1202.3118 (Anders W. Sandvik)

Finite-size scaling in two-dimensional valence-bond-solids    [PDF]

Anders W. Sandvik

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.

View original: http://arxiv.org/abs/1202.3118