Detection of symmetry-protected topological phases in one dimension

2012/04/03 by Frank Pollmann, Ari M. Turner · 360 citations
Mathematics · Physics and Astronomy · #Advanced Condensed Matter Physics #Combinatorics #Discrete symmetry #Geometry #Homogeneous space #Mathematics #Matrix multiplication #Matrix product state #Physics #Physics of Superconductivity and Magnetism #Quantum #Quantum many-body systems #Quantum mechanics #String (physics) #Symmetry (geometry) #Symmetry protected topological order #Theoretical physics #Topological order #Topology (electrical circuits) #cond-mat.str-el

paper · pdf · doi:10.1103/physrevb.86.125441

published in Physical Review B 86(12) (American Physical Society) · 12 pages, 9 Figures

arxiv created 2012/04/03 · openalex publication_date 2012/09/24 · arxiv updated 2014/12/17 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

A topological phase is a phase of matter which cannot be characterized by a local order parameter. It has been shown that gapped symmetric phases in one-dimensional (1D) systems can be completely characterized using tools related to projective representations of the symmetry groups. We explain two ways to detect these symmetry protected topological phases in 1D. First, we give a numerical approach for directly extracting the projective representations from a matrix-product state representation. Second, we derive nonlocal order parameters for time-reversal and inversion symmetry, and discuss a generalized string order for local symmetries for which the regular string-order parameter cannot be applied. We furthermore point out that the nonlocal order parameter for these topological phases is actually related to topological surfaces.

Citations

• Entanglement spectrum of a topological phase in one dimension
• String order and symmetries in quantum spin lattices
• Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model

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• Phases of one-dimensional SU(N) cold atomic Fermi gases—From molecular Luttinger liquids to topological phases
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• Many-Body Topological Invariants for Fermionic Symmetry-Protected Topological Phases

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