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Quantum criticality in open quantum systems from the purification perspective

Yuchen Guo, Shuo Yang

Abstract

Open quantum systems host mixed-state phases that go beyond the symmetry-protected topological and spontaneous symmetry-breaking paradigms established for closed, pure-state systems. Developing a unified and physically transparent classification of such phases remains a central challenge. In this work, we introduce a purification-based framework that systematically characterizes all mixed-state phases in one-dimensional systems with $\mathbb{Z}_2^σ \times \mathbb{Z}_2^τ$ symmetry. By introducing an ancillary $κ$ chain and employing decorated domain-wall constructions, we derive eight purified fixed-point Hamiltonians labeled by topological indices $(μ_{στ},μ_{τκ},μ_{κσ}) \in \{\pm1\}^3$. Tracing out the ancilla recovers the full structure of mixed-state phases, including symmetric, strong-to-weak spontaneous symmetry breaking, average symmetry-protected topological phases, and their nontrivial combinations. Interpolations between the eight fixed points naturally define a three-dimensional phase diagram with a cube geometry. The edges correspond to elementary transitions associated with single topological indices, while the faces host intermediate phases arising from competing domain-wall decorations. Along the edges, we identify a class of critical behavior that connects distinct strong-to-weak symmetry-breaking patterns associated with distinct strong subgroups, highlighting a mechanism unique to mixed-state settings. Large-scale tensor-network simulations reveal a rich phase structure, including pyramid-shaped symmetry-breaking regions and a fully symmetry-broken phase at the cube center. Overall, our purification approach provides a geometrically transparent and physically complete classification of mixed-state phases, unified with a single $\mathbb{Z}_2^σ \times \mathbb{Z}_2^τ \times \mathbb{Z}_2^κ$ model.

Quantum criticality in open quantum systems from the purification perspective

Abstract

Open quantum systems host mixed-state phases that go beyond the symmetry-protected topological and spontaneous symmetry-breaking paradigms established for closed, pure-state systems. Developing a unified and physically transparent classification of such phases remains a central challenge. In this work, we introduce a purification-based framework that systematically characterizes all mixed-state phases in one-dimensional systems with symmetry. By introducing an ancillary chain and employing decorated domain-wall constructions, we derive eight purified fixed-point Hamiltonians labeled by topological indices . Tracing out the ancilla recovers the full structure of mixed-state phases, including symmetric, strong-to-weak spontaneous symmetry breaking, average symmetry-protected topological phases, and their nontrivial combinations. Interpolations between the eight fixed points naturally define a three-dimensional phase diagram with a cube geometry. The edges correspond to elementary transitions associated with single topological indices, while the faces host intermediate phases arising from competing domain-wall decorations. Along the edges, we identify a class of critical behavior that connects distinct strong-to-weak symmetry-breaking patterns associated with distinct strong subgroups, highlighting a mechanism unique to mixed-state settings. Large-scale tensor-network simulations reveal a rich phase structure, including pyramid-shaped symmetry-breaking regions and a fully symmetry-broken phase at the cube center. Overall, our purification approach provides a geometrically transparent and physically complete classification of mixed-state phases, unified with a single model.
Paper Structure (17 sections, 46 equations, 1 figure, 1 table)

This paper contains 17 sections, 46 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Purification of mixed-state quantum phases
  3. Trivial weakly symmetric states
  4. SWSSB
  5. Physical picture
  6. Fidelity correlator
  7. Rényi-$2$ correlator
  8. ASPT
  9. Explicit weakening of $G$ from an SPT state
  10. Decorated domain-wall construction from a trivial mixed state
  11. Double ASPT
  12. SWSSB from an SPT state
  13. Decorated domain-wall construction from an SWSSB state
  14. Eight mixed-state phases from a minimal $\mathbb{Z}_2^{\sigma}\times\mathbb{Z}_2^{\tau}\times\mathbb{Z}_2^{\kappa}$ purified model
  15. Purified model
  16. ...and 2 more sections

Figures (1)

  • Figure 1: Quantum circuit to generate $\ket{\psi_{\sigma\tau\kappa}^{(-1, -1, +1)}}$. Starting from the product state $\prod_i \ket{\uparrow}_i$, one layer of Hadamard gates ($H$) generates the symmetric trivial state $\ket{\psi_{\sigma\tau\kappa}^{(+1, +1, +1)}}=\prod_i \ket{\rightarrow}_i$, then two layers of CZ gates $U_{\sigma\tau}^{\rm DW}$ (dashed line) and $U_{\tau\kappa}^{\rm DW}$ (dotted line) subsequently impose the mixed anomaly between corresponding pairs, leading to $\ket{\psi_{\sigma\tau\kappa}^{(-1, -1, +1)}}$.