Duality, criticality, anomaly, and topology in quantum spin-1 chains

In quantum spin-1 chains, there is a nonlocal unitary transformation known as the Kennedy-Tasaki ${U}_{\mathrm{KT}}$, which defines duality between Haldane phase and ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry-breaking phase. this paper, we find that ${U}_{\mathrm{KT}}$ also topological Ising critical trivial phase, provides ``hidden symmetry breaking'' interpretation of criticality. Moreover, since relates different phases matter, argue model with self-duality (i.e., invariant under ${U}_{\mathrm{KT}}$) natural to be at or multicritical point. We study concrete examples demonstrate argument. particular, when $H$ Hamiltonian antiferromagnetic Heisenberg chain, prove self-dual $H+{U}_{\mathrm{KT}}H{U}_{\mathrm{KT}}$ exactly equivalent gapless spin-$1/2$ XY implies an emergent anomaly. On other hand, show are dual each meet point indeed self-dual.