Framing Anomaly in Lattice Chern-Simons-Maxwell Theory

Abstract

Framing anomaly is a key property of $(2+1)d$ chiral topological orders, for it reveals that the chirality is an intrinsic bulk property of the system, rather than a property of the boundary between two systems. Understanding framing anomaly in lattice models is particularly interesting, as concrete, solvable lattice models of chiral topological orders are rare. In a recent work, we defined and solved the $U(1)$ Chern-Simons-Maxwell theory on spacetime lattice, showing its chiral edge mode and the associated gravitational anomaly on boundary. In this work, we show its framing anomaly in the absence of boundary, by computing the expectation of a lattice version of the modular $T$ operator in the ground subspace on a spatial torus, from which we extract that $\langle T \rangle$ has a universal phase of $-2π/12$ as expected: $-2π/8$ from the Gauss-Milgram sum of the topological spins of the ground states, and $2π/24$ from the framing anomaly; we can also extract the $2π/24$ framing anomaly phase alone from the full spectrum of $T$ in the ground subspace by computing $\langle T^m \rangle$. This pins down the last and most crucial property required for a valid lattice definition of $U(1)$ Chern-Simons theory.

Figures (9)

• Figure 1: Our convention of modular $T$ operator actively brings the values of the fields at $(x, y)$ to $(x+yL_x/L_y, y)$. (In the literature the convention is often passive.)
• Figure 2: (Top) The cup product $(X\cup Y)_c$ for a lattice 1-form $X$ and 2-form $Y$ on a cube $c$ is a sum of three terms, each with the $X$ value on a purple link multiplied to the $Y$ value on an associated orange plaquette. (Bottom) $(Y\cup X)_c$ is a different sum of three terms.
• Figure 3: The lower part of the path integral constructs $e^{-\beta H}$, while the upper part defines the lattice modular $T$ operator, and they glue as indicated, to form $Z_\mathcal{T}=\mathbf{Tr}(T e^{-\beta H})$. (Periodicity understood in $x,y$-directions.)
• Figure 4: (Left) $(X \cup Y)_c$ for cube $c$ at the lower layer of the lattice $T$ operator receives an extra term in addition to the usual ones in \ref{['fig:cup']}, given by the product of $X$ on the purple link and $Y$ on the orange triangular plaquette. (Right) $(Y \cup X)_c$ for cube $c$ at the lower layer of the lattice $T$ operator also receives an extra term: now $X$ on the orange link multiplies not only to $Y$ on the purple square plaquette on the side as usual, but to the sum of $Y$ on both the square and the triangular purple plaquettes.
• Figure 5: The orange dots are phases of $Z_{\mathcal{T}}$ for $k=1$, $e^2=1$ at different $L$'s. The blue curve is the quadratic fit using the $L$'s from $L_1=129$ to $L_2=256$.