Topologically shadowed quantum criticality: A non-compact conformal manifold

TL;DR

This paper proposes a new class of topological quantum critical points characterized by a non-compact conformal manifold, constrained by the topological data of adjacent phases, and explores their scale invariance and non-local structures.

Contribution

It introduces a novel framework for topological quantum critical points with a non-compact conformal manifold constrained by topological shadowing effects.

Findings

Quantum dynamics at the critical point is determined by braiding angles of adjacent phases.

Two-loop RG calculations suggest the scale invariance of the proposed theories.

The critical theories form a continuous conformal manifold constrained by topological data.

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.