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Especially for a ferromagnet a magnetic field should have a field-induced band shift in the density of states but I wonder if this shift is big enough to be significant and affect the thermal conductivity.

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  • $\begingroup$ Influence of Magnetic Domain Walls and Magnetic Field on the Thermal Conductivity of Magnetic Nanowires pubs.acs.org/doi/abs/10.1021/nl502577y H.T. Huang et al., “Influence of magnetic domain walls and magnetic field on the thermal conductivity of magnetic nanowires,” Nano Letters, vol. 15, pp. 2773-2779, 2015. $\endgroup$ – user124459 Jul 25 '16 at 4:27
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Magnetic fields certainly can influence thermal conductivity. This shows up, not surprisingly, when there is a strong influence of the magnetic field on other properties, particularly electronic ones.

One (non-metal) example is 'Thermal conductivity tensor in YBa$_{2}$Cu$_{3}$O$_{7-x}$: Effects of a planar magnetic field' by R. Ocana and P. Esquinazi, Phys Rev B66 064525 (2002). The influence of magnetic fields on the superconducting state is well known, so you could easily see they could mess with the pairing states and change the (primarily electronic) thermal conductivity at low temperatures.

Another example is 'Giant Magnetic Field Effect on Thermal Conductivity of Magnetic Multilayers, Cu/Co/Cu/Ni(Fe)', H. Sato et al., J. Phys. Soc. Jpn. 62 431-434 (1993). Again, the magnetic interactions that lead to the large magnetoresistance changes should, and can, impact the electronic component of the thermal conductivity.

As a final example, 'Effect of a magnetic field on the thermal conductivity of lead telluride-tin telluride', T. Knittel and H. J. Goldsmid, J. Phys. C. 12 1891-1897 (1979). Again, at low temperature, the electronic thermal conductivity dominates, and the magnetic field will modulate it.

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I don't believe that the thermal conductivity of most metals is very sensitive to magnetic fields. Yes, there will be some field-induced band shifting in the case of an itinerant ferromagnet which, in principle, leads to a change in the density of states at the Fermi level, but that will typically be a very small effect.

If the magnetic field induced change in the thermal conductivity of a given metal were significant, then by the Wiedemann–Franz law the electrical conductivity of the metal should also change by the same proportion since both the (electronic) thermal conductivity and the electrical conductivity of a metal are sensitive to the electronic density of states at the Fermi level. Try doing an experiment yourself in which you connect some metal wire up to a sensitive ohmmeter and see if you can detect a change in the electrical resistance of the wire when you put it close to a strong magnet. For the magnetic fields that you can get from typical permanent magnets, I would guess that you won't be able to detect any change in the electrical resistance of the wire. By Wiedemann-Franz, that means that that any change in the thermal conductivity of the wire is also likely to be nil for all practical purposes.

Strong magnetic fields can indeed induce interband transitions in which electrons can "hop" from one electronic band to another, in which case the familiar, semiclassical model of electron dynamics breaks down. This effect is called magnetic breakthrough. However, for magnetic fields to induce such hopping the quantity $\hbar \omega _c$ (where $\omega _c$is the cyclotron frequency $\frac{\text{eB}}{m}$ and e=electron charge, B=magnetic field, m=effective mass of electron) must be comparable to any given interband energy gap. For even a large field of $B=10^4$ Gauss (typical refrigerator magnet about 50 to 100 Gauss), $\hbar \omega _c$ is only of the order of $10^{-4}$eV so you can see that very, very large magnetic fields are usually required to have much of an effect on electron dynamics in regards to significantly changing the physical properties of a metal such as changing its thermal conductivity.

(*Note: There are always conditions that can be found where a small magnetic field change can induce, for example, a superconducting metal to become a normal metal, thus resulting in a large change in the thermal conductivity as well as other physical properties. But in my answer above I assume that you're excluding such "special cases".)

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