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You can introduce the ``would-be'' bosonic mean field exactly, using the Hubbard-Stratonich (a.k.a partial bosonization) method, see wikipedia and Interacting fermions on a lattice and Hubbard-Stratonovich transformation and mean-field approximation . The mean field approximation correspond to performing the integral over the bosonic field using the ...


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Mean-field theory is exact (in the thermodynamic limit) in the case of long-range interaction (which is not the case for the nearest-neighbor Ising model). Therefore, mean-field theory is exact for BCS, where you have an effective long-range interaction. As for rigorous results, Bogoliubov rigorously proved that in the ground state (zero temperature) the ...


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Mean field theory is only good when fluctuations are small, which means that the free energy of a fluctuation must be much smaller than the total free energy. The free energy of the typical fluctuation is of order $kT$ and its size is determined by the correlation length $\xi$, and is of order $\xi^d$, with $d=$ dimension: $$F_{fluct}\sim ...


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I will try to answer as many questions as I can. I won't presume to give you complete exhaustive answers, but maybe they will be nonetheless useful to you. What variables determine the range of temperatures over which matter is liquid? My understanding of thermodynamics is that matter changes from a solid to a gas when some thermal vibrations create an ...


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Liquid: molecules form bonds with neighboring molecules for most of the time, but there are enough energy for the bonds to break momentarily and be formed again with another molecule "These explanations seem hand-wavy to me." What is the level of your knowledge of physics? Are you aware of the quantum mechanical nature of atoms and molecules, at a ...


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Yes, you are correct. The magnet would just float, perhaps even before entering the tube. You anticipated correctly that the strength and lack of dissipation of the eddy currents keep the magnet in place. This is well illustrated in a clip about levitating superconductors.


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There are often several different methods of synthesizing materials, and in lots of cases they arrive at the same result. Sometimes the experiments you want to do will depend on your growth method, though: for instance, the polycrystalline samples you get from solid state reactions can be good for x-ray or neutron diffraction studies. If you want to ...


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Well, $1/2\otimes1/2=0\oplus1$, so a system with two fermions has integer spin. But it is still a two fermion system, and therefore its wavefunction must be antisymmetric, as usual. This is not specific to Cooper pairs, but is basic Quantum Mechanics... [what is specific to Cooper pairs is that their size is $\gg a_0$, which means they are highly ...


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Yes, pairing is possible from a repulsive interaction. The reason behind this is that pairing has to occur in a certain angular momentum channel : $l=0$ for s-wave superconductivity, $l=1$ for p-wave, and so on. To see this, you can expand the repulsive $k$-dependent interaction on Legendre polynomials. Check this review that deals with the Kohn-Luttinger ...



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