Reflection and time reversal symmetry enriched topological phases of matter: path integrals, non-orientable manifolds, and anomalies

TL;DR

<3-5 sentence high-level summary>We address the classification of (2+1)D symmetry-enriched topological (SET) phases with time-reversal and/or spatial-reflection symmetry using a unified, topological path-integral framework. By combining the Turaev–Viro–Barrett–Westbury construction with SPT-state-sum ideas and extending to G-equivariant (and anti-unitary) structures, we obtain a broad, exact description of SET phases defined on general space-time manifolds, including non-orientable ones, and derive explicit formulas for ground-state degeneracy and anomaly detection. We show how Dehn-twist anomalies on non-orientable manifolds relate to 3+1)D SPT invariants Z(RP^4) and Z(CP^2), and outline how to determine which (3+1)D SPT hosts a given (2+1)D SET boundary. The paper provides numerous concrete examples (D(H), Z_N toric code, decorated toric code, D(S_3), gauged T-Pfaffian) that illustrate anomaly structure, symmetry fractionalization, and the boundary-bulk correspondence, highlighting the framework's power and limitations for understanding symmetry-enriched topological matter.

Abstract

We study symmetry-enriched topological (SET) phases in 2+1 space-time dimensions with spatial reflection and/or time-reversal symmetries. We provide a systematic construction of a wide class of reflection and time-reversal SET phases in terms of a topological path integral defined on general space-time manifolds. An important distinguishing feature of different topological phases with reflection and/or time-reversal symmetry is the value of the path integral on non-orientable space-time manifolds. We derive a simple general formula for the path integral on the manifold $Σ^2 \times S^1$, where $Σ^2$ is a two-dimensional non-orientable surface and $S^1$ is a circle. This also gives an expression for the ground state degeneracy of the SET on the surface $Σ^2$ that depends on the reflection symmetry fractionalization class, generalizing the Verlinde formula for ground state degeneracy on orientable surfaces. Consistency of the action of the mapping class group on non-orientable manifolds leads us to a constraint that can detect when a time-reversal or reflection SET phase is anomalous in (2+1)D and, thus, can only exist at the surface of a (3+1)D symmetry protected topological (SPT) state. Given a (2+1)D reflection and/or time-reversal SET phase, we further derive a general formula that determines which (3+1)D reflection and/or time-reversal SPT phase hosts the (2+1)D SET phase as its surface termination. A number of explicit examples are studied in detail.

Figures (43)

• Figure 1: Two triangles with opposite orientations. A triangle with positive orientation (right) is assigned $s(\Delta^2)=\openone$. A triangle with negative orientation (left) is assigned $s(\Delta^2)=\ast$.
• Figure 2: A cellulation of $\mathbb{RP}^2$ with two vertices labeled by group elements. $\mathbb{RP}^2$ is obtained from a square by identifying antipodal points along the boundary, i.e. the top edge is identified with the bottom edge in opposite directions and the left edge is identified with the right edge in opposite directions. When vertices of the triangulation of the square are identified, their corresponding group labels are equated through multiplication by ${\bf r}$, which reflects the orientation reversal involved in identifying antipodal points of the boundary to form $\mathbb{RP}^2$. The relative orientation of the triangles of the cellulation of $\mathbb{RP}^2$ is induced from triangulation of the square.
• Figure 3: (1+1)D SPT states. Ellipses enclosing neighboring spins indicate that the enclosed pair of spins form a spin singlet state: $\frac{1}{\sqrt{2}} ( |\uparrow \downarrow\rangle - |\downarrow \uparrow\rangle)$. Top: The "nontrivial" (1+1)D SPT state, analogous to the AKLT state describing the Haldane chain. Bottom: The "trivial" (1+1)D SPT state. Bond-centered reflection corresponds to a reflection about the dashed line.
• Figure 4: A pair of anyons $a$ and $\bar{a}$ created from vacuum, located in regions that are related by the reflection ${\bf r}$.
• Figure 5: (a) The left and right boundaries of the cylinder have topological charge $a$ and $\overline{a}$ respectively. Under the action of time reversal, ${\bf T}$, these get permuted to $\rho_{\bf T}(a) \equiv \,^{\bf{T}}a$ and $\rho_{\bf T}(\overline{a}) \equiv \,^{\bf{T}}\overline{a}$, respectively. (b) Reflection ${\mathbf{r}}$ takes the coordinate along the cylinder $x \rightarrow -x$, so that the topological charges at the boundary get permuted to $\rho_{\mathbf{r}}(\overline{a}) \equiv \, {}^{\mathbf{r}}{a}$ and $\rho_{\bf T}(\overline{a}) \equiv \,^{\bf{T}}\overline{a}$, respectively.