I'm trying to understand the symmetry content of the conductivity matrix: one information is, presence of time-reversal symmetry causes the off-diagonal terms to vanish. When this is broken (e.g. in Hall effect) off-diagonal terms become finite. (A side question is, why is the conductivity matrix mostly anti-symmetric?!) Apart from that, does it contain any information about the spin of the charge carriers?
If I take a spin-orbit coupled (SOC) system (where inversion symmetry is broken), will the SOC information be present in the conductivity matrix? If it is, then how? What other symmetries in the system are relevant for the conductivity matrix?
Excellent question, Bikash. It is the field of science.
Regards,
Eugene.
Very briefly, the anti-symmetry of the Hall conductivity tensor can be viewed as being a consequence of the Onsager reciprocity relationship, according to which σμυ(Η) = συμ(-Η) (see for instance p. 262 of the book Solid State Physics, by Ashcroft & Mermin) Considering the conductivity tensor as relating two vectors, in an isotropic system it would be isotropic were it not for the fact that magnetic field breaks the time-reversal symmetry of the problem. The issue can be discussed within the framework of the group theory, in terms of the symmetry group of the Hamiltonian of the problem combined with the Clebsch-Gordon coefficients associated with the direct product of the irreducible representations of the group that transform as vectors (see for instance the book Group Theory, by Dresselhaus & Dresselhaus (Springer, Berlin, 2008)). In this connection, it is generally taken for granted that, following the above Onsager reciprocity relationship, in the absence of an external magnetic field the conductivity tensor should be symmetric. There is an old argument by Casimir, in response to this very Onsager relationship, reproduced and discussed by J.F. Nye in his book Physical Properties of Crystals: Their Representation by Tensors and Matrices, reprinted [first published in 1957] (Clarendon Press, Oxford, 2006), showing the possibility of the existence of an anti-symmetric contribution to the conductivity tensor that on satisfying a certain relationship is not observable.
@ Mariusz Książek - It is unclear to me why you have decided to download your CV on this page, which may be due to a technical error. I think that in your own interest you should consider to remove it, as it does not reflect positively on you; it is rather counter-productive.
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