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 transformation U₊ₓ, which defines a duality between the Haldane phase and the Z₂Z₂ symmetry-breaking phase. In this paper, we find that U₊ₓ also defines a duality between a topological Ising critical phase and a trivial Ising critical phase, which provides a ``hidden symmetry breaking'' interpretation of the topological criticality. Moreover, since the duality relates different phases of matter, we argue that a model with self-duality (i. e. , invariant under U₊ₓ) is natural to be at a critical or multicritical point. We study concrete examples to demonstrate this argument. In particular, when H is the Hamiltonian of the spin-1 antiferromagnetic Heisenberg chain, we prove that the self-dual model H+U₊ₓHU₊ₓ is exactly equivalent to a gapless spin-1/2 XY chain, which also implies an emergent quantum anomaly. On the other hand, we show that the topological and trivial Ising critical phases that are dual to each other meet at a multicritical point which is indeed self-dual.