  Motivated by the recent discovery of the Z_2 quantum spin liquid state in the
nearest neighbor Heisenberg model on the kagome lattice, we investigate the
"even-odd" effect occuring when this state is confined to infinitely long
cylinders of finite circumference. We pursue a dual analysis, where we map the
effective Z_2 gauge theory from the kagome lattice to a frustrated Ising model
on the dice lattice. Unexpectedly, we find that the latter theory, if
restricted to nearest neighbor interactions, is insufficient to capture this
effect. We provide an explanation of why further neighbor interactions are
needed via a high-temperature expansion of the effective Hamiltonian. We then
carry out projective symmetry group analysis to understand which second
neighbor interactions can be introduced while respecting the lattice
symmetries. Finally, we qualitatively compare our results to numerics by
computing the dimerization operator within our theory. Systems with odd
circumferences exhibit a non-vanishing dimerization that decays exponentially
with circumference.
