Folding approach to topological order enriched by mirror symmetry

We develop a folding approach to study two-dimensional symmetry-enriched topological (SET) phases with the mirror reflection symmetry. Our folding approach significantly transforms the mirror SETs, such that their properties can be conveniently studied through previously known tools: (i) it maps the nonlocal mirror symmetry to an onsite ${\mathbb{Z}}_2$ layer-exchange symmetry after folding the SET along the mirror axis, so that we can gauge the symmetry; (ii) it maps all mirror SET information into the boundary properties of the folded system, so that they can be studied by the anyon condensation theory—a general theory for studying gapped boundaries of topological orders; and (iii) it makes the mirror anomalies explicitly exposed in the boundary properties, i.e., strictly 2D SETs and those that can only live on the surface of a 3D system can be easily distinguished through the folding approach. With the folding approach, we derive a set of physical constraints on data that describes mirror SET, namely, mirror permutation and mirror symmetry fractionalization on the anyon excitations in the topological order. We conjecture that these constraints may be complete, in the sense that all solutions are realizable in physical systems. Several examples are discussed to justify this. Previously known general results on the classification and anomalies are also reproduced through our approach.

Figure 1

Mirror symmetric local unitary transformations (LUTs) and folding of mirror SETs. [(a) and (b)] Under mirror symmetric LUTs, the wave functions of any two strictly 2D mirror SETs can be transformed into one another, such that the difference occurs only near the mirror axis, implying that symmetry properties of mirror SETs are concentrated near the mirror axis. (c) We then fold the system, such that symmetry properties of mirror SETs become boundary properties of the folded double-layer system. [(d) and (e)] Similar mirror symmetric LUTs can be done for anomalous SETs, which live on the surface of a 3D mirror SPT, such that the 3D bulk is transformed into a product state, with the only exception being near the mirror plane. The inverted T-like junction decouples from the rest of the system. In addition, the mirror symmetry becomes onsite and protects a ${\mathbb{Z}}_2$ SPT state on the mirror plane. (f) Folding the bottom surface of the inverted T junction, symmetry properties of the anomalous SET are transformed into boundary properties between the double-layer system and the ${\mathbb{Z}}_2$ SPT state.

Figure 2

Definition of mirror-symmetry fractionalization. (a) A pair of anyons, including an anyon $a$ and its antiparticle $\overline{a}$, located symmetrically on the two sides of the mirror axis. Mirror symmetry $\mathbf{M}$ maps $a$ to ${\rho }_m\left(a\right)$, such that the wave function respects the mirror symmetry only if $\overline{a}={\rho }_m\left(a\right)$. (b) The mirror eigenvalue of the two-anyon wave function defines the fractional quantum number $\mu \left(a\right)=\pm 1$.

Figure 3

Anyon condensation theory describing a gapped boundary $\mathcal{T}$, between topological orders $\mathcal{D}$ and $\mathcal{U}$.

Figure 4

The left panel depicts the double-layer cylinder geometry with a layer-exchanging branch cut (dashed line). The white arrow represents the anyon flux $X_a^{\pm }$ threading through the cylinder. The topological spin ${\theta }_{X_a^{\pm }}$ of $X_a^{\pm }$ is equal to the Berry phase accumulated when one end of the cylinder is rotated by $2\pi$. The right panel depicts a single-layer cylinder geometry carrying the topological order described by $\mathcal{C}$ that is topologically equivalent to the left panel. The anyon flux threading through the single-layer cylinder is given by $a$.

Figure 5

The left panel depicts the double-layer cylinder geometry with a layer-exchanging branch cut (dashed line). The white arrow represents the anyon flux $X_a^{\pm }$ threading through the cylinder. The Wilson line operator $W_{{(c,c)}^{+}}$ is indicated by the red line. The right panel depicts a single-layer cylinder geometry carrying the topological order described by $\mathcal{C}$ that is topologically equivalent to the left panel. The anyon flux threading through the single-layer cylinder is given by $a$. The application of the Wilson line operator $W_{{(c,c)}^{+}}$ on the left panel is topologically equivalent to applying the Wilson line operator $W_c$ on the single-layer cylinder geometry. The Wilson line operator $W_c$ brings an anyon $c\in \mathcal{C}$ around the cylinder.