Entanglement spectrum of a topological phase in one dimension

2009/10/09 by Frank Pollmann, Erez Berg, Ari M. Turner +1 · 1 voice · 9 citations
Physics and Astronomy · #Physics of Superconductivity and Magnetism #Quantum and electron transport phenomena #Quantum many-body systems #cond-mat.str-el

paper · pdf · doi:10.1103/physrevb.81.064439

arxiv published 2009/10/09 · openalex publication_date 2010/02/26 · arxiv updated 2010/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the Haldane phase of S=1 chains is characterized by a double degeneracy of the entanglement spectrum. The degeneracy is protected by a set of symmetries (either the dihedral group of \ensuremathπ rotations about two orthogonal axes, time-reversal symmetry, or bond centered inversion symmetry), and cannot be lifted unless either a phase boundary to another, ``topologically trivial,'' phase is crossed, or the symmetry is broken. More generally, these results offer a scheme to classify gapped phases of one-dimensional systems. Physically, the degeneracy of the entanglement spectrum can be observed by adiabatically weakening a bond to zero, which leaves the two disconnected halves of the system in a finitely entangled state.

Citations

• String Order and Symmetries in Quantum Spin Lattices
• Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model

Cited by

• Projective Representations, Bogomolov Multiplier, and Their Applications in Physics
• Parent Hamiltonian and intrinsic phase transition in non-Hermitian photonic systems
• Instability of the Haldane Phase: Roles of Charge Fluctuations and Hund's Coupling
• Electrostatically-induced topological phase transitions in polyacetylene molecules
• Charge pumps, pivot Hamiltonians and symmetry-protected topological phases
• Homomorphism, substructure and ideal: Elementary but rigorous aspects of renormalization group or hierarchical structure of topological orders
• Topological crystals and soliton lattices in a Gross-Neveu model with Hilbert-space fragmentation
• Diagnosing 2D symmetry protected topological states via mixed state anomaly
• Higgs-confinement transitions in QCD from symmetry protected topological phases

Discussions

• Following this classic paper on the relation between symmetry protected toplogical phases (SPTs), matrix product states (MPS's), and entanglement spectrum degeneracies, I have a very basic confusion a [stackexchange, 3 points]