  General conditions in which disordered, spin liquid, and valence-bond ordered
phases occur in quantum Ising antiferromagnets are studied using the prototype
Kagome lattice spin models. A range of quantum dynamical processes in the Ising
model, with and without total Ising spin conserved, are analytically shown to
yield all three characteristic quantum paramagnetic phases in the Kagome
system. Special emphasis is given to the XXZ model that can be sensibly
compared to the Kagome lattice Heisenberg antiferromagnet. It is explicitly
demonstrated that the total-spin-conserving dynamics can yield a resonant
valence bond (RVB) liquid phase with very short-ranged correlations, but also a
valence-bond ordered phase compatible with the one proposed to explain the
seemingly gapless singlet states of the Heisenberg antiferromagnet on the
Kagome lattice. Likely consequences for generic spin models are discussed. The
analysis combines compact U(1) gauge theory, duality transformations,
lattice-field-theoretical methods, and variational approach.
