Duality, Criticality, Anomaly, and Topology in Quantum Spin-1 Chains

Abstract

In quantum spin-1 chains, there is a nonlocal unitary transformation known as the Kennedy-Tasaki transformation $U_{\text{KT}}$, which defines a duality between the Haldane phase and the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry-breaking phase. In this paper, we find that $U_{\text{KT}}$ also defines a duality between a topological Ising critical phase and a trivial Ising critical phase, which provides a "hidden symmetry breaking" interpretation for the topological criticality. Moreover, since the duality relates different phases of matter, we argue that a model with self-duality (i.e., invariant under $U_{\text{KT}}$) 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 $H+U_{\text{KT}}H{U}_{\text{KT}}$ 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 criticalities that are dual to each other meet at a multicritical point which is indeed self-dual.

arXiv comment16 pages, 10 figures

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• Decorated Defect Construction of Gapless-SPT States
• Universal entanglement spectrum in gapless symmetry protected topological states
• Intrinsically/Purely Gapless-SPT from Non-Invertible Duality Transformations
• Generating Lattice Non-invertible Symmetries
• Quantum phase transition between topologically distinct quantum critical points
• Observation of counterflow superfluidity in a two-component Mott insulator
• Definition and Classification of Fermi Surface Anomalies
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• Topological pump of $SU(Q)$ quantum chain and Diophantine equation
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• Incommensurate gapless ferromagnetism connecting competing symmetry-enriched deconfined quantum phase transitions
• Duality and hidden symmetry breaking in the q-deformed Affleck-Kennedy-Lieb-Tasaki model
• Topology and edge modes surviving criticality in non-Hermitian Floquet systems