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 operatorsinto a bilinear form of a new set of dimer fermion operators. We constructparent Hamltonians of the columnar- and the staggered-VBC on the squarelattice, for which the VBC is an eigenstate in all regimes and the exact groundstate in some region of the phase diagram. We study the depart from the exactVBC regime upon tuning the anisotropy by means of the hierarchical mean fieldtheory and exact diagonalization on finite clusters. In both Hamiltonians, theVBC phase extends over the exact regime and transits to a columnarantiferromagnet (CAFM) through a window of intermediate phases, revealing anintriguing competition of correlation lengths at the VBC-CAFM transition. Themethod can be readily applied to construct other VBC parent Hamiltonians indifferent lattices and dimensions.
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