London penetration depth in d-wave superconductors in the presence of a soft internal phase mode

We have recently shown that a d-wave superconductor, in presence of strong correlations,
hosts a soft internal phase mode (EPL 119, 27004 (2017)). Here, we first demonstrate, within
a Ginzburg-Landau theory, that the presence of this soft phase mode affects the behavior of
the London penetration depth in a significant way, and, that this mode must be accounted
for to understand experiments better. Next, within a t-J like model, we compute the London
penetration depth explicitly and show how the results relate to some recent experiments on cuprate
superconductors.