Proof that the maximally localized Wannier functions are real
  The maximally localized Wannier functions play a very important role in the study of chemical bonding, ballistic transport and strongly-correlated system, etc. A significant development in this branch was made in 1997 and conjectured that the maximally localized Wannier functions are real. In this paper, we prove the conjecture. A key to this proof is that the real parts of complex Wannier functions equal to the algebraic average of two real Wannier functions and the spread functional of one of the two real Wannier functions is less than one of complex Wannier functions. If one starts with initial real Wannier functions, the gradients of spread functional to get the maximally localized Wannier functions are only calculated in half of Brillouin zone and more localized Wannier functions are obtained. 
Sangryol Ri, Suil Ri
cond-mat.mtrl-sci