Bootstrapping the Simplest Deconfined Quantum Critical Point

TL;DR

This paper uses conformal bootstrap techniques to analyze the $CP^{2}$ model, providing bounds on operator dimensions that support its description as a deconfined quantum critical point, with results matching large $N$ predictions and lattice data.

Contribution

It applies conformal bootstrap to the $CP^{2}$ model to derive operator dimension bounds, supporting its role as a deconfined quantum critical point and validating large $N$ predictions.

Findings

Bootstrap bounds match large $N$ predictions for monopole operator dimensions.

Predicted scaling dimensions for lowest spinning monopole operators.

Supports the $CP^{2}$ model as a deconfined quantum critical point.

Abstract

We study the $N=3$ case of the $CP^{N-1}$ model, which is a field theory of $N$ complex scalars in $3d$ coupled to an Abelian gauge field with $SU(N) \times U(1)$ global symmetry. Recent evidence suggests the $N=2$ theory is not critical, which makes the $N=3$ theory the simplest possibility of deconfined quantum criticality. We apply the conformal bootstrap to correlators of charge $q=0,1,2$ scalar operators under the $U(1)$ symmetry, which gives us access also to $q=3,4$ operators. After imposing that only the lowest $q=0,1,2$ scalar operators are relevant, we find that the bootstrap bounds are saturated by the large $N$ prediction for $q=1,2,3,4$ scalar monopole operator scaling dimensions, which were shown earlier to be accurate even for small $N$, as well as a lattice prediction for the $q=0$ non-monopole scalar operator. We also predict the scaling dimensions of the lowest spinning monopole operators, which we match to the large charge prediction for spinning operators. This suggests that the critical $CP^{2}$ model is described by this bootstrap bound.