Conductivity between Luttinger Liquids in the Confinement Regime andc-Axis Conductivity in the Cuprate Superconductors

We calculate the interliquid conductivity for Luttinger liquids, within the so-called “confinement” regime where interliquid hopping is completely incoherent. We argue that the interliquid conductivity behaves as ${\sigma }_{\perp }\left(\omega \right)\sim {\omega }^{4\alpha }$, where $\alpha$ is the Luttinger liquid exponent. We discuss the effect of finite temperature, which is found to introduce a coherent weight into the conductivity. These results are in good agreement with experimental measurements of the frequency dependent normal state $c$-axis conductivity in the high-temperature superconductors YB${\mathrm{a}}_2$C${\mathrm{u}}_3$O ${}_{7-\delta }$ and La ${}_{2-x}$Sr ${}_x$Cu${\mathrm{O}}_4$.