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Say we have the electrical conductivity tensor expressed as a 3x3 matrix. I've seen that if it's cubic material then the conductivity tensor reduces to just the diagonal terms and these are equal, meaning that the conductivity is isotropic. However, if some of the off-diagonal terms are not zero, what direction do they correspond to?

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The conductivity tensor is defined by $$ J_i = \sigma_{ij}E_j$$ For example, if $\sigma_{xy}$ is not zero, then an electric field in the $y$ direction produces a current in the $x$ direction. So the current is perpendicular to the electric field. This can happen in a conductor in an external magnetic, as in the Hall effect. If your system has time reversal symmetry then the off-diagonal elements must be zero.

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