Anomalous Symmetry Fractionalization and Surface Topological Order

TL;DR

The paper addresses how symmetry enriched topological phases can harbor anomalies that obstruct purely 2D realizations. It develops a gauge-theoretic diagnostic using $H^4(G,U(1))$ to detect anomalies and links these SET obstructions to 3D SPT bulk order, establishing a bulk–boundary correspondence. By constructing a decorated Walker-Wang model, the authors realize anomalous surface projective semions for $G=Z_2 imesZ_2$ and demonstrate nontrivial 3-loop braiding as a robust bulk invariant. An alternative $O(5)$ nonlinear sigma-model in 3D corroborates the SPT classification, and the work outlines extensions to anti-unitary symmetries and broader SETs with discrete unitary groups.

Abstract

In addition to possessing fractional statistics, anyon excitations of a 2D topologically ordered state can realize symmetry in distinct ways , leading to a variety of symmetry enriched topological (SET) phases. While the symmetry fractionalization must be consistent with the fusion and braiding rules of the anyons, not all ostensibly consistent symmetry fractionalizations can be realized in 2D systems. Instead, certain `anomalous' SETs can only occur on the surface of a 3D symmetry protected topological (SPT) phase. In this paper we describe a procedure for determining whether an SET of a discrete, onsite, unitary symmetry group $G$ is anomalous or not. The basic idea is to gauge the symmetry and expose the anomaly as an obstruction to a consistent topological theory combining both the original anyons and the gauge fluxes. Utilizing a result of Etingof, Nikshych, and Ostrik, we point out that a class of obstructions are captured by the fourth cohomology group $H^4( G, \,U(1))$, which also precisely labels the set of 3D SPT phases, with symmetry group $G$. We thus establish a general bulk-boundary correspondence between the anomalous SET and the 3d bulk SPT whose surface termination realizes it. We illustrate this idea using the chiral spin liquid ($U(1)_2$) topological order with a reduced symmetry $\mathbb{Z}_2 \times \mathbb{Z}_2 \subset SO(3)$, which can act on the semion quasiparticle in an anomalous way. We construct exactly solved 3d SPT models realizing the anomalous surface terminations, and demonstrate that they are non-trivial by computing three loop braiding statistics. Possible extensions to anti-unitary symmetries are also discussed.

Figures (14)

• Figure 1: The fusion rule $\Omega_x \times \Omega_x = s$ derived from the symmetry action on the semion in theory X (defined in equation \ref{['ps']}): (a) Bringing a gauge flux $\Omega_x$ around the center semion is equivalent to acting locally on it with the corresponding symmetry $g_x$; (b) Bringing two $\Omega_x$ gauge fluxes around the center semion gives rise to a $-1$ phase factor, because in theory X a semion carries half of the corresponding $g_x$ charge. This $-1$ can be reproduced by bringing another semion around the center one, giving rise to the $s$ on the right hand side of the fusion rule.
• Figure 2: Braiding and fusion statistics of the semion theory: (a) The exchange of two semions leads to a phase factor of $i$; (b) The two ways of fusing three semions differ by a sign.
• Figure 3: Pentagon equation for the fusion of four gauge fluxes.
• Figure 4: Some configurations in the semion ground state. The vertex condition ensures that only configurations with an even number of edges on which $n_i=1$ (shown in blue in the Figure) can meet at a vertex, such that the ground state is a superposition of loops. The relative amplitudes of these loop confiurations are given by the phase factor $\Theta (P) \Phi_{O,O'} \Phi_{U,U'}$ in Eq. (\ref{['SemPlaqH']}).
• Figure 5: (a) We define our model is defined on the point-split cubic lattice, because the Hamiltonian is simplest for a lattice with only trivalent vertices. (b) The plaquette term is defined with respect to a fixed angle of projection. In this angle, there are two edges ($O$ and $U$) that are projected into each plaquette from above and below, respectively, shown in red. These (and their partner edges $O'$ and $U'$, shown here in blue) are used to define the phase factors $\Phi_{O, O'}, \Phi_{U, U'}$ in the plaquette operator.