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We employ a plaquette basis-generated by coupling the four spins in a$2\times2$ lattice to a well-defined total angular momentum-for the study ofHeisenberg ladders with antiferromagnetic coupling. Matrix elements of theHamiltonian in this basis are evaluated using standard techniques inangular-momentum (Racah) algebra. We show by exact diagonalization of small($2\times4$ and $2\times6$) systems that in excess of 90% of the ground-stateprobability is contained in a very small number of basis states. These fewbasis states can be used to define a severely truncated basis which we use toapproximate low-lying exact eigenstates. We show how, in this low-energy basis,the isotropic spin-1/2 Heisenberg ladder can be mapped onto an anisotropicspin-1 ladder for which the coupling along the rungs is much stronger than thecoupling between the rungs. The mapping thereby generates two distinct energyscales which greatly facilitates understanding the dynamics of the originalspin-1/2 ladder. Moreover, we use these insights to define an effectivelow-energy Hamiltonian in accordance to the newly developed COntractorREnormalization group (CORE) method. We show how a simple range-2 COREapproximation to the effective Hamiltonian to be used with our truncated basisreproduces the low-energy spectrum of the exact $2\times6$ theory at the $\alt1%$ level.Comment: 12 pages with two postscript figure

Plaquette basis for the study of Heisenberg ladders