Symmetry fractionalization and twist defects

TL;DR

This work develops a framework for symmetry enriched topological order in two dimensions when a finite group $G$ permutes anyon types, introducing a twisted group cohomology class $[\omega] \in H^2_{\rm twisted}(G, \mathcal{A})$ to capture symmetry fractionalization and twist defects. It constructs exactly solved lattice Hamiltonians built from $|G|$ copies of a $\mathbb{Z}_n$ gauge theory with a $G$-twist and analyzes how gauging $G$ reveals a topological order given by a group extension $E$ of $G$ by $\mathbb{Z}_n$, with distinct $[\omega]$ producing nonisomorphic $E$ and thereby distinct SETs. A key insight is that for permuting symmetries, the distinctions between SETs are not visible in single-defect data but appear in the fusion and associativity data (F-matrices) of defect pairs, and in the gauged theory’s quasiparticle statistics. The paper provides a concrete $\mathbb{Z}_4$ gauge theory with $\mathbb{Z}_2$ symmetry illustrating two inequivalent SETs corresponding to $E=\mathbb{D}_8$ and $E=\mathbb{Q}_8$, which share defect fusion rules at the level of superselection sectors but differ in F-matrix data. Altogether, the results establish that braided $G$-crossed categories underpin a robust classification of 2D SETs with permuting symmetries, with practical diagnostics via defect braiding and gauged topological order.

Abstract

Topological order in two dimensions can be described in terms of deconfined quasiparticle excitations - anyons - and their braiding statistics. However, it has recently been realized that this data does not completely describe the situation in the presence of an unbroken global symmetry. In this case, there can be multiple distinct quantum phases with the same anyons and statistics, but with different patterns of symmetry fractionalization - termed symmetry enriched topological (SET) order. When the global symmetry group $G$, which we take to be discrete, does not change topological superselection sectors - i.e. does not change one type of anyon into a different type of anyon - one can imagine a local version of the action of $G$ around each anyon. This leads to projective representations and a group cohomology description of symmetry fractionalization, with $H^2(G,{\cal A})$ being the relevant group. In this paper, we treat the general case of a symmetry group $G$ possibly permuting anyon types. We show that despite the lack of a local action of $G$, one can still make sense of a so-called twisted group cohomology description of symmetry fractionalization, and show how this data is encoded in the associativity of fusion rules of the extrinsic `twist' defects of the symmetry. Furthermore, building on work of Hermele, we construct a wide class of exactly solved models which exhibit this twisted symmetry fractionalization, and connect them to our formal framework.

Figures (18)

• Figure 1: Lattice on which our model is defined. The links carry $\mathbb{Z}_n$ labels.
• Figure 2: A diagram showing the flux piercing the plaquette $(f,g,h)$. Note that it spans layers $f$, $fg$ and $fgh$
• Figure 3: A tetrahedron spanning the layers $\{e,f,fg,fgh \}$. The total $\mathbb{Z}_n$ flux emanating out of this tetrahedron must be trivial (eq. \ref{['eq:cocycle1']}) in order to avoid a degenerate set of frustrated ground states.
• Figure 4: Vertical links are colored red and horizontal links are colored black. Collections of vertical links connecting points with the same x-y coordinate constitute supervertices, labeled $V_1$ and $V_2$. Plaquettes made entirely out of horizontal links that project to the same plaquette form a superplaquette, labeled $P$ in the figure. Plaquettes containing at least one vertical link are deemed vertical. The dotted lines identify vertices and help offset the figure. Certain links are dashed in order to add perspective.
• Figure 5: Placement of the $G$ gauge variable relative to the supervertex and superlink