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.

We show numerically that the "deconfined" quantum critical point between the Néel antiferromagnet and the columnar valence-bond solid, for a square lattice of spin 1=2, has an emergent SO(5) symmetry. This symmetry allows the Néel vector and the valence-bond solid order parameter to be rotated into each other. It is a remarkable (2 þ 1)-dimensional analogue of the SOð4Þ ¼ ½SUð2Þ × SUð2Þ =Z 2 symmetry that appears in the scaling limit for the spin-1=2 Heisenberg chain. The emergent SO(5) symmetry is strong evidence that the phase transition in the (2 þ 1)-dimensional system is truly continuous, despite the violations of finite-size scaling observed previously in this problem. It also implies surprising relations between correlation functions at the transition. The symmetry enhancement is expected to apply generally to the critical two-component Abelian Higgs model (noncompact CP 1 model). The result indicates that in three dimensions there is an SO (5) Many condensed matter systems show higher symmetry in the infrared than they do in the ultraviolet. The liquid-gas critical point is a classical example: although there is no microscopic Z 2 symmetry exchanging liquidlike and gaslike configurations, the fixed point has an emergent Z 2 symmetry and is simply the Ising fixed point. Microscopically, this fixed point is perturbed by operators that break the Z 2 symmetry, but it nevertheless governs the critical behavior because these perturbations are irrelevant under the renormalization group.To reach this critical point two variables, say temperature and pressure, must be tuned. The spin-1=2 Heisenberg chain provides an example of emergent symmetry without such fine-tuning in a quantum setting. The ground state of this model is well known to be critical. Its microscopic symmetries are SU(2) spin rotations, together with spatial symmetries. However the scaling limit of the spin-1=2 chain is the SUð2Þ 1 Wess-Zumino-Witten conformal field theory [1], and this has an SOð4Þ ¼ ½SUð2Þ × SUð2Þ =Z 2 symmetry that is much larger than the global symmetry present microscopically.Physically, this arises as follows [2]. The Néel vectorÑ has three components. There is also a spin-Peierls parameter φ that distinguishes between the two different patterns of dimer (singlet) order and which changes sign under appropriate reflections or translations. We may form the four-component superspinΦ ¼ ðÑ; φÞ, and the emergent SO(4) corresponds to rotations of this vector. Although the dimer and Néel order parameters are utterly inequivalent microscopically, a symmetry between them arises in the infrared. Technically, this again relies on the SO(4)-breaking perturbations of the conformal field theory (CFT) being irrelevant or marginally irrelevant.Naively, one might expect this phenomenon to be special to one spatial dimension, where the enlarged symmetry is related to special properties of 2D conformal invariance (the doubling of conserved currents [1]). We show here however that an analogous symmetry enhancement occurs for the spin-1=2 magnet ...

We consider the statistical mechanics of a class of models involving close-packed loops with fugacity n on three-dimensional lattices. The models exhibit phases of two types as a coupling constant is varied: in one, all loops are finite, and in the other, some loops are infinitely extended. We show that the loop models are discretisations of CP n−1 σ models. The finite and infinite loop phases represent, respectively, disordered and ordered phases of the σ model, and we discuss the relationship between loop properties and σ model correlators. On large scales, loops are Brownian in an ordered phase and have a non-trivial fractal dimension at a critical point. We simulate the models, finding continuous transitions between the two phases for n = 1, 2, 3 and first order transitions for n ≥ 4. We also give a renormalisation group treatment of the CP n−1 model that shows how a continuous transition can survive for values of n larger than (but close to) two, despite the presence of a cubic invariant in the Landau-Ginzburg description. The results we obtain are of broader relevance to a variety of problems, including SU (n) quantum magnets in (2+1) dimensions, Anderson localisation in symmetry class C, and the statistics of random curves in three dimensions.

