Detecting Higher Berry Phases via Boundary Scattering

We introduce a boundary–scattering framework to detect higher Berry phases without evaluating bulk Berry curvatures. For a one–dimensional family with parameter space $X \simeq S^3$, the topological response is encoded entirely in the single–particle reflection matrix $R(\lambda)$. Through dimensional reduction, we show that the bulk higher Berry phase (the second Chern number) equals the third winding number $\nu_3[R]$ of the boundary reflection, yielding a purely scattering–based invariant. We develop a field–theoretic treatment and a lattice prescription that computes $R$ from the boundary Green's function via a continued–fraction scheme. Computing $\nu_3[R]$ produces an integer invariant interpretable as the Chern number pumped per cycle. We further identify a "Chern–number current" that peaks when boundary modes transmit in an open–boundary setup. Numerical studies show the quantized response is robust to moderate onsite disorder.