C-axis optical sum in underdoped superconducting cuprates

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In conventional metals, the total optical spectral weight under the real part of the dynamical conductivity remains unchanged in going from normal to superconducting state. In the underdoped cuprates, however, experiments found that the interlayer conductivity no longer respects this sum rule. Here, we find that a recently proposed phenomenological model of the pseudogap state which is based on ideas of a resonating valence bond spin liquid naturally leads to such a sum-rule violation. For the interplane charge transfer, a coherent tunneling model is used. We also obtain analytic results based on a simplification of the theory which reduces it to an arc model. This provides further insight into the effect of the opening of a pseudogap on the $c$-axis optical conductivity $\text{Re}[{\ensuremath{\sigma}}_{c}(\ensuremath{\omega})]$. The missing area under $\text{Re}[{\ensuremath{\sigma}}_{c}(\ensuremath{\omega})]$ normalized to the superfluid density, which is found to be one in the Fermi-liquid limit with no pseudogap, is considerably reduced when the pseudogap becomes large and the size of the Luttinger pockets or arcs is small.