Soft symmetries of topological orders

TL;DR

This work reveals soft symmetries in topological orders as elements of $Aut_{br}(\mathcal{C})$ that fix all anyons yet act nontrivially on higher-genus ground state spaces. It shows that gauged SPT defects generate invertible codimension-1 defects whose torus partition function is indistinguishable from the trivial case, but which affect genus-$g$ sectors, yielding a nontrivial $Aut_{sf}(\mathcal{C})$ action and a concrete physical mechanism for soft symmetries. The authors provide an explicit lattice realization in a (2+1)D finite gauge theory, demonstrate a non-permuting $\mathbb{Z}_2$ soft symmetry acting on genus-$g$ spaces, and discuss implications for gapped boundaries, oblique SSB in (1+1)D, and higher-dimensional analogues such as a (3+1)D $Q_8$ gauge theory with nontrivial 3-group structure. Together, these results advance the classification of symmetry-enriched topological phases and point to new constructions and open questions for soft symmetries, including possible on-site realizations and generalization to other gauge groups and dimensions.

Abstract

(2+1)D topological orders possess emergent symmetries given by a group $\text{Aut}(\mathcal{C})$, which consists of the braided tensor autoequivalences of the modular tensor category $\mathcal{C}$ that describes the anyons. In this paper we discuss cases where $\text{Aut}(\mathcal{C})$ has elements that neither permute anyons nor are associated to any symmetry fractionalization but are still non-trivial, which we refer to as soft symmetries. We point out that one can construct topological defects corresponding to such exotic symmetry actions by decorating with a certain class of gauged SPT states that cannot be distinguished by their torus partition function. This gives a physical interpretation to work by Davydov on soft braided tensor autoequivalences. This has a number of important implications for the classification of gapped boundaries, non-invertible spontaneous symmetry breaking, and the general classification of symmetry-enriched topological phases of matter. We also demonstrate analogous phenomena in higher dimensions, such as (3+1)D gauge theory with gauge group given by the quaternion group $Q_8$.

Figures (10)

• Figure 1: Labels and fusion decomposition defining an orthonormal basis for states on genus $g$ surfaces. The $\otimes$ signifies the fact that the loops are non-contractible on the genus $g$ surface.
• Figure 2: A point junction of the three symmetry defects emits an Abelian anyon $\mathcal{O}_3(\varphi_1,\varphi_2,\varphi_3)\in\mathcal{A}$. This represents the 2-group structure formed by the 0-form symmetries and the Abelian anyons.
• Figure 3: The anyon $x$ crossing through a pair of tri-junctions of symmetry defects. Initially $x$ is crossing the three defects $\varphi_1,\varphi_2,\varphi_3$, and finally $x$ crosses a single defect $\varphi_{123}$. There are two ways to arrive at the final configuration; passing $x$ along the top or bottom of the picture. The phase factor for each path is represented by the thick blue arrow. Equating the phase factors for consistency results in \ref{['eq:etaeta=Metaeta']}.
• Figure 4: The action of a gauged SPT defect on magnetic defects. (a): The 0-form symmetry generated by the gauged SPT defect acts on the magnetic defect by attaching the $(d-1)$D gauged SPT defect to the magnetic defect. This can be understood as the twisted compactification of SPT with respect to the circle with holonomy $g\in G$. (b): The 0-form gauged SPT defect in $(d+1)$D spacetime acts on a point junction of the $(d-1)$ magnetic defects by a phase factor given by the SPT cocycle. The figure shows the case of $d=2$, where the (1+1)D SPT defect acts on the junction of two magnetic defects by a 2-cocycle $\omega(g,h)$.
• Figure 5: The edges nearby a vertex $v$ and face $f$.