Symmetry Fractionalization, Defects, and Gauging of Topological Phases

TL;DR

The paper builds a comprehensive framework for symmetry in 2+1D topological phases by introducing the topological symmetry group Aut(\mathcal{C}) and a systematic cohomology-based classification of symmetry fractionalization via Obstruction [\textswab{O}] ∈ H^3_{[\rho]}(G,\mathcal{A}) and fractionalization classes in H^2_{[\rho]}(G,\mathcal{A}). It then develops a general theory of extrinsic defects using a G-crossed braided tensor category \mathcal{C}_{G}^{\times}, enabling precise computations of defect types, fusion, braiding, and generalized Verlinde relations, along with conditions (heptagon equations) for consistency and obstructions. The authors show how gauging G leads to a new topological phase (\mathcal{C}_{G}^{\times})^{G}, whose data—charges, fusion, twists, and S-matrix—are determined from the G-crossed input, including a universal relation for the total quantum dimension and chiral central charge. A suite of examples, including non-permutation actions and the three-fermion theory with S_3 symmetry, demonstrates the framework and connects to prior work on PSG, CFT orbifolds, and topological Bose condensation. Overall, the work provides a practical, physics-oriented skeletonization of G-crossed BTCs, enabling complete SET classifications and gauging constructions for a broad class of 2+1D topological phases with symmetry.

Abstract

We examine the interplay of symmetry and topological order in $2+1$ dimensional topological phases of matter. We present a definition of the \it topological symmetry \rm group, which characterizes the symmetry of the emergent topological quantum numbers of a topological phase, and we describe its relation with the microscopic symmetry of the underlying physical system. We derive a general framework to characterize and classify symmetry fractionalization in topological phases, including phases that are non-Abelian and symmetries that permute the quasiparticle types and/or are anti-unitary. We develop a theory of extrinsic defects (fluxes) associated with elements of the symmetry group, which provides a general classification of symmetry-enriched topological phases derived from a topological phase of matter $\mathcal{C}$ with symmetry group $G$. The algebraic theory of the defects, known as a $G$-crossed braided tensor category $\mathcal{C}_{G}^{\times}$, allows one to compute many properties, such as the number of topologically distinct types of defects associated with each group element, their fusion rules, quantum dimensions, zero modes, braiding exchange transformations, a generalized Verlinde formula for the defects, and modular transformations of the $G$-crossed extensions of topological phases. We also examine the promotion of the global symmetry to a local gauge invariance, wherein the extrinsic $G$-defects are turned into deconfined quasiparticle excitations, which results in a different topological phase $(\mathcal{C}_{G}^{\times})^{G}$. A number of instructive and/or physically relevant examples are studied in detail.

Figures (22)

• Figure 1: The Pentagon equation enforces the condition that different sequences of $F$-moves from the same starting fusion basis decomposition to the same ending decomposition gives the same result. Eq. (\ref{['eq:pentagon']}) is obtained by imposing the condition that the above diagram commutes.
• Figure 2: The Hexagon equations enforce the condition that braiding is compatible with fusion, in the sense that different sequences of $F$-moves and $R$-moves from the same starting configuration to the same ending configuration give the same result. Eqs. (\ref{['eq:hexagon+']}) and (\ref{['eq:hexagon-']}) are obtained by imposing the condition that the above diagram commutes.
• Figure 3: The global on-site symmetry action on states containing quasiparticles takes the form given in Eq. (\ref{['eq:symmetryLocalization_n']}), where the global action $R_{\bf g}$ factorizes into the global symmetry action operator $\rho_{\bf g}$, which acts only on the topological quantum numbers, and local transformations $U^{(j)}_{\bf g}$, each of which only acts nontrivially within a region $\mathcal{R}_{j}$ well-localized around the $j$th quasiparticle carrying topological charge $a_j$.
• Figure 4: The action of a global "locality preserving" symmetry operator $R_\mathbf{g}$ on a state with quasiparticles may move the locations where the quasiparticles are localized, from the regions $\mathcal{R}_{j}$ to the regions $^{\bf g}\mathcal{R}_{j}$. The locality preserving property ensures that the regions $^{\bf g}\mathcal{R}_{j}$ are mutually disjoint for distinct $j$ whenever the regions $\mathcal{R}_{j}$ are mutually disjoint for distinct $j$. Additionally, the locality preserving symmetry action induces unitary transformations $U_\mathbf{g}^{{\bf g}(j)}$ that are, respectively, localized in the regions $^{\bf g}\mathcal{R}_{j}$, together with a global transformation $\rho_\mathbf{g}$ which strictly acts on the topological quantum numbers.
• Figure 5: (a) When the system is cut along a line $C$, quasiparticles cannot propagate across the cut. (b) The system can be reglued together along $C$ in a manner that conjugates bond/plaquette operators straddling the cut by a local ${\bf g}$-symmetry action on one side of the cut, as indicated by red dots. The result is a ${\bf g}$ and ${\bf g}^{-1}$ pair of defects at the end-points of a defect branch line (replacing the cut). (c) Such a construction effectively implements a ${\bf g}$-symmetry transformation on quasiparticles that propagate across the defect branch line, around the defects. For example, a quasiparticle $a$ will be transformed into $\rho_{\bf g} (a)$ when it encircles the ${\bf g}$-defect in a counterclockwise fashion. For symmetries that are not on-site, such as translational or rotational symmetries, ${\bf g}$-defects correspond to lattice dislocations or disclinations, respectively. (d) For more general systems, the defect construction can be generalized by defining regions $C_{l}$ and $C_{r}$ on either side of the cut line, such that terms in the Hamiltonian that straddle the cut line are localized within $C_{l} \cup C_{r}$. These regions will typically have width $w$ that is a few correlation lengths $\xi$.

Theorems & Definitions (8)

• Definition 1
• Definition 2
• Theorem D.1: [turaev2010kirillov2004]
• Theorem D.2: [ENO2009]
• Definition 3
• Theorem D.3: [DGNO2009]
• Theorem D.4
• Theorem D.5