1202.2722 (R. F. Bishop et al.)

The frustrated Heisenberg antiferromagnet on the checkerboard lattice:
the $J_{1}$--$J_{2}$ model    [PDF]

R. F. Bishop, P. H. Y. Li, D. J. J. Farnell, J. Richter, C. E. Campbell

We study the ground-state (gs) phases of the spin-half anisotropic planar
pyrochlore (or crossed chain) model using the coupled cluster method (CCM). The
model is a frustrated antiferromagnetic (AFM) $J_{1}$--$J_{2}$ system on the
checkerboard lattice, with nearest-neighbor exchange bonds $J_{1}>0$ and
next-nearest-neighbor bonds $J_{2} \equiv \kappa J_{1} > 0$. Using various AFM
classical ground states as CCM model states we present results for their gs
energy, average on-site magnetization, and susceptibilities to plaquette
valence-bond crystal (PVBC) and crossed-dimer valence-bond crystal (CDVBC)
ordering. We show that the state with Neel ordering is the gs phase for $\kappa
< \kappa_{c_1} \approx 0.80 \pm 0.01$, but that none of the fourfold set of AFM
states selected by quantum fluctuations at $O(1/s)$ in a large-$s$ analysis
(where $s$ is the spin quantum number) from the infinitely degenerate set of
AFM states that form the gs phase for the classical version of the model (for
$\kappa>1$) survives the quantum fluctuations to form a stable
magnetically-ordered gs phase for the spin-half case. The Neel state becomes
susceptible to PVBC ordering at or very near to $\kappa = \kappa_{c_1}$, and
the fourfold AFM states become infinitely susceptible to PVBC ordering at
$\kappa = \kappa_{c_2} \approx 1.22 \pm 0.02$. In turn, we find that these
states become infinitely susceptible to CDVBC ordering for all values of
$\kappa$ above a certain critical value at or very near to $\kappa =
\kappa_{c_2}$. We thus find a Neel-ordered gs phase for $\kappa<\kappa_{c_1}$,
a PVBC-ordered phase for $\kappa_{c_1} < \kappa < \kappa_{c_2}$, and a
CDVBC-ordered phase for $\kappa > \kappa_{c_2}$. Both transitions are probably
direct ones, although we cannot exclude very narrow coexistence regions
confined to $0.79 \lesssim \kappa \lesssim 0.81$ and $1.20 \lesssim \kappa
\lesssim 1.22$ respectively.

View original: http://arxiv.org/abs/1202.2722