A gauge transformation can be performed | n ~ ~ ( R ) ⟩ ⟩ = e − − i β β ( R ) | n ( R ) ⟩ ⟩ to a new set of states that differ from the original ones only by an R -dependent phase factor. This modifies the open-path Berry phase to be γ γ ~ ~ n ( t ) = γ γ n ( t ) + β β ( t ) − − β β ( 0 ) . For a closed path, continuity requires that β β ( T ) − − β β ( 0 ) = 2 π π m ( m an integer), and it follows that γ γ n is invariant, modulo 2 π π , under an arbitrary gauge transformation.