Finite-size scaling and boundary effects in two-dimensional valence-bond solids

Abstract

Various lattice geometries and boundary conditions are used to investigate valence-bond-solid (VBS) ordering in the ground state of an $S=1/2$ square-lattice quantum spin model---the $J$-$Q$ model, in which four- or six-spin interactions $Q$ are added to the standard Heisenberg exchange $J$. Ground-state results for finite systems (with up to thousands of spins) are obtained using an unbiased projector quantum Monte Carlo method. It is found that great care has to be taken when extrapolating the order parameter to infinite lattice size, in particular, in cylinder geometry. Even though strong VBS order exists in two dimensions, and is established clearly with increasing system size on $L\ifmmode\times\else\texttimes\fi{}L$ lattices (or ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}$ lattices with a fixed aspect ratio ${L}_{x}/{L}_{y}$ of order 1), only short-range VBS correlations are observed on long cylinders (when ${L}_{x}\ensuremath{\rightarrow}\ensuremath{\infty}$ at fixed ${L}_{y}$). The correlation length increases with the cylinder width, until long-range order sets in at a ``critical'' width. This width is very large even when the 2D order is relatively strong. For example, for a system in which the order parameter is $70%$ of the largest possible value, ${L}_{y}=8$ is required for ordering. Extrapolations of the VBS order parameter based on correlation functions (the square of the order parameter) for small $L\ifmmode\times\else\texttimes\fi{}L$ lattices can also be misleading. For a $20%$-ordered system, results for $L$ up to $\ensuremath{\approx}$20 appear to extrapolate clearly to a vanishing order parameter, while for larger lattices the scaling behavior crosses over and extrapolates to a nonzero value (with exponentially small finite-size corrections). The VBS order parameter also exhibits interesting edge effects related the known emergent U(1) symmetry close to a ``deconfined'' critical point, which, if not considered properly, can lead to wrong conclusions for the thermodynamic limit. The observed finite-size behavior for small $L\ifmmode\times\else\texttimes\fi{}L$ lattices and long cylinders is very similar to that predicted for a ${Z}_{2}$ spin liquid. The results therefore raise concerns about recent numerical work claiming ${Z}_{2}$ spin-liquid ground states in 2D frustrated quantum spin systems, in particular, the Heisenberg model with nearest and next-nearest-neighbor couplings. Based on the results presented so far, a VBS state in this system cannot be ruled out.