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 symmetry. The continuum description is complemented by a lattice model analysis informed by the SymTFT framework.
Symmetry topological field theory (SymTFT) gives a holographic correspondence between systems with a global symmetry and a higher-dimensional topological field theory. In this framework, classification of gapped phases of matter in spacetime dimension 1+1D correspond to classifications of mechanisms to confine the SymTFT by condensing anyons. In this work, we extend these results to characterize gapless symmetry-protected topological states: symmetry-enriched gapless phases or critical points that exhibit edge modes protected by symmetry and topology. We establish a one-to-one correspondence between 1+1D bosonic gSPTs, and partially-confined boundaries of 2+1D SymTFTs. From general physical considerations, we determine the set of data and consistency conditions required to define a 1+1D gSPT, and show that this data precisely matches that of symmetry-preserving partial confinement (or partially gapped boundaries) of 2+1D quantum double models. We illustrate this correspondence through a dimensional reduction (thin-slab) construction, which enables a physically-intuitive derivation of how properties of the gSPT such as edge modes, emergent anomalies, and stability to perturbations arise from the SymTFT perspective.ditions required to define a 1+1D gSPT and show that they fully determine the physics of the gSPT including edge modes and emergent anomaly.
Topological holography is a conjectured correspondence between the symmetry charges and defects of a
D-dimensional system with the anyons in a
(D+1)-dimensional topological order: the symmetry topological field theory (SymTFT). Topological holography is conjectured to capture the topological aspects of symmetry in gapped and gapless systems, with different phases corresponding to different gapped boundaries (anyon condensations) of the SymTFT. This correspondence was previously considered primarily for bosonic systems, excluding many phases of condensed matter systems involving fermionic electrons. In this work, we extend the SymTFT framework to establish a topological holography correspondence for fermionic systems. We demonstrate that this fermionic SymTFT framework captures the known properties of
1+1D fermion gapped phases and critical points, including the classification, edge-modes, and stacking rules of fermionic symmetry-protected topological phases (SPTs), and computation of partition functions of fermionic conformal field theories (CFTs). Beyond merely reproducing known properties, we show that the SymTFT approach can additionally serve as a practical tool for discovering new physics, and use this framework to construct a new example of a fermionic intrinsically gapless SPT phase characterized by an emergent fermionic anomaly.
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
Z2σ×Z2τ symmetry. By introducing an ancillary
κ chain and employing decorated domain-wall constructions, we derive eight purified fixed-point Hamiltonians labeled by topological indices
(μστ,μτκ,μκσ)∈{±1}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
Z2σ×Z2τ×Z2κ model.
Classifying gapless phases of matter remains a fundamental challenge in condensed matter theory. The authors pioneer here a systematic classification of 1+1D bosonic gapless symmetry-protected topological phases via the Symmetry Topological Field Theory (SymTFT) framework. They derive, from physical considerations, the algebra of symmetry operators in a 1+1D bosonic gSPT and show that it precisely matches the structure of condensable algebras in the SymTFT. This study marks the first systematic classification of gapless quantum phases in the SymTFT paradigm.
Symmetry topological field theory (SymTFT), or topological holography, offers a unifying framework for describing quantum phases of matter and phase transitions between them. While this approach has seen remarkable success in describing gapped and gapless pure-state phases in
1+1d, its applicability to open quantum systems remains entirely unexplored. In this work, we propose a natural extension of the SymTFT framework to mixed-state phases by introducing the \textit{symmetry taco}: a bilayer topological order in
2+1d whose folded geometry naturally encapsulates both strong and weak symmetries of the
1+1d theory. We use this perspective to identify a series of correspondences, including a one-to-one map between intrinsically gapless SPTs (igSPTs) and certain gapped SPTs, and a mapping between igSPTs and intrinsically average SPTs (iASPTs) arising in
1+1d mixed states. More broadly, our framework yields a classification of short-range correlated
G-symmetric Choi states in
1+1d, provides a route for systematically generating mixed-state SPTs via local decoherence of igSPTs, and allows us to identify a new mixed-state ``anomaly". Besides folding in mixed-state phases into the SymTFT paradigm, the symmetry taco opens new avenues for exploring dualities, anomalies, and non-equilibrium criticality in mixed-state quantum matter.