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While reading about the BHZ model used to describe HgTe quantum well topological insulators, I read at many places that the effective Hamiltonian (which is a 4 x 4 matrix) can be written in block diagonal form and the lower 2x2 block can be derived from upper 2x2 block as follows:

[H(k)][/lower]=[H(-k)][/*]

This effective Hamiltonian is said to be Time reversal symmetric and then using Cramer's degeneracy, it is said that the dispersion relations for upspin and down spin should intersect at [k][/x]=0.

I want to just show this through simple mathematical steps, but I am unable to get this result. In order to show time reversal invariance, I tried the following equation:

[T][/-1]HT=H, where T is the Time reversal symmetry operator.

but I am not sure what form of T should be used. I tried to use the following form:

T=-i x [0 [σ][/y];[σ][/y] 0]K {K is complex conjugation which is a 4x4 matrix with [0][/2x2] in the diagonals and Pauli matrix in y as off diagonal elements.}

But this is not giving me that BHZ Hamiltonian is time reversal symmetric.

Can anybody help me where I am going wrong?

Thanks

Regards

Minato

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# Time reversal symmetry in Topological insulators of HgTe quantum Wells

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