Intrinsic Zak Phase of Solitons: A Geometric Correspondence with the Center-of-Mass Position

Abstract

The center-of-mass position of a nonlinear soliton in a lattice is commonly attributed to the interplay between nonlinearity and lattice confinement. Here, we reveal its geometric origin by establishing an exact relation between the soliton position and an intrinsic Zak phase. Unlike the Zak phase of a linear Bloch band, this intrinsic Zak phase arises from the nonlinear self-consistent band of the lattice reconstructed by the soliton density. We construct a composite periodic function to define this phase and demonstrate that its topological winding during nonlinear Thouless pumping dictates the quantized soliton displacement. Our results establish band geometry as a fundamental descriptor of nonlinear soliton position in a lattice and provide a geometric framework for their topological transport.