Solving the puzzle of an unconventional phase transition for a 2d dimerized quantum Heisenberg model

Motivated by the indication of a new critical theory for the spin-1/2 Heisenberg model with a spatially staggered anisotropy on the square lattice as suggested in \cite{Wenzel08}, we re-investigate the phase transition of this model induced by dimerization using first principle Monte Carlo simulations. We focus on studying the finite-size scaling of $\rho_{s1} 2L$ and $\rho_{s2} 2L$, where $L$ stands for the spatial box size used in the simulations and $\rho_{si}$ with $i \in \{1,2\}$ is the spin-stiffness in the $i$-direction. Remarkably, while we do observe a large correction to scaling for the observable $\rho_{s1}2L$ as proposed in \cite{Fritz11}, the data for $\rho_{s2}2L$ exhibit a good scaling behavior without any indication of a large correction. As a consequence, we are able to obtain a numerical value for the critical exponent $\nu$ which is consistent with the known O(3) result with moderate computational effort. Specifically, the numerical value of $\nu$ we determine by fitting the data points of $\rho_{s2}2L$ to their expected scaling form is given by $\nu=0.7120(16)$, which agrees quantitatively with the most accurate known Monte Carlo O(3) result $\nu = 0.7112(5)$. Finally, while we can also obtain a result of $\nu$ from the observable second Binder ratio $Q_2$ which is consistent with $\nu=0.7112(5)$, the uncertainty of $\nu$ calculated from $Q_2$ is more than twice as large as that of $\nu$ determined from $\rho_{s2}2L$.