  We utilize the topological holographic framework to characterize and gain
insights into the nature of quantum critical points and gapless phases in
fermionic quantum systems. Topological holography is a general framework that
describes the generalized global symmetry and the symmetry charges of a local
quantum system in terms of a slab of a topological order, termed as the
symmetry topological field theory (SymTFT), in one higher dimension. In this
work, we consider a generalization of the topological holographic picture for
$(1+1)d$ fermionic quantum phases of matter. We discuss how spin structures are
encoded in the SymTFT and establish the connection between the formal
fermionization formula in quantum field theory and the choice of fermionic
gapped boundary conditions of the SymTFT. We demonstrate the identification and
the characterization of the fermionic gapped phases and phase transitions
through detailed analysis of various examples, including the fermionic systems
with $\mathbb{Z}_{2}^{F}$, $\mathbb{Z}_{2} \times \mathbb{Z}_{2}^{F}$,
$\mathbb{Z}_{4}^{F}$, and the fermionic version of the non-invertible
$\text{Rep}(S_{3})$ symmetry. Our work uncovers many exotic fermionic quantum
critical points and gapless phases, including two kinds of fermionic symmetry
enriched quantum critical points, a fermionic gapless symmetry protected
topological (SPT) phase, and a fermionic gapless spontaneous symmetry breaking
(SSB) phase that breaks the fermionic non-invertible symmetry.
