A two-leg ladder is a nice place to discuss Z₂× Z₂ symmetry with sufficient numerical accuracy. Using a Z₄ Berry phases, several phases of bosonic ladders are discussed in relation to the bulk-edge correspondence. The paper “Z₂× Z₂ symmetry and Z₄ Berry phase of bosonic ladders” by Yoshihito Kuno and Yasuhiro Hatsugai, has been accepted for publication in Phys. Rev. A on 10 July 2023.　It is published as Phys. Rev. A 108, 013319 – Published 27 July 2023. See also arXiv:2304.09533