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Mixed-state Phases from Higher-order SSPTs with Kramers-Wannier Symmetry

Aswin Parayil Mana, Zijian Song, Fei Yan, Tzu-Chieh Wei

Abstract

Mixed-state phases have recently attracted significant attention as a generalization beyond their pure-state counterparts. Prominent examples include mixed-state symmetry-protected topological (mSPT) phases and the strong-to-weak symmetry breaking (SWSSB) phases. It has been shown recently that mSPT phases admit a holographic dual description in terms of higher-order subsystem SPT phases. In this work, we investigate the mixed-state phases obtained by tracing out the bulk degrees of freedom of higher-order subsystem SPT phases protected by non-invertible symmetries. We find that the resulting mixed states exhibit the coexistence of the symmetry-protected topological order and SWSSB. We also use the interface as a probe to characterize the mixed state phases, and specifically, when there is no local modification to preserve the symmetries across the interface, the two sides of the interface are in distinct phases.

Mixed-state Phases from Higher-order SSPTs with Kramers-Wannier Symmetry

Abstract

Mixed-state phases have recently attracted significant attention as a generalization beyond their pure-state counterparts. Prominent examples include mixed-state symmetry-protected topological (mSPT) phases and the strong-to-weak symmetry breaking (SWSSB) phases. It has been shown recently that mSPT phases admit a holographic dual description in terms of higher-order subsystem SPT phases. In this work, we investigate the mixed-state phases obtained by tracing out the bulk degrees of freedom of higher-order subsystem SPT phases protected by non-invertible symmetries. We find that the resulting mixed states exhibit the coexistence of the symmetry-protected topological order and SWSSB. We also use the interface as a probe to characterize the mixed state phases, and specifically, when there is no local modification to preserve the symmetries across the interface, the two sides of the interface are in distinct phases.
Paper Structure (52 sections, 134 equations, 4 figures, 8 tables)

This paper contains 52 sections, 134 equations, 4 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Review of the holographic picture for mSPT phases
  3. Toy model of DASPTs
  4. Overview of the 2D cluster state
  5. 1D DASPT from 2D cluster state
  6. DASPTs from non-invertible SSPTs
  7. Overview of 2D blue state
  8. 1D DASPT from 2D blue state
  9. 1D DASPT from 2D blue$^{k_0;k_1}$ state
  10. Interfaces between different phases
  11. Interface between trivial and $\rho^{\rm A}_{\text{1D-DASPT}}$ states: different phases w.r.t. invertible symmetries
  12. Interface between $\rho^{\rm A}_{\text{1D-DASPT}}$ and $\rho^{\rm A}_{\text{blue}}$ states: same phase w.r.t. invertible symmetries
  13. Interface between $\rho^{\rm A}_{\text{1D-DASPT}}$ and $\rho^{\rm A}_{\text{blue}^{k_0;k_1}}$ states: same phase w.r.t invertible symmetries
  14. Conclusion
  15. Extra examples
  16. ...and 37 more sections

Figures (4)

  • Figure 1: Arrangement of qubits for the 2D SSPT Hamiltonian in \ref{['eq:H2DSSPT']}. Each unit cell contains two types ($L$ and $R$) of $\sigma$ and $\tau$ spins. The green regions specify the different interactions in the SSPT Hamiltonian \ref{['eq:H2DSSPT']}. Each yellow region contains two $L$ and $R$ spin chains.
  • Figure 2: Bipartite lattice with open boundary condition in the $y$ direction and periodic boundary condition in the $x$ direction. The top boundary is at $y=k_0+\frac{1}{2}$. We trace out all the d.o.f. except those on region $\rm A$ defined to be the rows at $y=k_0$ and $y=k_0+\frac{1}{2}$. Green region contains the d.o.f. that we trace over. One can set the boundary row reference $k_0$ to zero without loss of generality.
  • Figure 3: Interface between two 2D states $\ket{\Psi_{\rm I}}$ and $\ket{\Psi_{\rm II}}$. We impose Dirichlet boundary condition with the boundary state to be the product state $\ket{+}^{\Delta_{v_b}\cap \rm A}$. After tracing out the bulk d.o.f. inside the green region, we get interfaces between the corresponding 1D states. The left vertical line is at $x=l+\frac{1}{2}$ and the right vertical line is at $L_x+\frac{1}{2}$.
  • Figure 4: The vertices in the region $\rm A$ are at $y=k$ and $y=k+\frac{1}{2}$. The green line connecting red and blue vertices in region $\rm A$ indicate the $CZ$ gates that remain after tracing out $\rm A^c$. The green region contains the d.o.f. that we trace over.