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Motivated by the recent discovery of the Z_2 quantum spin liquid state in thenearest neighbor Heisenberg model on the kagome lattice, we investigate the"even-odd" effect occuring when this state is confined to infinitely longcylinders of finite circumference. We pursue a dual analysis, where we map theeffective Z_2 gauge theory from the kagome lattice to a frustrated Ising modelon the dice lattice. Unexpectedly, we find that the latter theory, ifrestricted to nearest neighbor interactions, is insufficient to capture thiseffect. We provide an explanation of why further neighbor interactions areneeded via a high-temperature expansion of the effective Hamiltonian. We thencarry out projective symmetry group analysis to understand which secondneighbor interactions can be introduced while respecting the latticesymmetries. Finally, we qualitatively compare our results to numerics bycomputing the dimerization operator within our theory. Systems with oddcircumferences exhibit a non-vanishing dimerization that decays exponentiallywith circumference.

Finite size effects in the Z_2 spin liquid on the kagome lattice