  Motivated by a recent experiment on Rb2Cu3SnF12, where spin-1/2 Cu2+ moments
reside on the layers of Kagome-like lattices, we investigate quantum ground
states of the antiferromagnetic Heisenberg model on a series of deformed Kagome
lattices. The deformation is characterized by a weaker exchange coupling
(alpha*J) on certain lattice links appropriate for Rb2Cu3SnF12 with alpha=1
corresponding to the ideal Kagome lattice. In particular, we study possible
valence bond solid phases using the perturbation theory around isolated dimer
limits, dimer series expansion, and self-consistent bond operator mean field
theory. It is shown that the valence bond solid phase with a 36-site unit cell
of the ideal Kagome lattice is quite sensitive to a small lattice distortion as
the kind discovered in Rb2Cu3SnF12. As a result, we find that a more likely
quantum ground state in Rb2Cu3SnF12 is the valence bond solid phase with a
12-site unit cell, where six dimers form a pinwheel structure, leading to
strong modification of the elementary triplet and singlet excitation spectra in
the deformed Kagome lattices.
