Linear independence of nearest-neighbor valence-bond states on the kagome lattice and construction of S U ( 2 ) -invariant spin- 1 2 Hamiltonian with a Sutherland-Rokhsar-Kivelson quantum liquid ground state

A class of local $SU(2)$-invariant spin-$\frac{1}{2}$ Hamiltonians is studied that has ground states within the space of nearest-neighbor valence-bond states on the kagome lattice. Cases include ``generalized Klein'' models without obvious non-valence-bond ground states, as well as a ``resonating valence-bond'' Hamiltonian whose unique ground states within the nearest-neighbor valence-bond space are four topologically degenerate ``Sutherland-Rokhsar-Kivelson'' (SRK) -type wave functions, which are expected to describe a gapped ${\mathbb{Z}}_{2}$ spin liquid. The proof of this uniqueness is intimately related to the linear independence of the nearest-neighbor valence-bond states on quite general and arbitrarily large kagome lattices, which is also established in this work. It is argued that the SRK ground states are also unique within the entire Hilbert space depending on properties of the generalized Klein models. Applications of the strategies developed in this work to other lattice types are also discussed.