Deconfined Quantum Criticality, Scaling Violations, and Classical Loop Models

Numerical studies of the transition between Neel and valence bond solid phases in two-dimensional quantum antiferromagnets give strong evidence for the remarkable scenario of deconfined criticality, but display strong violations of finite-size scaling that are not yet understood. We show how to realize the universal physics of the Neel-valence-bond-solid (VBS) transition in a three-dimensional classical loop model ( this model includes the subtle interference effect that suppresses hedgehog defects in the Neel order parameter). We use the loop model for simulations of unprecedentedly large systems ( up to linear size L = 512). Our results are compatible with a continuous transition at which both Neel and VBS order parameters are critical, and we do not see conventional signs of first-order behavior. However, we show that the scaling violations are stronger than previously realized and are incompatible with conventional finite-size scaling, even if allowance is made for a weakly or marginally irrelevant scaling variable. In particular, different approaches to determining the anomalous dimensions. VBS and Neel yield very different results. The assumption of conventional finite-size scaling leads to estimates that drift to negative values at large sizes, in violation of the unitarity bounds. In contrast, the decay with distance of critical correlators on scales much smaller than system size is consistent with large positive anomalous dimensions. Barring an unexpected reversal in behavior at still larger sizes, this implies that the transition, if continuous, must show unconventional finite-size scaling, for example, from an additional dangerously irrelevant scaling variable. Another possibility is an anomalously weak first-order transition. By analyzing the renormalization group flows for the noncompact CPn-1 field theory (the n-component Abelian Higgs model) between two and four dimensions, we give the simplest scenario by which an anomalously weak first-order transition can arise without fine-tuning of the Hamiltonian.