In book it says" If a field in x direction induce any current in y direction,by exploiting the symmetry,one can predict a same current will arise in -y direction,only consistent possibility is zero current so the conductivity tensor is diagonal in cubic symmetry lattice"
What hell he is talking? why field in x direction can induce current in y direction?And if so,why will be same current induced in -y direction? why those stuff related to symmetry? I have no clue about this
It's an ugly explanation of the symmetry principle: a field in x-direction CANNOT induce any current in y-direction (or z-) because otherwise a mirror symmetry of the lattice would be broken. As a result in a cubic lattice, a filed in x-direction produce ONLY x-directed current, y-field -- y-current, and z-field -- z-current, i.e., the conductivity tensor is diagonal.
Crystals are not isotropic media, like majority of liquids. Thus the electric current density in them satisfies the relation j=\sigma E, where j is vector of current density, E is electric field vector, and \sigma is generally the 3x3 tensor of conductivity. \sigma in cubic crystals is not only diagonal, its all three diagonal elements are equal to each other. This need not to be the case in other symmetries, for example in tetragonal crystals \sigma_xx=\sigma_yy but \sigma_zz usually has different value. In still lower symmetry you may expect more non-zero elements of \sigma. If you don't see why in crystals the vectors j and E don't have to be parallel, think about a bunch of parallel insulated wires, not aligned with power source contacts attached to their ends.
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