A classification of phases of bosonic quantum lattice systems in one dimension

Abstract

We study invertible states of 1d bosonic quantum lattice systems. We show that every invertible 1d state is in a trivial phase: after tensoring with some unentangled ancillas it can be disentangled by a fuzzy analog of a finite-depth quantum circuit. If an invertible state has symmetries, it may be impossible to disentangle it in a way that preserves the symmetries, even after adding unentagled ancillas. We show that in the case of a finite unitary symmetry G the only obstruction is an index valued in degree-2 cohomology of $G$. We show that two invertible $G$-invariant states are in the same phase if and only if their indices coincide.

arXiv commentErrors in the proofs of Lemma 4.5 and Theorem 1 have been corrected. The new version also includes an appendix on a multiplicative version of the Lieb-Robinson bound

References in corpus (9)

• An Area Law for One Dimensional Quantum Systems
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Topological phases of fermions in one dimension
• Quasi-adiabatic Continuation of Quantum States: The Stability of Topological Ground State Degeneracy and Emergent Gauge Invariance
• Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems
• A ${\mathbb{Z}}_2$-index of symmetry protected topological phases with time reversal symmetry for quantum spin chains
• Hall conductance and the statistics of flux insertions in gapped interacting lattice systems
• The classification of symmetry protected topological phases of one-dimensional fermion systems
• Automorphic equivalence preserves the split property

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• Homotopical Foundations of Parametrized Quantum Spin Systems
• An index for two-dimensional SPT states
• Long-range nonstabilizerness and phases of matter
• A unified diagrammatic approach to topological fixed point models
• Disentangling modular Walker-Wang models via fermionic invertible boundaries
• Charge pumps, pivot Hamiltonians and symmetry-protected topological phases
• SPT indices emerging from translation invariance in two dimensional quantum spin systems
• Entanglement area law and Lieb-Schultz-Mattis theorem in long-range interacting systems, and symmetry-enforced long-range entanglement
• Universal Decay of Mutual Information and Conditional Mutual Information in Gapped Pure- and Mixed-State Quantum Matter
• The index of a pair of pure states and the interacting integer quantum Hall effect