Quantum transition points in the J-Q model--the test bed of the deconfined critical point theory--and the SU(2)-symmetric discrete noncompact CP(1) representation of the deconfined critical action are directly compared by the flowgram method. We find that the flows of two systems coincide in a broad region of linear system sizes (10 < L < 50 for the J-Q model), implying that the deconfined critical point theory correctly captures the mesoscopic physics of competition between the antiferromagnetic and valence-bond orders in quantum spin systems. At larger sizes, however, we observe significant deviations between the two flows which both demonstrate strong violations of scale invariance. This reliably rules out the second-order transition scenario in at least one of the two models and suggests the most likely explanation for the nature of the transition in the J-Q model.

We study the low-temperature physics of the SU(2)-symmetric spin-1/2 Heisenberg antiferromagnet on a pyrochlore lattice and find "fingerprint" evidence for the thermal spin-ice state in this frustrated quantum magnet. Our conclusions are based on the results of bold diagrammatic Monte Carlo simulations, with good convergence of the skeleton series down to the temperature T /J = 1/6. The identification of the spin-ice state is done through a remarkably accurate microscopic correspondence for static structure factor between the quantum Heisenberg, classical Heisenberg, and Ising models at all accessible temperatures, and the characteristic bowtie pattern with pinch points observed at T /J = 1/6. The dynamic structure factor at real frequencies (obtained by the analytic continuation of numerical data) is consistent with diffusive spinon dynamics at the pinch points.

We argue that the four-state Potts antiferromagnet has a finite-temperature phase transition on any Eulerian plane triangulation in which one sublattice consists of vertices of degree 4. We furthermore predict the universality class of this transition. We then present transfer-matrix and Monte Carlo data confirming these predictions for the cases of the Union Jack and bisected hexagonal lattices.