Fermi arcs and isotope effect of the magnetic penetration depth in underdoped cuprates

The isotope coefficient $\beta$ of the magnetic penetration depth in the superconducting state is studied at T=0 for a $d$-CDW and a nodal metal model. Disregarding superconductivity the Fermi surface of the first model possesses arcs whereas the second model has no arcs. We show that a large increase of $\beta$ in the pseudogap region is generically incompatible with Fermi arcs in the pseudogap state. Thus only the second model shows a large increase of $\beta$ with decreasing doping. The required electron-phonon coupling is small and compatible with first-principles calculations based on the local density approximation (LDA).