  We establish exact relations between the winding of "energy" (eigenvalue of
Hamiltonian) on the complex plane as momentum traverses the Brillouin zone with
periodic boundary condition, and the presence of "skin modes" with open
boundary condition in non-hermitian systems. We show that the nonzero winding
with respect to any complex reference energy leads to the presence of skin
modes, and vice versa. We also show that both the nonzero winding and the
presence of skin modes share the common physical origin that is the
non-vanishing current through the system.
