E}^j+b_{ij}{B}^i{B}^j)$
correspondence to electric and magnetic susceptibility.
Could somebody clarify, why?

If a ferromagnetic material is immersed in an alternating magnetic field at frequency $\mathrm{\omega <!--\; \omega \; -->}$, the material will follow a hysteresis cycle at that frequency. But if that frequency is high enough, the spins will oscillate more and more, and I think it will somehow increase the temperature of the material (because in a classical picture: more frequency $\Rightarrow <!--\; \Rightarrow \; -->$ more velocity $\Rightarrow <!--\; \Rightarrow \; -->$ more kinetic energy $\Rightarrow <!--\; \Rightarrow \; -->$ more temperature.)
As hysteresis only appear below the Curie temperature, if higher frequencies make the metal get hotter, there will be a critical frequency ${\mathrm{\omega <!--\; \omega \; -->}}_c$, above which hysteresis won't appear. Does such frequency exists for every ferromagnetic material?

Manganese has five unpaired electrons, but Iron has four, then why is Iron ferromagnetic and Manganese paramagnetic? What's that property I'm missing?

I understand that the magnetization must saturate as more and more domains are aligned. But B is still directly proportional to H, and hence it must increase linearly with H. But every book that teaches B−H curve, says that it saturates after some time. How can this be so?