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Entanglement spectrum of a topological phase in one dimension

Frank Pollmann, Erez Berg, Ari M. Turner, Masaki Oshikawa

TL;DR

The paper investigates symmetry-protected topological order in one-dimensional spin-1 systems by linking the Haldane phase to a twofold-degenerate entanglement spectrum. Using matrix-product states and the projective representations of protecting symmetries, it shows that inversion, time-reversal, or dihedral D2 symmetries enforce a nontrivial degeneracy that persists until a phase transition. The authors illustrate with AKLT-like constructions and iTEBD simulations, and they outline a general scheme to classify all gapped 1D phases via the structure of these projective representations, including extensions to DM interactions and related bosonic models. They also connect the degeneracy to a measurable residual entanglement of $ \ln(2) $ upon adiabatic bond weakening, offering a potential experimental signature of the Haldane phase.

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 $π$-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.

Entanglement spectrum of a topological phase in one dimension

TL;DR

The paper investigates symmetry-protected topological order in one-dimensional spin-1 systems by linking the Haldane phase to a twofold-degenerate entanglement spectrum. Using matrix-product states and the projective representations of protecting symmetries, it shows that inversion, time-reversal, or dihedral D2 symmetries enforce a nontrivial degeneracy that persists until a phase transition. The authors illustrate with AKLT-like constructions and iTEBD simulations, and they outline a general scheme to classify all gapped 1D phases via the structure of these projective representations, including extensions to DM interactions and related bosonic models. They also connect the degeneracy to a measurable residual entanglement of upon adiabatic bond weakening, offering a potential experimental signature of the Haldane phase.

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 -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.

Paper Structure

This paper contains 15 sections, 35 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model Hamiltonians
  3. Matrix product state representation
  4. Definitions
  5. Symmetries in matrix product states
  6. Degeneracies in the entanglement spectrum
  7. Inversion symmetry
  8. Time reversal symmetry
  9. Sets of Rotations
  10. Examples
  11. Classification of gapped phases in one dimension
  12. Adiabatic bond weakening
  13. Summary
  14. Generalization of the symmetry-protected Haldane phase
  15. Schmidt decomposition of a $\phi _{\mathcal{I}}=\pi$ state

Figures (4)

  • Figure 1: The colormaps show the entanglement entropy $S$ for different spin-1 models: Panel (a) shows the data for Hamiltonian $H_0$ in ( \ref{['H0']}), panel (b) for $H_0$ plus a term which breaks the $e^{-i\pi S^{y}}\times$TR symmetry [Eq. (\ref{['eq:break_TR']})], and panel (c) for $H_0$ plus a term which breaks the $e^{-i\pi S^{y}}\times$TR and inversion symmetry [Eq. (\ref{['eq:break_inv']})]. The blue lines indicate a diverging entanglement entropy as a signature of a continuous phase transition. The phase diagrams contain four different phases: A trivial insulating phase (TRI) for large $U_{zz}$, two symmetry breaking antiferromagnetic phases $Z_2^z$ and $Z_2^y$, and a Haldane phase (which is absent in the last panel).
  • Figure 2: Entanglement spectrum of Hamiltonian $H_0$ in (\ref{['H0']}) for $B_x=0$ (only the lower part of the spectrum is shown). The dots show the multiplicity of the Schmidt values, which is even in the entire Haldane phase.
  • Figure 3: The colormaps show the difference between the two largest Schmidt values $|\lambda_1-\lambda_2|$ for different spin-1 models. Panel (a) corresponds to the original Hamiltonian $H_0$ in (\ref{['H0']}), panel (b) to $H_0$ plus a term that breaks the time reversal symmetry [Eq. (\ref{['eq:break_TR']})], and panel (c) to $H_0$ plus a term which breaks time reversal and inversion symmetry [Eq. (\ref{['eq:break_inv']})]. The quantity $|\lambda_1-\lambda_2|$ is zero only in the Haldane phase.
  • Figure 4: Half-chain entanglement entropy of the model Hamiltonian (\ref{['H0']}) at a bond which is slowly weakened as a function of time $J_{\text{weak}}=J-t\Gamma$ and $\Gamma=J^2/40$. The entanglement entropy of the resulting state is $\ln2$ in the Haldane phase and zero otherwise.