Emergent SO(5)SO(5) Symmetry at the N\'eel to Valence-Bond-Solid Transition

We show numerically that the `deconfined' quantum critical point between the N\'eel antiferromagnet and the columnar valence-bond-solid, for a square lattice of spin-1/2s, has an emergent $SO(5)$ symmetry. This symmetry allows the N\'eel vector and the valence-bond-solid order parameter to be rotated into each other. It is a remarkable 2+1-dimensional analogue of the $SO(4)= [SU(2)\times SU(2)]/Z_2$ symmetry that appears in the scaling limit for the spin-1/2 Heisenberg chain. The emergent $SO(5)$ is strong evidence that the phase transition in the 2+1D 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 (non-compact $CP^1$ model). The result indicates that in three dimensions there is an $SO(5)$-symmetric conformal field theory which has no relevant singlet operators, so is radically different to conventional Wilson-Fisher-type conformal field theories.Comment: 4+epsilon pages, 6 figure

Length Distributions in Loop Soups

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) $\sigma$ models together with replica techniques. The resulting joint length distribution for macroscopic loops is Poisson-Dirichlet with a parameter $\theta$ 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.Comment: 4.5 page

Deconfined Quantum Criticality, Scaling Violations, and Classical Loop Models

Numerical studies of the N\'eel to valence-bond solid phase transition in 2D 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 realise the universal physics of the Neel-VBS transition in a 3D classical loop model (this includes the interference effect that suppresses N\'eel hedgehogs). We use this model to simulate unprecedentedly large systems (of size $L\leq 512$). Our results are compatible with a direct continuous transition at which both order parameters are critical, and we do not see conventional signs of first-order behaviour. However, we find that the scaling violations are stronger than previously realised and are incompatible with conventional finite-size scaling over the size range studied, even if allowance is made for a weakly/marginally irrelevant scaling variable. In particular, different determinations of the anomalous dimensions $\eta_\text{VBS}$ and $\eta_\text{N\'eel}$ yield very different results. The assumption of conventional finite-size scaling gives estimates which drift to negative values at large $L$, in violation of unitarity bounds. In contrast, the behaviour of correlators on scales much smaller than $L$ is consistent with large positive anomalous dimensions. Barring an unexpected reversal in behaviour at still larger sizes, this implies that the transition, if continuous, must show unconventional finite-size scaling, e.g. from a dangerously irrelevant scaling variable. Another possibility is an anomalously weak first-order transition. By analysing the renormalisation group flows for the non-compact $CP^{n-1}$ model ($n$-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.Comment: 20 pages, 19 figure

Growth in systems of vesicles and membranes

We present a theoretical study for the intermediate stages of the growth of membranes and vesicles in supersaturated solutions of amphiphilic molecules. The problem presents important differences with the growth of droplets in the classical theory of Lifshitz-Slyozov-Wagner, because the aggregates are extensive only in two dimensions, but still grow in a three dimensional bath. The balance between curvature and edge energy favours the nucleation of small planar membranes, but as they grow beyond a critical size they close themselves to form vesicles. We obtain a system of coupled equations describing the growth of planar membranes and vesicles, which is solved numerically for different initial conditions. Finally, the range of parameters relevant in experimental situations is discussed.Comment: 13 pages and 5 postscript figures. To appear in Phys. Rev

3D loop models and the CP^{n-1} sigma model

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 $n\geq 5$. The results are relevant to line defects in random media, as well as to Anderson localization and $(2+1)$-dimensional quantum magnets.Comment: Published versio