Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries

TL;DR

The paper develops a comprehensive, SymTFT-driven framework to classify (1+1)d phases with fusion-category (including non-invertible) symmetries by mapping phases to condensable algebras in the Drinfeld center $\mathcal Z(\mathcal S)$ and organizing deformations in a Hasse diagram. It introduces intrinsically gapless SPT and SSB phases (igSPT/igSSB) and provides concrete non-invertible examples, notably with $\mathsf{Rep}(D_8)$, where igSPT/igSSB arise and are characterized via specific condensable algebras and boundary data. The work generalizes gapped SPT/SSB classifications to gapless settings, clarifies the role of symmetry gaps $\Delta_{\mathcal S}$, and explains decomposition through gauging trivially acting non-invertible symmetries using a club-sandwich (KT) transformation. Together, these results offer a principled language for phase transitions and symmetry realization in (1+1)d and suggest pathways to higher dimensions with richer fusion-category symmetries.

Abstract

We discuss (1+1)d gapless phases with non-invertible global symmetries, also referred to as categorical symmetries. This includes gapless phases showing properties analogous to gapped symmetry protected topological (SPT) phases, known as gapless SPT (or gSPT) phases; and gapless phases showing properties analogous to gapped spontaneous symmetry broken (SSB) phases, that we refer to as gapless SSB (or gSSB) phases. We fit these gapless phases, along with gapped SPT and SSB phases, into a phase diagram describing possible deformations connecting them. This phase diagram is partially ordered and defines a so-called Hasse diagram. Based on these deformations, we identify gapless phases exhibiting symmetry protected criticality, that we refer to as intrinsically gapless SPT (igSPT) and intrinsically gapless SSB (igSSB) phases. This includes the first examples of igSPT and igSSB phases with non-invertible symmetries. Central to this analysis is the Symmetry Topological Field Theory (SymTFT), where each phase corresponds to a condensable algebra in the Drinfeld center of the symmetry category. On a mathematical note, gSPT phases are classified by functors between fusion categories, generalizing the fact that gapped SPT phases are classified by fiber functors; and gSSB phases are classified by functors from fusion to multi-fusion categories. Finally, our framework can be applied to understand gauging of trivially acting non-invertible symmetries, including possible patterns of decomposition arising due to such gaugings.

Figures (2)

• Figure 1: Hasse diagram for $\mathcal{Z} (\mathsf{Vec}_{\mathbb{Z}_4})$: On the left hand side we show the Hasse diagram of condensable algebras. In each box we show a condensable algebra. The lowest level are the maximal, i.e. Lagrangian, algebras. Picking one of these as the symmetry Lagrangian algebra $\mathcal{L}_{{\mathcal{S}}}$ that fixes the symmetry ${\mathcal{S}}$ allows the classification of all ${\mathcal{S}}$-symmetric phases (right hand side).
• Figure 2: Hasse diagram for $\mathcal{Z} (\mathsf{Vec}_{S_3})$: The top figure shows the Hasse diagram of the condensable algebras. The lowest level are the maximal, i.e. Lagrangian, algebras. Picking one of these as the symmetry Lagrangian algebra that fixes the symmetry ${\mathcal{S}}$ allows classification of all phases: this is done for $\mathsf{Vec}_{S_3}$ in the middle figure and $\mathsf{Rep}(S_3)$ in the bottom figure. There are no igSPTs for these symmetries.