Finite-size scaling in frustrated Heisenberg models

Numerical studies of the 2D frustrated antiferromagnets are hindered by the absence of large-scale quantum Monte Carlo methods and by large finite-size effects in other methods. Using the effective $\sigma$-model we demonstrate that the most significant finite-size effects can be eliminated by an appropriate choice of the cluster aspect ratio, allowing for much more precise estimates of observables already in small systems. We show that such a ``magic'' aspect ratio depends on the boundary conditions, with a simple and convenient choice being precisely the geometry optimal for the DMRG method. Combining the improved DMRG accuracy with the use of non-traditional clusters for rapidly converging finite-size scaling, we study the ordering in the square- and triangular-lattice Heisenberg models. We demonstrate the vanishing of the leading finite-size effect $\sim O(1/L)$ in the order parameter $M$ for the sequence of clusters with the ``magic'' aspect ratio ($L_x/L_y$ close to 2), in agreement with the effective $\sigma$-model. We determine the thermodynamic limit of $M$ for the square lattice with an error comparable to quantum Monte Carlo. For the triangular lattice, we verify the existence of three-sublattice magnetic order, and estimate the order parameter to be $M = 0.205(15)$.