Symmetry topological field theory for fermionic quantum phases and phase transitions

Symmetry topological field theory (SymTFT) provides a topological holographic framework for describing generalized symmetries and symmetry charges in quantum systems, using a slab of topological field theory in one higher dimension. In this work, we extend the SymTFT framework to fermionic quantum phases of matter. We discuss how spin structures are incorporated within SymTFT and establish the connection between the fermionization formula in quantum field theory and the choice of fermionic gapped boundary conditions. Through various examples, we characterize fermionic gapped phases and phase transitions, revealing several exotic quantum critical points and gapless phases. These include fermionic symmetry-enriched quantum critical points, fermionic gapless symmetry-protected topological phases, and fermionic gapless spontaneous symmetry breaking phases that involve the breaking of fermionic non-invertible symmetries.