Fermionic quantum criticality through the lens of topological holography

TL;DR

This work develops a fermionic extension of topological holography by generalizing the sandwich construction to include fermionic gapped boundaries and spin-structure dependence. It establishes explicit fermionization formulas that connect standard bosonic boundary data to fermionic theories via Arf-invariant weighted sums and encodes spin structures in the bulk SymTFT, enabling non-perturbative analysis of fermionic gapped phases and phase transitions. Through detailed analyses of $\mathbb{Z}_2^{F}$, $\mathbb{Z}_2 \times \mathbb{Z}_2^{F}$, $\mathbb{Z}_{4}^{F}$, and non-invertible $\text{SRep}(\mathcal{H}_{S_{3}})$ symmetries, the paper reveals a zoo of exotic fermionic quantum critical points, including two types of fermionic symmetry-enriched quantum critical points (SEQCPs) and gapless SPT/SSB phases, all organized via bulk twist defects and boundary condensing data. The results provide non-perturbative CFT descriptions (e.g., Ising, Majorana, $m=5$ minimal models) for diverse fermionic transitions and demonstrate how dualities and twisted gauging manifest as changes in the symmetry boundary, yielding a unifying perspective for fermionic QCPs and gapless phases with both invertible and non-invertible symmetries. The framework lays foundations for systematic classification and computation of fermionic phases in (1+1)D and points to extensions to higher dimensions and more intricate fusion-category symmetries.

Abstract

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.

Figures (7)

• Figure 1: The sandwich picture in the topological holography. A (1+1)$d$ theory can be viewed as a sandwich built from a topological order $\mathcal{T}$ in the bulk with a topological gapped boundary conditions on the left and a potentially non-topological boundary condition on the right.
• Figure 2: Performing the fermionization procedure in the sandwich picture corresponds to change the symmetry boundary from a bosonic gapped boundary $\mathcal{A}_{b}$ to a fermionic gapped boundary $\mathcal{A}_{f}$.
• Figure 3: The fermionic partition functions with different spin structures are represented as the twisted partition functions $Z_{F}[m,n]$ in the anyon basis. We denote the representative anyon in the bulk SymTFT that implements the fermion parity symmetry as $\mu_\rho$ and define a map $\iota$ from the spin structures to the bulk anyons (see main text). The twisted partition function is equivalent to the partition function of the $(2+1)$D SymTFT on a solid torus with an insertion of a composite anyon $\iota(m) \times \mathcal{A}_{f}$ along the time-direction, wrapping around by $\iota(n)$ in the spatial direction. The temporal cycle is the vertical direction and the spatial cycle is horizontal.
• Figure 4: Summary of the phase transitions in the fermionic systems with $\mathbb{Z}_2 \times \mathbb{Z}_{2}^{F}$ symmetry discussed in Sec. \ref{['sec:z2z2fqcp']} and Sec. \ref{['sec:fseqcp']}. The red arrow indicate the input boundary critical theory on the physical boundary of the sandwich, which is the $\text{Ising}^{2}$ CFT.
• Figure 5: The sandwich pictures for two different types of the fermionic SEQCPs. (a) The first type of fermionic SEQCPs is obtain from the stacking of the usual SSB transitions with a SPT state. It corresponds to an insertion of a SPT twist defect $\mathcal{D}_{\textbf{T}}$ in the sandwich picture with the symmetry boundary that can "absorb" the SPT defect. (b) The second type of fermionic SEQCPs is obtain from the a twisted gauging $\textbf{TS}$. It corresponds to an insertion of a twist defect $\mathcal{D}_{\textbf{T}} \circ \mathcal{D}_{\textbf{S}}$, where $\mathcal{D}_{\textbf{S}}$ is a defect that implements the $\textbf{S}$ gauging.