Relative Anomalies in (2+1)D Symmetry Enriched Topological States

TL;DR

The paper develops a general, data-driven method to compute relative symmetry anomalies for symmetry-enriched topological states in (2+1)D, incorporating unitary, anti-unitary, and space-time reflection symmetries, as well as anyon permutation. Central to the approach is a G-crossed braided tensor category framework extended to include symmetry defects, defect fusion/twisting by a 2-cocycle, and a relative 4-cocycle obstruction that encodes the bulk SPT needed to cancel the anomaly. The authors derive a universal relative anomaly formula, reproduce known results for Z2^T and U(1)-related symmetries, and extend to new cases such as Z4^T and mixed Z2×Z2^T anomalies, providing concrete examples across toric codes, doubled semion theories, and abelian Chern-Simons theories. This work yields a comprehensive diagnostic toolkit for classifying SETs and predicting allowed surface phenomena based on bulk SPT data, with implications for condensed matter and topological quantum computation.

Abstract

Certain patterns of symmetry fractionalization in topologically ordered phases of matter are anomalous, in the sense that they can only occur at the surface of a higher dimensional symmetry-protected topological (SPT) state. An important question is to determine how to compute this anomaly, which means determining which SPT hosts a given symmetry-enriched topological order at its surface. While special cases are known, a general method to compute the anomaly has so far been lacking. In this paper we propose a general method to compute relative anomalies between different symmetry fractionalization classes of a given (2+1)D topological order. This method applies to all types of symmetry actions, including anyon-permuting symmetries and general space-time reflection symmetries. We demonstrate compatibility of the relative anomaly formula with previous results for diagnosing anomalies for $\mathbb{Z}_2^{\bf T}$ space-time reflection symmetry (e.g. where time-reversal squares to the identity) and mixed anomalies for $U(1) \times \mathbb{Z}_2^{\bf T}$ and $U(1) \rtimes \mathbb{Z}_2^{\bf T}$ symmetries. We also study a number of additional examples, including cases where space-time reflection symmetries are intertwined in non-trivial ways with unitary symmetries, such as $\mathbb{Z}_4^{\bf T}$ and mixed anomalies for $\mathbb{Z}_2 \times \mathbb{Z}_2^{\bf T}$ symmetry, and unitary $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry with non-trivial anyon permutations.

Figures (12)

• Figure 1: Left: A ${\bf g}$ defect in space. Branch cut is depicted by dashed line; there can be topologically distinct endpoints, labeled by $a_{\bf g}$. Right: ${\bf g}$ defect is a two-dimensional sheet in space-time; an anyon $x$ is permuted to $\,^{\bf g} x$ upon crossing the sheet.
• Figure 2: Top left: Fusion of defect branch sheets. Changing the symmetry fractionalization class by a $2$-cocyle $\textswab{t}({\bf g}, {\bf h})$ changes the fusion of the defect worldsheets by the appearance of a Wilson line of $\textswab{t}({\bf g}, {\bf h})$ at the trijunction. Top right: Equivalent diagrammatic representation, where $a_{\bf g}$, $b_{\bf h}$ are the end-points of the ${\bf g}$ and ${\bf h}$ line defects. The fusion rule of the new theory thus becomes $a_{\bf g} \times b_{\bf h} = \textswab{t}({\bf g}, {\bf h}) \sum_{c_{\bf gh}} N_{ab}^c c_{\bf gh}$. We define $c_{\bf g h}'= \textswab{t}({\bf g}, {\bf h}) c_{\bf gh}$. Bottom panel: associativity of the defect fusion implies the 2-cocyle condition for $\textswab{t}({\bf g}, {\bf h})$.
• Figure 3: Splitting and fusing defect sheets ${\bf g}$ and ${\bf h}$ leaves behind an anyon loop for $\textswab{t}({\bf g}, {\bf h})$. Invertibility of the process thus requires $d_{\textswab{t}({\bf g}, {\bf h})} = 1$.
• Figure 4: In the original theory, sliding an anyon line through the defect trijunction gave a phase $\eta_x({\bf g}, {\bf h})$. In the new theory, the anyon line $x$ must also pass through the Abelian anyon line associated with $\textswab{t}({\bf g}, {\bf h})$, which picks up the mutual braiding phase $M_{x \hbox{\textswab{t}}({\bf g}, {\bf h})}$ between $x$ and $\textswab{t}({\bf g}, {\bf h})$, as illustrated.
• Figure 5: 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.