Green's-function windings are not phase invariants in interacting fermion chains

Windings of the zero-frequency single-particle Green's function are widely used to diagnose interacting topological phases of one-dimensional fermion chains. Such windings are known to change through Green's-function zeros, or under a change of the Hamiltonian, without a phase transition. We show that the winding of the elementary fermion field is not constant on a gapped symmetric phase even modulo the interacting classification, and even when both states compared have zero-free Green's functions. An exact identity moves a finite-depth circuit of symmetric gates from the field onto the state and the Hamiltonian, so dressing the field is the same as moving the state within its phase at constant many-body spectrum. With such dressings the winding changes by one in the trivial phase of a Majorana chain of class BDI, takes every value modulo eight in the trivial phase of the Fidkowski–Kitaev eight-chain model, and goes from minus one to minus two inside the Kitaev phase, so that even its parity, the Green's-function form of the class-D index, is not a phase label. In the phase of the fully dimerized spinful Peierls–Hubbard chain of Manmana and co-workers it changes by two, also with bond inversion kept. Every path between such endpoints crosses a Green's-function zero, here at constant spectrum, so invariance along zero-free gapped paths is untouched. We also prove sufficient conditions for invariance: Gaussian dressings of any state, finite-component dressings of cluster product states, and Clifford monomial dressings of stabilizer states, the last two modulo eight. A computed winding therefore cannot by itself certify an interacting phase.