Question

Is the band structure of an electronic crystal always symmetric around the center of the Brillouin zone?

Answer 1

What are the necessary and/or sufficient conditions that lead to the band structure of an electronic crystal being symmetric around the center of the Brillouin zone? What symmetries of the Hamiltonian will lead to such a symmetric band structure?  I'm specifically interested in the inversion P and time-reversal T symmetries and whether their existence lead to a symmetric band structure and why.  For example, If we have time-reversal symmetry [H,T]=0, then it is necessary that the states ψ(k,↑) and Tψ(k,↑)=ψ∗(−k,↓) have the same energies. Does it mean the band structure has to be symmetric around k=0? If not, does Kramers' theorem (that say all states must be at least doubly degenerate in a TR-symmetric spin-1/2 system) still hold if the band is not symmetric and therefore E(k)≠E(−k)? (I think yes, because the spin degeneracy guarantees a two-fold degeneracy)

Answer 2

when both time reversal invariance (TT) and parity (PP) are symmetries of the Hamiltonian (H^H^), we will have four fold degeneracy in principle (suppose we have an eigenstate Ψ(r⃗ )Ψ(r→), then TΨ(r⃗ )TΨ(r→), PΨ(r⃗ )PΨ(r→) and PTΨ(r⃗ )PTΨ(r→) are all degenerate unless the eigenstate itself is symmetric)