Topologically shadowed quantum criticality: A non-compact conformal manifold

Abstract

We put forward a proposal for topological quantum critical points (tQCPs) separating non-invertible chiral topological orders in $(2+1)$ dimensions. We conjecture that these tQCPs can be captured by a family of scale-invariant field theories forming a non-compact scale-invariant manifold. A central feature of our proposal is topological shadowing: the underlying critical theory is rigorously constrained by the global topological data of the two adjacent gapped phases. These theories can be further projected into quantum field theories with universal non-local structures. Specifically, we show that the quantum dynamics of the $U(1)$ symmetric critical point uniquely characterized by a topological angle $\Theta_{\text{cft}}$ -- which is defined by a commutator between two Wilson loop operators on a torus -- is determined by the braiding angles $\Theta_{1,2}$ of the adjacent gapped phases via the relation $\Theta_{\text{cft}}^{-1} =\frac{1}{2}[\Theta_1^{-1} + \Theta_2^{-1}]$. Despite the non-locality, our renormalization group calculations (up to two-loop order) strongly suggest that the theory shall maintain exact scale invariance. This establishes, without supersymmetry, a continuous manifold of fixed points that naturally becomes a conformal manifold when the local structure is further enforced.