A gapped one-particle density matrix does not determine the topological phase

Local topological markers computed from the band-flattened one-particle density matrix (OPDM) have been proposed as diagnostics of interacting topological phases, with the statement that an open OPDM gap suffices; their authors noted that a failure would need a gapped-OPDM state that cannot be joined to a Gaussian state without closing the gap, and did not know such a state. We construct such states. One layer of commuting, on-site, charge-conserving gates, each coupling an orbital to an ancilla pair holding one fermion, maps the OPDM ρ of any state to λ²ρ + (1 − λ²) diag ρ on the original orbitals, with λ the cosine of the gate angle, at constant many-body spectrum and Hall conductance. At full dephasing, for an integer mean charge on every site, the dressed state has the two-point functions of an on-site product state, so two gapped ground states with the same gapped, on-site OPDM can have Hall conductances e²/h and 0. For translation-invariant two-band insulators the marker drops to zero at an exact threshold; certified witnesses in the Qi–Wu–Zhang model and, with the time-reversal-symmetric version of the dressing, in the Kitaev chain have OPDM gaps above 0.33 and 0.13 and markers 0 inside nontrivial phases. The dressed Hamiltonians are free models rotated by an on-site unitary, strongly interacting only in the bare-fermion basis; the bare-fermion OPDM is not covariant under such unitaries. Statements restricted to weak interactions, or to states joined to a Gaussian state within their phase, are not contradicted.