A $(2+1)\mathrm{D}$ topologically ordered phase may or may not have a gappable edge, even if its chiral central charge $c_{-}$ is vanishing. Recently, it was discovered that a quantity regarded as a “higher” version of chiral central charge gives a further obstruction beyond $c_{-}$ to gapping out the edge. In this Letter, we show that the higher central charges can be characterized by the expectation value of the partial rotation operator acting on the wave function of the topologically ordered state. This allows us to extract the higher central charge from a single wave function, which can be evaluated on a quantum computer. Our characterization of the higher central charge is analytically derived from the modular properties of edge conformal field theory, as well as the numerical results with the $\nu =1/2$ bosonic Laughlin state and the non-Abelian gapped phase of the Kitaev honeycomb model, which corresponds to $\mathrm{U}(1{)}_2$ and Ising topological order, respectively. The Letter establishes a numerical method to obtain a set of obstructions to the gappable edge of $(2+1)\mathrm{D}$ bosonic topological order beyond $c_{-}$, which enables us to completely determine if a $(2+1)\mathrm{D}$ bosonic Abelian topological order has a gappable edge or not. We also point out that the expectation values of the partial rotation on a single wave function put a constraint on the low-energy spectrum of the bulk-boundary system of $(2+1)\mathrm{D}$ bosonic topological order, reminiscent of the Lieb-Schultz-Mattis-type theorems.

• Received 25 March 2023
• Revised 7 October 2023
• Accepted 18 November 2023

DOI:https://doi.org/10.1103/PhysRevLett.132.016602