Symmetry of the Bloch Ground State

[Jan Paniev]

I was reading Hasan & Kane's review on topological insulators and right in the beginning, page 3, they say that the Bloch ground state is U(N) invariant. I do not see that. Would anyone be able to show it or point to a reference?

Thanks,
Jan.

 

[DrDu]

Any Hartree or Hartree Fock wavefunction is invariant with respect to a separate unitary U(N) and U(M) transformation of the N occupied and M unoccupied orbitals, respectively, among each other. The Blochfunctions are special insofar as they diagonalize the Hartree Fock effective single particle operators which have no fundamental significance, however.
A popular unitary transformation transforms the Bloch functions to the so called localized Brillouin functions (I hope, I remember the name correctly).

 

[Jan Paniev]

Thanks a lot for the answer. Would you know in which book I would be able to find the details?

Jan.

 

[DrDu]

Sorry, the name is not Brillouin function but Wannier function.
They should be named in any book on solid state theory.
The invariance of many particle wavefunctions in quantum chemistry is discussed in depth in
Roy McWeeny, Methods of Molecular Quantum Mechanics, 1989.
Basically it is nothing more than the invariance of the determinant under unitary transformations as prooved in elementary linear algebra.