A Real-Space Formulation of the Zak Phase via Weyl m-Functions

We establish a new, real-space formula for the Zak phase for one dimensional periodic Jacobi operators in terms of the Weyl $m_{+}$-function that does not rely on Floquet-Bloch theory. This novel representation highlights the dependence of the Zak phase on boundary terms. Moreover, we show how to recover the classical quantisation of the Zak phase for periodic Jacobi operators with inversion symmetric fundamental cells.