  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.
