Resonating Valence Bond Ground States on Corner-sharing Simplices

The Hubbard model in the $U\to\infty$ limit has been known to have resonating valence bond (RVB) ground states on certain corner-sharing simplex lattices. Examples include both the quasi-1D sawtooth lattice with open boundary and a larger class of higher dimensional lattices without boundaries. The two types of results were obtained by different approaches which do not apply to one another. In the second class of lattices, the simplest simplex is a tetrahedron. We hereby generalize both results by studying the singly hole-doped system on the quasi-1D lattice of a tetrahedron chain, which can be considered a stripe of the pyrochlore or checkerboard lattices. The energy level ordering of irreducible representations of each tetrahedron shows that a chain of them has exponentially degenerate partial RVB or dimer-monomer ground states where each tetrahedron hosts one spin-$1/2$ monomer and one spin-$0$ dimer. The exact ground states in the infinitely long chain limit are analytically solved by introducing basis transformations between local Hilbert spaces of neighboring tetrahedra, and its energy agrees with the extrapolation of numerical exact diagonalization results of finite sized systems.