Theory of optical conductivity in detwinned YBa${}_2$Cu${}_3$O${}_y$

We study theoretically the chain and in-plane optical conductivities of detwinned YBa${}_2$Cu${}_3$O${}_y$. We elaborate that the chain is superconducting for $y>6.67$ while insulating for $y<6.67$ due to the competition between the plane-chain coupling and the antiferromagnetic order, corresponding to a superconductor-insulator transition. Stemming also from the coupling between the plane and chain, a new peak emerges at a low frequency in the in-plane spectra in the superconducting state, while it disappears in the normal state. Our scenario accounts satisfactorily for very recent experiments.