Consistent scaling exponents at the deconfined quantum-critical point

We report a quantum Monte Carlo study of the phase transition between antiferromagnetic and valence-bond solid ground states in the square-lattice $S=1/2$ $J$-$Q$ model. The critical correlation function of the $Q$ terms gives a scaling dimension corresponding to the value $ u = 0.455 pm 0.002$ of the correlation-length exponent. This value agrees with previous (less precise) results from conventional methods, e.g., finite-size scaling of the near-critical order parameters. We also study the $Q$-derivatives of the Binder cumulants of the order parameters for $L^2$ lattices with $L$ up to $448$. The slope grows as $L^{1/ u}$ with a value of $ u$ consistent with the scaling dimension of the $Q$ term. There are no indications of runaway flow to a first-order phase transition. The mutually consistent estimates of $ u$ provide compelling support for a continuous deconfined quantum-critical point.