Entropic order parameters and topological holography

TL;DR

This work establishes SymTFT/topological holography as a unified framework to define and compute entropic order parameters for phases with (partially) broken symmetries, including both invertible and non-invertible cases. By encoding symmetry data in a 3d topological boundary (SymTFT) and using intertwiners and a conditional expectation to project onto invariant content, the authors show that relative entropy between vacua serves as a robust order parameter that directly reflects the fusion-category data, such as quantum dimensions. They illustrate the approach with concrete examples: Abelian groups (Z2, Z2×Z2) and non-Abelian (S3) invertible symmetries, plus non-invertible cases from Rep(S3) and the Ising TY category, revealing when vacua are distinguishable and how SPT/SSB phases are encoded in boundary conditions and twisted sectors. The results highlight a deep link between entropic distinguishability of vacua and the categorical structure of the symmetry, offering a principled route to analyze gapped and potentially gapless phases across dimensions. The framework thus provides a versatile tool for understanding symmetry breaking, topological order, and holographic dualities in quantum field theory and condensed matter systems.

Abstract

We show that the symmetry topological field theory (SymTFT) construction, also known as the topological holography, provides a natural and intuitive framework for the entropic order parameter characterising phases with (partially) broken symmetries. Various examples of group and non-invertible symmetries are studied. In particular, the origin of the distinguishability of the vacua resulting from spontaneously broken non-invertible symmetries is made manifest with an information-theoretic perspective, where certain operators in the SymTFT are excluded from observation.

Figures (5)

• Figure 1: The 2d QFT with symmetry category $\mathcal{C}$ is represented as a 'sandwich' consisting of i) the 3d SymTFT in which the topological lines form $\mathcal{Z}(\mathcal{C})$, ii) the symmetry boundary where the SymTFT has Dirichlet boundary condition, and iii) the physical boundary which is not necessarily topological.
• Figure 2: In the SymTFT construction, a local operator $O \in \mathcal{F}$ is represented as a triple $(x_{O}, L_{O}, \widetilde{O})$, where $L_{O}$ is a topological line in the SymTFT, $x_{O}$ is the topological junction of $L_{O}$ on the symmetry boundary, and $\widetilde{O}$ is its (generally non-topological) junction on the physical boundary.
• Figure 3: The action of $a$ in the symmetry category $\mathcal{C}$ on a (genuinely local) operator $O \in \mathcal{F}$ amounts to pushing the corresponding topological defect line on the symmetry boundary through the junction $x_{O}$. This action can leave behind another defect line $b$ attaching the junction to the line $a$. Note that actions of the global symmetry are performed solely on the symmetry boundary and cannot change the bulk topological line $L_{O}$. The latter labels the multiplet in which the operator $O$ transforms. In particular, if $L_{O}$ is the trivial line, the corresponding operator $O$ is invariant under the symmetry and belongs to $\mathcal{O}$.
• Figure 4: SymTFT representation of the intertwiner $\mathcal{I}_{RR^{\prime}}$ supported on the region $R$ and its complement $R^{\prime}$. It is evident from (a) that $\mathcal{I}_{RR^{\prime}}$ is invariant under the global symmetry and belongs to $\mathcal{O}$. In (b), the bulk topological line $L$ is merged into the symmetry boundary using the bulk-to-boundary map $F$; $\bar{L}$ is the orientation reversal of $L$. In (c), the line $\bar{L}$ and the associated junction are sent to infinity; the operator resulting in this limit is denoted as $\mathcal{I}_{R}$, which lives in the $F(L)$-twisted sector. We will mainly focus on the case where $F(L)$ is the identity line, for which $\mathcal{I}_{R}$ belongs to $\mathcal{F}(R)$ but not to $\mathcal{O}(R)$.
• Figure 5: (a) The 'string order parameter' corresponds to the twisted-sector intertwiner in the $\mathbb{Z}_{2}$ SPT phase. (b) Gauging $\mathbb{Z}_{2}$ on the symmetry boundary converts the Dirichlet boundary condition to Neumann and results in a dual SSB phase, where the string order parameter is mapped to an ordinary one associated with the untwisted intertwiner.