Topological holography for fermions

TL;DR

This work develops a fermionic extension of symmetry topological field theory (SymTFT) to codify 1+1D fermionic phases with finite symmetry G^F. By introducing a 2+1D fermionic gauge theory and a fermionic reference boundary, the authors implement a sandwich construction that captures symmetry defects, fermion parity, spin-structure dependence, edge modes, and partition functions for gapped fermionic phases, fermionic SPTs, and gapless phases such as the Majorana CFT. They classify 1+1D fermionic SPTs within this framework, derive stacking rules via diagonal and fermionic condensations in D(G^F), and show how bosonization corresponds to changing the reference boundary, recovering Ising and Z_4 bosonic analogs for anomalous fermionic symmetries. Beyond reproducing known phases, the approach yields a practical route to discover new physics, exemplified by intrinsically fermionic intrinsically gapless SPTs (igSPTs) with emergent fermionic anomalies and Majorana edge modes. The results provide a non-perturbative, topological handle on fermionic symmetry, including gapless regimes, with potential extensions to higher dimensions and non-invertible fermionic symmetries.

Abstract

Topological holography is a conjectured correspondence between the symmetry charges and defects of a $D$-dimensional system with the anyons in a $(D+1)$-dimensional topological order: the symmetry topological field theory (SymTFT). Topological holography is conjectured to capture the topological aspects of symmetry in gapped and gapless systems, with different phases corresponding to different gapped boundaries (anyon condensations) of the SymTFT. This correspondence was previously considered primarily for bosonic systems, excluding many phases of condensed matter systems involving fermionic electrons. In this work, we extend the SymTFT framework to establish a topological holography correspondence for fermionic systems. We demonstrate that this fermionic SymTFT framework captures the known properties of $1+1D$ fermion gapped phases and critical points, including the classification, edge-modes, and stacking rules of fermionic symmetry-protected topological phases (SPTs), and computation of partition functions of fermionic conformal field theories (CFTs). Beyond merely reproducing known properties, we show that the SymTFT approach can additionally serve as a practical tool for discovering new physics, and use this framework to construct a new example of a fermionic intrinsically gapless SPT phase characterized by an emergent fermionic anomaly.

Figures (20)

• Figure 1: The SymTFT setup. (a). The SymTFT for a general symmetry category $\mathcal{A}$ is the Drinfeld center $\sf{Z}[\mathcal{A}]$. The sandwich (left) reduces to the original system $\sf{T}$ (right) when viewed as an effective $1+1D$ system. (b). A non-trivial topological defect line near the reference boundary becomes a symmetry defect operator $\sf{D}_a$ of $\sf{T}$ after the dimensional reduction.
• Figure 2: Local and non-local charges in SymTFT. (a). A gauge charge line in the SymTFT can be absorbed by the reference boundary. This corresponds to a local symmetry charge of $\sf{T}$ after dimensional reduction. The red dot represents an excitation created by the anyon line at $\sf{B_{phys}}$. (b). A general anyon of the SymTFT is a dyon, the associated defect line can not be fully absorbed by $\sf{B_{ref}}$, instead it becomes a flux line on $\sf{B_{ref}}$(blue). After dimensional reduction this becomes a charge that lives on the end of a symmetry defect, i.e. a non-local charge or a string order parameter.
• Figure 3: Edge modes in SymTFT. The physical boundary has two regions of different condensations separated by a point defect, represented by a black box. The all-flux condensation $\mathcal{A}^0$ corresponds to the trivial symmetric phase. $\mathcal{A}_{\sf{T}}$ is the condensation that corresponds to a non-trivial $1+1D$ phase $\sf{T}$. If there is an anyon string that can end on the point defect as well as the reference boundary, then this anyon line becomes a local zero mode supported on the edge of $\sf{T}$ after dimensional reduction.
• Figure 4: Edge modes of the cluster chain from SymTFT. The SymTFT bulk is $D(\mathbb{Z}_2\times \mathbb{Z}_2)$-two copies of toric codes. The trivial paramagnet phase of $\mathbb{Z}_2\times \mathbb{Z}_2$ is represented by the canonical magnetic condensation $\mathcal{A}^0=\langle m_1,m_2\rangle$. The non-trivial SPT phase, i.e. the cluster chain, is represented by the magnetic condensation $\mathcal{A}^1=\langle e_1m_2,e_2m_1\rangle$. The two panels show two linearly independent edge modes in the SymTFT. Notice these two edge modes anti-commute, giving rise to a two-fold GSD for every edge.
• Figure 5: $\mathbb{Z}_2^F$-SymTFT setup. Top: An $f$-line can end on $\sf{B_{ref}}$, with the endpoint dressed by a local fermion operator $c$, represented by a blue dot. Notice there is no excitation created on $\sf{B_{ref}}$. The excitation is created at the other end of the $f$-line, represented by a red dot. After dimensional reduction this vertical line becomes a local fermionic operator. Bottom: The local fermion operators $c$ attached to $f$-lines give anti-commutation relation between two vertical $f$-lines. After dimensional reduction, this becomes the standard anti-commutation relation between local fermionic operators.