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arXiv:2503.10311 · Physical review. B./Physical review. B

Strong-to-weak spontaneous symmetry breaking and average symmetry protected topological order in the doubled Hilbert space

Yoshihito Kuno, Takahiro Orito, Ikuo Ichinose

Published May 15, 2025• 3 citations
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Abstract

There are two kinds of symmetries in quantum mixed states (QMSs) -- called strong and weak symmetries-- and studying their spontaneous symmetry breaking (SSB) pattern helps classify QMSs, which is an ongoing important issue. The authors study here a decohered cluster model with ${Z}_{2}\ensuremath{\bigotimes}{Z}_{2}$ symmetry and uncover two kinds of symmetry-protected topological (SPT) mixed states. Using doubled Hilbert formalism, they demonstrate a transition between the two states, named strong and double average SPT states, where the latter features coexisting SPT and SSB.

DOI
10.1103/physrevb.111.174110

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There are two kinds of symmetries in quantum mixed states (QMSs) -- called strong and weak symmetries-- and studying their spontaneous symmetry breaking (SSB) pattern helps classify QMSs, which is an ongoing important issue. The authors study here a decohered cluster model with Z2\ensuremath⨂Z2{Z}_{2}\ensuremath{\bigotimes}{Z}_{2}Z2​\ensuremath⨂Z2​ symmetry and uncover two kinds of symmetry-protected topological (SPT) mixed states. Using doubled Hilbert formalism, they demonstrate a transition between the two states, named strong and double average SPT states, where the latter features coexisting SPT and SSB.
The interplay of symmetry and topology in quantum many-body mixed states has recently garnered significant interest. In a phenomenon not seen in pure states, mixed states can exhibit average symmetries—symmetries that act on component states while leaving the ensemble invariant. In this work, we systematically characterize symmetry-protected topological (SPT) phases of short-range entangled (SRE) mixed states of spin systems—protected by both average and exact symmetries—by studying their pure Choi states in a doubled Hilbert space, where the familiar notions and tools for SRE and SPT pure states apply. This advantage of the doubled space comes with a price: extra symmetries as well as subtleties around how hermiticity and positivity of the original density matrix constrain the possible SPT invariants. Nevertheless, the doubled-space perspective allows us to obtain a systematic classification of mixed-state SPT (MSPT) phases. We also investigate the robustness of MSPT invariants under symmetric finite-depth quantum channels, the bulk-boundary correspondence for MSPT phases, and the consequences of the MSPT invariants for the separability of mixed states and the symmetry-protected sign problem. In addition to MSPT phases, we study the patterns of spontaneous symmetry breaking (SSB) of mixed states, including the phenomenon of exact-to-average SSB, and the order parameters that detect them. Mixed-state SSB is related to an ingappability constraint on symmetric Lindbladian dynamics.
The fate of average symmetry-protected topological (ASPT) states based on the one-dimensional cluster state is investigated by using the Choi mapping, namely, the doubled Hilbert-space formalism. By introducing two effective spins, τττ and ηηη, on each doubled rung, we obtain a transparent representation of the ASPT parent Hamiltonian and systematically identify the conserved stabilizers and the corresponding orders. We further study symmetry-preserving operations that deform the ASPT by combining qualitative effective-Hamiltonian and systematic parent-Hamiltonian analyses with numerical matrix-product-state filtering calculations. We find that although the ASPT remains robust over a broad range of decoherence strengths, strong-to-weak spontaneous symmetry breaking (SWSSB) becomes exact in the maximal-decoherence limit. In addition, we consider symmetry preserving no-click postselection, which removes the exact conservation law supporting the Rényi-2 string order of the ASPT and drives the system toward weak-symmetry spontaneous symmetry breaking (WSSB). Again, the ASPT remains robust over a broad range of postselection strengths, with WSSB becoming pronounced only near the projective limit and exact at the limit itself.
We develop a general framework for studying phases of mixed states with strong and weak symmetries, including non-invertible or categorical symmetries. The central idea is to consider a purification of the mixed state density matrix, which lives in a doubled Hilbert space. We propose a systematic classification of phases in this doubled Hilbert space, relying crucially on the Symmetry Topological Field Theory (SymTFT) approach. This framework applies not only to group symmetries but also, importantly, to non-invertible symmetries. We illustrate the approach in 1+1d to classify phases with strong (non-)invertible symmetries, which include strong-to-weak spontaneous symmetry breaking (SWSSB) phases and mixed strong/weak symmetry-protected topological phases (SPTs). We also develop an approach for studying symmetries that involve a combination of strong and weak symmetries. A noteworthy example of this has weak non-invertible Kramers-Wannier duality symmetry and strong Z2\mathbb{Z}_2Z2​ symmetry. The continuum description is complemented by a lattice model analysis informed by the SymTFT framework.
We study the effect of subsystem decoherence to an extended cluster state which is a symmetry protected topological (SPT) phase. The model includes many subsystem Z2Z_2Z2​ symmetries. We report that subsystem decoherence induces local charge fluctuations, leading to a mixed SPT state in the unaffected subsystems. If we start from the extended cluster state, hierarchical mixed-state SPT phases emerge in response to step-by-step subsystem decoherences. These mixed-state SPT phases keep strong symmetries the symmetry of which is protecting symmetries for the initial cluster SPT. Moreover, these SPTs can be characterized by Rényi-2 string orders. Then, as the subsystems are progressively decohered, the hierarchy of mixed-state SPT phases terminates in a Z2Z_2Z2​ strong-to-weak spontaneous symmetry breaking (SWSSB) state on the final remaining subsystem, where a long-range entangled state appears, namely a glassy Greenberger-Horne-Zeilinger (GHZ) state. Our work demonstrates that decoherence is not merely a destructive process, but can induce and organize series of nontrivial mixed-states. This reveals a systematic route from mixed-state SPT order to SWSSB with the glassy GHZ-type long-range entanglement.
We explore the effect of decoherence on many-body mixed states from a purification perspective. Here, quantum channels map to unitary transformations acting on a purified state defined within an extended Hilbert space. First, we exploit this approach to analyze the phenomenon of spontaneous strong-to-weak symmetry breaking (sw-SSB) triggered by strongly symmetric local quantum channels. We find that sw-SSB for mixed states relates to proximity to a symmetry-protected topological (SPT) order in the purified state. Remarkably, the measurement-induced long-range order in the purified SPT state, as characterized by the R\'enyi-2 correlator, mirrors the ensuing long-range order in the mixed state due to sw-SSB. Moreover, we establish a 1-to-1 correspondence between various sw-SSB order parameters for the mixed state, and strange correlators in the purification, which signify the SPT order. This purification perspective is further extended to explore intrinsic mixed-state topological order and decoherent symmetry-protected topological phases.
Edge bit in an average symmetry protected topological (ASPT) mixed state is studied. The state is protected by one strong Z2Z_2Z2​ and one weak (average) Z2Z_2Z2​ symmetries. As analogous objects of pure symmetry protected topological (SPT) states, the ASPT possesses edge bits. In particular, the analogous operator response exists, that is, symmetry fractionalization. The fractionalization preserves the presence of the ASPT in the bulk, and the fractionalized edge operators acting on the edge bits of the ASPT. %analogous to the ones in the pure SPTs. In this work, based on the cluster model and by employing Choi mapping, we discuss generic features of the edge bits and numerically clarify the behavior of the edge bits and their robustness for varying decoherence and perturbative interactions. By using an operator-space mutual information (OSMI), we track the flow of quantum correlations between the two edges. Remarkably, even in the ASPT regime, a finite portion of the initial edge-to-edge correlation survives.