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  • $\begingroup$ In 2005.08994 (which I haven't studied), is the gauge field just a background field? A background field is prescribed, so changing the field changes the model -- like changing the coefficients in the Hamiltonian, which is what the Peierls substitution seems to be doing. You can still have a family of models that is collectively gauge-invariant, in the sense that a gauge transform of the background field permutes the models in that family with each other, and the proof uses a matter-only "gauge tranfsormation" like you highlighted. Is that the paper's context? $\endgroup$
    – Chiral Anomaly
    Commented May 14, 2021 at 13:23
  • $\begingroup$ The gauge fields in both papers are cavity fields. 2005.08994 covers a tight-binding model within a photon-populated cavity. $\endgroup$
    – Ruslan Mushkaev
    Commented May 14, 2021 at 13:31