Fermionic Non-Invertible Symmetries in (1+1)d: Gapped and Gapless Phases, Transitions, and Symmetry TFTs

TL;DR

<3-5 sentence high-level summary> The paper develops a categorical framework for fermionic non-invertible symmetries in 1+1d, using fusion π-supercategories and a bosonic SymTFT to classify gapped and gapless phases, as well as phase transitions, via condensable algebras in the symmetry's Drinfeld center. It shows that fermionic phases are obtained by fermionization of bosonic phases and stacking with Arf TFT, connecting to generalized KT transformations and Jordan-Wigner-type mappings. The authors provide detailed constructions for a range of finite fermionic symmetries (Z2^f, Z4^f, Rep(S3)^f, and Z2×Z2^f with Gu-Wen and beyond), including explicit boundary data, bulk-boundary maps, and Ising/Majorana-type CFT realizations of gapless transitions. The framework unifies gapped/gapless phases and transitions, clarifies the role of Arf stacking, and yields concrete phase diagrams (Hasse diagrams) that illuminate how condensations drive symmetry breaking, SPT/SSB phenomena, and critical theories in 1+1d fermionic systems.

Abstract

We study fermionic non-invertible symmetries in (1+1)d, which are generalized global symmetries that mix fermion parity symmetry with other invertible and non-invertible internal symmetries. Such symmetries are described by fermionic fusion supercategories, which are fusion $π$-supercategories with a choice of fermion parity. The aim of this paper is to flesh out the categorical Landau paradigm for fermionic symmetries. We use the formalism of Symmetry Topological Field Theory (SymTFT) to study possible gapped and gapless phases for such symmetries, along with possible deformations between these phases, which are organized into a Hasse phase diagram. The phases can be characterized in terms of sets of condensed, confined and deconfined generalized symmetry charges, reminiscent of notions familiar from superconductivity. Many of the gapless phases also serve as phase transitions between gapped phases. The associated fermionic conformal field theories (CFTs) can be obtained by performing generalized fermionic Kennedy-Tasaki (KT) transformations on bosonic CFTs describing simpler transitions. The fermionic non-invertible symmetries along with their charges and phases discussed here can be obtained from those of bosonic non-invertible symmetries via fermionization or Jordan-Wigner transformation, which is discussed in detail.

Figures (3)

• Figure 1: The fermionization and bosonization in 1+1d. JW stands for the Jordan-Wigner transformation and GSO stands for the GSO projection.
• Figure 2: The interface ${\mathcal{I}}$ implements a map of lines from $\mathcal{Z}'$ to $\mathcal{Z}({\mathcal{S}}_{f})$. A line $a\in {\mathcal{A}}$ can end end on the interface from the left. Meanwhile a simple line $a'\in \mathcal{Z}'$ maps to a possibly non-simple line $\mathcal{Z}_{\mathcal{L}}(a')\in \mathcal{Z}({\mathcal{S}}_f)$.
• Figure 3: A 2d ${\mathcal{S}}_f$ symmetric fermionic TFT $\mathfrak{T}_f$ constructed as the interval compactification of the SymTFT $\mathfrak{Z}({\mathcal{S}}_f)$ with symmetry boundary $\mathfrak{B}^{\text{sym}}_{{\mathcal{S}}_f}$ and the bosonic topological boundary $\mathfrak{B}^{\text{phys}}={\mathcal{I}}$ on the other end.