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

TL;DR

This study investigates valence-bond-solid (VBS) order in the ground state of a 2D quantum spin model, revealing how finite-size effects, boundary conditions, and system geometry influence the detection of VBS order and challenging claims of Z2 spin liquid states.

Contribution

The paper provides a detailed analysis of finite-size and boundary effects on VBS order in the J-Q model, emphasizing the importance of system geometry in interpreting quantum ground states.

Findings

Strong 2D VBS order is confirmed with increasing system size.

Long cylinders show only short-range correlations unless width exceeds a critical value.

Finite-size effects can mislead extrapolations of order parameters in small lattices.

Abstract

Various lattice geometries and boundaries 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 4- or 6-spin interactions Q are added to the Heisenberg exchange J. Ground state results for finite systems (with up to thousands of spins) are obtained using a projector QMC method. Great care has to be taken when extrapolating the order parameter to infinite size, in particular in cylinder geometry. Even though strong 2D VBS order exists and is established clearly with increasing system size on L*L lattices (or Lx* Ly lattices with a fixed Lx/Ly), only short-range VBS correlations are observed on long cylinders (when Lx -> infinity at fixed Ly). The correlation length increases with Ly, until long-range order sets in at a "critical" Ly. This width is large even when the 2D order is strong, e.g, for a system where the order parameter is 70% of the largest possible value, Ly=8 is required for ordering. Correlation functions for small L*L lattices can also be misleading. For a 20%-ordered system results for L up to 20 appear to extrapolate to a vanishing order parameter, while for larger L the behavior crosses over and extrapolates to a non-zero value (with exponentially small finite size corrections). The VBS order also exhibits interesting edge effects related to emergent U(1) symmetry, which, if not considered properly, can lead to wrong conclusions for the thermodynamic limit. The finite-size behavior for small L*L lattices and long cylinders is similar to that predicted for a Z2 spin liquid. The results raise concerns about recent works claiming Z2 spin liquid ground states in frustrated 2D systems, in particular, the Heisenberg model with nearest and next-nearest-neighbor couplings. VBS state in this system cannot be ruled out.