Measuring the inertial mass of a vortex in the cuprate superconductors

Scanning tunneling microscopy measurements by Hoffman {\em et al}.~[1] on the vortex lattice of Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ exhibit halos in the vicinity of each vortex with modulations in the tunneling conductance at a period close to 4 lattice spacings. We discuss these observations in the light of a novel theory for superfluids in the vicinity of a Mott transition~[2]. Such superfluids have a `quantum order' which is reflected in a degeneracy in the vortex excitation spectrum, and leads to modulations when a vortex is localized. Our interpretation implies that the size of the halos is determined by the zero-point fluctuations of the quantum vortices. We will show how the existing scanning tunneling microscopy measurements, when combined with measurements of the London penetration length, lead to a quantitative estimate of the vortex inertial mass. \begin{thebibliography}{99} \bibitem{1} J.~E.~Hoffman {\em et al.}, Science {\bf 295}, 466 (2002). \bibitem{2} L.~Balents, L.~Bartosch, A.~Burkov, S.~Sachdev, and K.~Sengupta, cond-mat/0408329, cond-mat/0409470. \end{thebibliography}