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

TL;DR

This paper explores fermionic non-invertible symmetries in (1+1)d, classifying phases and phase transitions using Symmetry TFTs, and introduces a categorical framework for understanding these symmetries and their associated phases.

Contribution

It develops a categorical Landau paradigm for fermionic symmetries and applies SymTFT formalism to classify gapped and gapless phases, including phase transitions, in fermionic systems.

Findings

Classification of fermionic phases via symmetry charges

Identification of gapless phases as phase transitions

Construction of fermionic CFTs from bosonic models

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 $\pi$-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.