Many statistical mechanics problems can be framed in terms of random curves; we consider a class of three-dimensional loop models that are prototypes for such ensembles. The models show transitions between phases with infinite loops and short-loop phases. We map them to CP(n-1) sigma models, where n is the loop fugacity. Using Monte Carlo simulations, we find continuous transitions for n=1, 2, 3, and first order transitions for n≥5. The results are relevant to line defects in random media, as well as to Anderson localization and (2+1)-dimensional quantum magnets.

The universal behaviour of two-dimensional loop models can change dramatically when loops are allowed to cross. We study models with crossings both analytically and with extensive Monte Carlo simulations. Our main focus (the 'completely packed loop model with crossings') is a simple generalisation of well-known models which shows an interesting phase diagram with continuous phase transitions of a new kind. These separate the unusual 'Goldstone' phase observed previously from phases with short loops. Using mappings to Z_2 lattice gauge theory, we show that the continuum description of the model is a replica limit of the sigma model on real projective space (RP^{n-1}). This field theory sustains Z_2 point defects which proliferate at the transition. In addition to studying the new critical points, we characterise the universal properties of the Goldstone phase in detail, comparing renormalisation group (RG) calculations with numerical data on systems of linear size up to L=10^6 at loop fugacity n=1. (Very large sizes are necessary because of the logarithmic form of correlation functions and other observables.) The model is relevant to polymers on the verge of collapse, and a particular point in parameter space maps to self-avoiding trails at their \Theta-point; we use the RG treatment of a perturbed sigma model to resolve some perplexing features in the previous literature on trails. Finally, one of the phase transitions considered here is a close analogue of those in disordered electronic systems --- specifically, Anderson metal-insulator transitions --- and provides a simpler context in which to study the properties of these poorly-understood (central-charge-zero) critical points.Comment: Published version. 22 pages, 16 figure

We investigate approximate emergent nonabelian symmetry in a class of weakly first order 'deconfined' phase transitions using Monte Carlo simulations and a renormalization group analysis. We study a transition in a 3D classical loop model that is analogous to a deconfined 2+1D quantum phase transition in a magnet with reduced lattice symmetry. The transition is between the Néel phase and a twofold degenerate valence bond solid (lattice-symmetry-breaking) phase. The combined order parameter at the transition is effectively a four-component superspin. It has been argued that in some weakly first order 'pseudocritical' deconfined phase transitions, the renormalization group flow can take the system very close to the ordered fixed point of the symmetric O(N ) sigma model, where N is the total number of 'soft' order parameter components, despite the fact that O(N ) is not a microscopic symmetry. This yields a first order transition with unconventional phenomenology. We argue that this occurs in the present model, with N = 4. This means that there is a regime of lengthscales in which the transition resembles a 'spin-flop' transition in the ordered O(4) sigma model. We give numerical evidence for (i) the first order nature of the transition, (ii) the emergence of O(4) symmetry to an accurate approximation, and (iii) the existence of a regime in which the emergent O(4) is 'spontaneously broken', with distinctive features in the order parameter probability distribution. These results may be relevant for other models studied in the literature, including 2+1D QED with two flavours, the 'easy-plane' deconfined critical point, and the Néel-VBS transition on the rectangular lattice. CONTENTS

Statistical lattice ensembles of loops in three or more dimensions typically have phases in which the longest loops fill a finite fraction of the system. In such phases it is natural to ask about the distribution of loop lengths. We show how to calculate moments of these distributions using CP(n-1) or RP(n-1) and O(n) σ models together with replica techniques. The resulting joint length distribution for macroscopic loops is Poisson-Dirichlet with a parameter θ fixed by the loop fugacity and by symmetries of the ensemble. We also discuss features of the length distribution for shorter loops, and use numerical simulations to test and illustrate our conclusions.

The Rosenzweig-Porter model is a one-parameter family of random matrices with three different phases: ergodic, extended non-ergodic and localized. We characterize numerically each of these phases and the transitions between them. We focus on several quantities that exhibit non-analytical behaviour and show that they obey the scaling hypothesis. Based on this, we argue that non-ergodic chaotic and ergodic regimes are separated by a continuous phase transition, similarly to the transition between non-ergodic chaotic and localized phases.