Deconfined Quantum Criticality, Scaling Violations, and Classical Loop Models

Abstract

Numerical studies of the transition between Néel 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 Néel-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 Néel order parameter). We use the loop model for simulations of unprecedentedly large systems (up to linear size L ¼ 512). Our results are compatible with a continuous transition at which both Néel 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 finitesize scaling, even if allowance is made for a weakly or marginally irrelevant scaling variable. In particular, different approaches to determining the anomalous dimensions η VBS and η Néel 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 CP n−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.

New Easy-Plane CPN−1 Fixed Points

“…Based on various numerical studies it is now believed that most likely N s < 2, so that Eq. 2 has a FP for all values of N (see [18] for a nice summary). A central issue we address here is the fate of these FPs when easy-plane anisotropy v is introduced.…”

Universality class of the two-dimensional polymer collapse transition

“…(7) (see Refs. [12,22,39] for more detail). We take a model with N -component complex vectors Z located on the links of the lattice, with fixed length | Z| 2 = N .…”

Quantum criticality with two length scales

“…We show that, contrary to past assumptions, ξ can govern the finite-size scaling even of magnetic properties that are sensitive only to ξ in the thermodynamic limit. Our simulations of the J-Q model at low temperatures and in the lowest S = 1 (two-spinon) excited state demonstrate complete agreement with the two-length scaling hypothesis, with no other anomalous scaling corrections remaining.Finite-size scaling forms Consider first a system with a single divergent correlation length ξ ∝ |δ| −ν , where δ = g − g c is the distance to a phase transition driven by quantum fluctuations 2 arising from non-commuting interactions controlled by g at T = 0 and g c is the critical value of g. In finite-size scaling theory [29], for a system of linear sizeclose to δ = 0 a singular quantity A takes the formwhere the exponents κ, ν, ω are tied to the universality class, κ also depends on A, and f ap-alternatively, T = 0) so that scaling arguments depending on T have been eliminated.The form (1) fails for some properties of the J-Q model [18,19,22] and other DQC candidate systems [26,24,25]. A prominent example is the spin stiffness, which for an infinite 2D…”