Codimension-2 defects and higher symmetries in (3+1)D topological phases

TL;DR

This work advances the understanding of codimension‑2 defects, termed twist strings, in (3+1)D topological phases by developing multiple constructive frameworks (layer constructions, gauging of (1+1)D SPT phases, and condensation defects) and by analyzing their rich interplay with flux strings and point charges. It demonstrates that invertible twist strings generate a 3‑group symmetry combining 0‑form, 1‑form, and 2‑form components, with concrete realizations in Abelian and non‑Abelian discrete gauge theories, including D(G) theories and A6, as well as both bosonic and fermionic sectors. The paper also presents a portfolio of exactly solvable lattice models for twist strings in (2+1)D and (3+1)D toric codes, explores non‑topological (geometric) twist strings, and constructs codimension‑3 non‑Abelian point defects at twist endpoints. Furthermore, it relates layer‑construction twist strings to gauging (1+1)D SPT phases through modular invariants and 3‑group data, offering a path toward a categorical 3‑group (and eventually fusion 3‑categories) description of defects in (3+1)D topological order with potential implications for fault‑tolerant quantum computation. Overall, the work broadens the landscape of higher symmetries in 3+1D topological matter and provides concrete tools for classifying and manipulating defects in these systems.

Abstract

(3+1)D topological phases of matter can host a broad class of non-trivial topological defects of codimension-1, 2, and 3, of which the well-known point charges and flux loops are special cases. The complete algebraic structure of these defects defines a higher category, and can be viewed as an emergent higher symmetry. This plays a crucial role both in the classification of phases of matter and the possible fault-tolerant logical operations in topological quantum error correcting codes. In this paper, we study several examples of such higher codimension defects from distinct perspectives. We mainly study a class of invertible codimension-2 topological defects, which we refer to as twist strings. We provide a number of general constructions for twist strings, in terms of gauging lower dimensional invertible phases, layer constructions, and condensation defects. We study some special examples in the context of $\mathbb{Z}_2$ gauge theory with fermionic charges, in $\mathbb{Z}_2 \times \mathbb{Z}_2$ gauge theory with bosonic charges, and also in non-Abelian discrete gauge theories based on dihedral ($D_n$) and alternating ($A_6$) groups. The intersection between twist strings and Abelian flux loops sources Abelian point charges, which defines an $H^4$ cohomology class that characterizes part of an underlying 3-group symmetry of the topological order. The equations involving background gauge fields for the 3-group symmetry have been explicitly written down for various cases. We also study examples of twist strings interacting with non-Abelian flux loops (defining part of a non-invertible higher symmetry), examples of non-invertible codimension-2 defects, and examples of interplay of codimension-2 defects with codimension-1 defects. We also find an example of geometric, not fully topological, twist strings in (3+1)D $A_6$ gauge theory.

Figures (42)

• Figure 1: The vertical (green) line represents the topological twist string (codimension-2 defect), and the horizontal (red) line is the flux string. When the flux string is moved across the twist string, a point charge appears from their intersection.
• Figure 2: Illustration of the generic properties of a non-topological twist string (green line). There exist a certain flux string with an attached charge $\pi$, such that the transformation of this charge depends on the detailed geometric trajectory of the flux string and the charge. (a) If the worldline of the attached charge $\pi$ (blue) intersects with the geometric twist string, the attached charge will be transformed into $\pi'$ (purple). (b) If the worldline of the attached charge avoids the twist string, the attached charge remains the same.
• Figure 3: Layers of (2+1)D anyon theories are prepared, and then pairs of particle excitations in adjacent layers are condensed.
• Figure 4: (a) Sweeping the codimension-$k$ defect on a codimension-$(k-1)$ submanifold (grey sheet) $W^{d-k+1}$ gives rise to an emergent $(k-1)$-form symmetry. The swept region is an open manifold $W^{d-k+1}_{cut}$ (light green), while the codimension-$k$ defect (green) is located on its boundary $\Sigma^{d-k} = \partial W_{cut}^{d-k+1}$. (b) For an invertible defect, the corresponding $(k-1)$-form symmetry operator is a constant-depth local quantum circuit, since the sweeping of the codimension-$k$ invertible defect $A$ can be done in two steps in $O(1)$ time: (1) Parallel creation of a defect $A$ and its inverse $\bar{A}$ separated with $O(1)$ distance. (2) Parallel annihilation of the defect $A$ with its neighboring inverse $\bar{A}$ on the other side. Note that the box in the figure represents a 3-torus with periodic boundary conditions.
• Figure 5: Gauging ${\mathbb Z}_2$ symmetry Y16. Left: each dot represents a qubit on a vertex. We consider the symmetric sector of the Hilbert space: $\prod_v X_v = 1$. Symmetric operators are generated by the single $X_v$ and the product of adjacent $Z_v$. Right: The Hilbert space contains qubits at all edges, with the gauge constraint $\prod_{e\subset f} Z_e = 1$ for each face $f$. For non-simply connected manifolds, there are additional constraints that the product of $Z_e$ along any cycle equals $+1$.