Anomaly indicators for topological orders with $U\left(1\right)$ and time-reversal symmetry

We study anomalies in time-reversal-(${\mathbb{Z}}_2^T$) and $U\left(1\right)$-symmetric topological orders. In this context, an anomalous topological order is one that cannot be realized in a strictly $(2+1)$-dimensional system but can be realized on the surface of a $(3+1)$-dimensional [($3+1$)D] symmetry-protected topological (SPT) phase. To detect these anomalies we propose several anomaly indicators, functions that take as input the algebraic data of a symmetric topological order and that output a number indicating the presence or absence of an anomaly. We construct such indicators for both structures of the full symmetry group, i.e., $U\left(1\right)⋊{\mathbb{Z}}_2^T$ and $U\left(1\right)\times {\mathbb{Z}}_2^T$, and for both bosonic and fermionic topological orders. In all cases we conjecture that our indicators are complete in the sense that the anomalies they detect are in one-to-one correspondence with the known classification of $(3+1)\mathrm{D}$ SPT phases with the same symmetry. We also show that one of our indicators for bosonic topological orders has a mathematical interpretation as a partition function for the bulk $(3+1)\mathrm{D}$ SPT phase on a particular manifold and in the presence of a particular background gauge field for the $U\left(1\right)$ symmetry.

Figure 1

(a) Threading a thin $2\pi$-flux tube in a two-dimensional system. (b) Threading a thin $2\pi$-flux tube through a three-dimensional bosonic topological insulator slab featuring a symmetric topologically ordered top surface and a symmetry-breaking (and short-range entangled) bottom surface with ${\sigma }_H=1$. In both figures, the blue circle represents the location of the flux anyon $f$.

Figure 2

A 3D electronic topological insulator in a thickened torus (or Corbino donut) geometry with fluxes ${\mathrm{\Phi }}_x$ and ${\mathrm{\Phi }}_y$ threading the $x$ and $y$ cycles of the torus. Also pictured is the string operator $W_o^y$ (the red line) wrapping the $y$ cycle of the outer surface of the torus.