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Decorated Defect Construction of Gapless-SPT States

Linhao Li, Masaki Oshikawa, Yunqin Zheng

Abstract

Symmetry protected topological (SPT) phases are one of the simplest, yet nontrivial, gapped systems that go beyond the Landau paradigm. In this work, we study an extension of the notion of SPT for gapless systems, namely, gapless symmetry protected topological states. We construct several simple gapless-SPT models using the decorated defect construction, which allow analytical understanding of non-trivial topological features including the symmetry charge under twisted boundary conditions, and boundary (quasi)-degeneracy under open boundary conditions. We also comment on the stability of the gapless-SPT models under symmetric perturbations, and apply small-scale exact diagonalization when direct analytic understanding is not available.

Decorated Defect Construction of Gapless-SPT States

Abstract

Symmetry protected topological (SPT) phases are one of the simplest, yet nontrivial, gapped systems that go beyond the Landau paradigm. In this work, we study an extension of the notion of SPT for gapless systems, namely, gapless symmetry protected topological states. We construct several simple gapless-SPT models using the decorated defect construction, which allow analytical understanding of non-trivial topological features including the symmetry charge under twisted boundary conditions, and boundary (quasi)-degeneracy under open boundary conditions. We also comment on the stability of the gapless-SPT models under symmetric perturbations, and apply small-scale exact diagonalization when direct analytic understanding is not available.
Paper Structure (54 sections, 117 equations, 6 figures, 4 tables)

This paper contains 54 sections, 117 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction and Summary
  2. Gapped Quantum Matter
  3. Properties of Gapless SPT States
  4. Decorated Defect Construction
  5. Constructing Gapped SPT
  6. Constructing Gapless-SPT
  7. Probing gSPT and igSPT
  8. Organization of the Paper
  9. gSPT: $(1+1)$d Spin Chains With $\mathbb{Z}_2\times \mathbb{Z}_2$ Symmetry
  10. Spin Chain Construction
  11. Domain Wall Decoration
  12. The Spacetime Picture:
  13. The Hamiltonian Picture:
  14. $\mathbb{Z}_2^A\times \mathbb{Z}_2^G$ gSPT
  15. More On $U_{DW}$
  16. ...and 39 more sections

Figures (6)

  • Figure 1: Phase diagram of non-intrinsically and intrinsically gapless-SPT. The horizontal axis is the strength of $G$-defect fluctuation. For the non-intrinsic case (left panel), the $G$-defects can be fully proliferated and one obtains $\Gamma$ gapped SPT. For the intrinsic case, one can only fluctuate the $G$-defects to the critical point. Further increase the fluctuation will not drive the system to $\Gamma$ symmetric gapped SPT phase.
  • Figure 2: Triangulation of 2d spacetime. The black and red solid links are where the background field $g_{ij}=0,1$ respectively. The red dashed line in the dual lattice is the spacetime trajectory of the $\mathbb{Z}_2$ domain wall $[g]$, i.e. $\mathbb{Z}_2$ symmetry defect line. Flatness of $g$ ensures that $[g]$ forms loops.
  • Figure 3: Phase diagram of $\mathbb{Z}_2^G$ Ising CFT (before decoration) and $\mathbb{Z}_2^A\times \mathbb{Z}_2^G$ gSPT (after decoration). The horizontal axis represents the transverse field $\lambda$.
  • Figure 4: The phase diagram of \ref{['KTHam']} when $\theta$=0.
  • Figure 5: $\mathbb{Z}_4^\Gamma$ charge of the ground state under PBC, relative $\mathbb{Z}_4^\Gamma$ charge of the ground state under $\mathbb{Z}_2^A$ TBC, relative $\mathbb{Z}_2^A$ charges of the ground state under $\mathbb{Z}_4^\Gamma$ TBC, and the gap between the ground state and first excited state under PBC and two TBC's. The horizontal axis is the perturbation strength \ref{['Z4perturbation']}. The system size is $L=11$.
  • ...and 1 more figures