  We present a general method to construct translation-invariant and SU(2)
symmetric antiferromagnetic parent Hamiltonians of valence bond crystals (VBC).
The method is based on a canonical mapping transforming S=1/2 spin operators
into a bilinear form of a new set of dimer fermion operators. We construct
parent Hamltonians of the columnar- and the staggered-VBC on the square
lattice, for which the VBC is an eigenstate in all regimes and the exact ground
state in some region of the phase diagram. We study the depart from the exact
VBC regime upon tuning the anisotropy by means of the hierarchical mean field
theory and exact diagonalization on finite clusters. In both Hamiltonians, the
VBC phase extends over the exact regime and transits to a columnar
antiferromagnet (CAFM) through a window of intermediate phases, revealing an
intriguing competition of correlation lengths at the VBC-CAFM transition. The
method can be readily applied to construct other VBC parent Hamiltonians in
different lattices and dimensions.
