  Despite of the success of the slave-boson theory in capturing qualitative
physics of high-temperature superconductors like cuprates, it fails to
reproduce the correct temperature-dependent behavior of superfluid density, let
alone the independence of the linear temperature term on doping in the
underdoped regimes of hole-doped cuprate, a common experimental observation in
different cuprates. It remains puzzling up to now in spite of intensive
theoretical efforts. For electron-doped case, even qualitative treatment is not
reported at present time. Here we revisit these problems and provide an
alternative superfluid density formulation by using the London relation instead
of employing the paramagnetic current-current correlation function. The
obtained formula, on the one hand, provides the correct temperature-dependent
behavior of the superfluid density in the whole temperature regime, on the
other hand, makes the doping dependence of the linear temperature term
substantially weaken and a possible interpretation for its independence on
doping is proposed. As an application, electron-doped cuprate is studied, whose
result qualitatively agrees with existing experiments and successfully explains
the origin of $d$- to anisotropic $s$-wave transition across the optimal
doping. