The Symmetry Taco: Equivalences between Gapped, Gapless, and Mixed-State SPTs

TL;DR

The paper extends the symmetry topological field theory (SymTFT) framework to mixed-state 1+1d SPTs by introducing the symmetry taco (SymTaco), a folded bilayer 2+1d topological order that encodes strong and weak symmetries. It establishes a tetraptych of correspondences linking intrinsically gapless SPTs (igSPTs), folded gapped SPTs, and intrinsically average SPTs (iASPTs) through anyon condensation and Choi-state dualities, enabling a unified holographic description of gapped, gapless, and mixed-state phases. A central insight is that folding imposes positivity and Hermiticity constraints on density matrices, which precisely match folded-boundary data in the SymTaco, thus connecting igSPTs, iASPTs, and folded SPTs. The framework yields practical constructions (e.g., stabilizer codes for Abelian cases), reveals a new mixed-state anomaly, and provides systematic routes to generate mixed-state phases from known pure-state SPTs, with broad implications for dualities, gauging, and non-equilibrium criticality in open quantum matter.

Abstract

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 $1+1$d, 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 $2+1$d whose folded geometry naturally encapsulates both strong and weak symmetries of the $1+1$d 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 $1+1$d mixed states. More broadly, our framework yields a classification of short-range correlated $G$-symmetric Choi states in $1+1$d, 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.

Figures (6)

• Figure 1: The Symmetry Taco (symtaco) framework: (a) starting with a $2+1$d topological order described by the quantum double $\mathcal{D}(G)$, condense a subset of anyons--specified by a condensable subalgebra $\mathcal{A}$--in a subregion to obtain a new topological order $\mathcal{D}(G)/\mathcal{A}$. (b) Folding this theory along the mirror axis (dashed black line) produces the SymTaco. This setup describes gapped boundaries from the folded $\mathcal{D}(G) \times \overline{\mathcal{D}(G)} \cong \mathcal{D}(G\times G)$ topological order to vacuum. Such folded boundaries characterize a subset of $1+1$d gapped phases with $G\times G$ symmetry. (c) Breaking the SymTaco in half along the dashed black line produces a gapless boundary from the $\mathcal{D}(G)$ topological order to vacuum. Such boundaries characterize $G$-symmetric intrinsically gapless SPT phases. (d) Constraints imposed by folding along the dashed black line precisely correspond to the positivity and Hermiticity constraints on Choi states of density matrices. In the Choi space, folding corresponds to maximal decoherence which couples the two layers, represented here via the bold red lines. The SymTaco hence provides the SymTFT for $1+1$d $G$-symmetric mixed-state SPTs. In all cases, the symmetry boundary (from $\mathcal{D}(G)$ to vacuum) is given by the canonical charge condensed boundary. See the summary of results \ref{['sec:summary']} for details.
• Figure 2: (a) Thin-slab construction of SymTFT for general $1+1$d phases. (b) Illustration of bulk-boundary correspondence for symmetry operator $\prod \mathcal{X}_i$ and local order parameter $\mathcal{Z}_i^k$ for $\mathbb{Z}_n$ symmetry (shown here for $k=1$).
• Figure 3: Thin-slab construction of the SymTFT corresponding to the $\mathbb{Z}_4$ igSPT (dotted pink line), which is obtained by condensing the non-Lagrangian subalgebra generated by $\langle e^2 m^2 \rangle$ in the $\mathcal{D}(\mathbb{Z}_4)$ TO. The string order parameter shown here has long range order, and encodes Eq. \ref{['eq:igspt_stringorder']}.
• Figure 4: SymTFT for the Choi state. (a) Distinction between strong and weak symmetry operators. (b) Distinction between strong and weak order parameters. The bold red lines connecting the layers represent decoherence ($a_+ \bar{a}_-$ anyon condensation), which has the effect of coupling the ket and bra spaces of $\ket{\rho}\rangle$.
• Figure 5: SymTaco correspondence between folded SPTs and mixed-state SPTs. Folding along the $\mathcal{D}(G)/\mathcal{A}$ TO corresponds to maximal decoherence of $\mathcal{D}(G)/\mathcal{A}$ in the Choi space. Folded SPTs with $G \times G$ symmetry can be interpreted as mixed-state SPTs with total $G_+ \times G_-$ symmetry. The bold red lines connecting the layers represent decoherence, which is implemented via condensation of anyons $a_+ \bar{a}_-$ in $\mathcal{D}(G)/\mathcal{A} \boxtimes \overline{\mathcal{D}(G)/\mathcal{A}}$.