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.

16 pages, 10 figures

References in corpus (12)

• String order and symmetries in quantum spin lattices
• One-Dimensional Symmetry Protected Topological Phases and their Transitions
• Spin nematic correlations in bilinear-biquadratic S=1 spin chains
• Intrinsically Gapless Topological Phases
• Symmetric-Gapped Surface States of Fractional Topological Insulators
• Class of exactly solvable SO(n) symmetric spin chains with matrix product ground states
• Stiefel Liquids: Possible Non-Lagrangian Quantum Criticality from Intertwined Orders
• Twisted boundary condition and Lieb-Schultz-Mattis ingappability for discrete symmetries
• Pivot Hamiltonians as generators of symmetry and entanglement
• Topological and symmetry-enriched random quantum critical points
• Rigorous Solution of the Spin-1 Quantum Ising Model with Single-ion Anisotropy
• Disentangling topological degeneracy in the entanglement spectrum of one-dimensional symmetry protected topological phases

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• 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
• Generalized Kramers-Wannier Duality from Bilinear Phase Map
• Topological pump of $SU(Q)$ quantum chain and Diophantine equation
• Symmetry-protected topological phases, conformal criticalities, and duality in exactly solvable SO($n$) spin chains
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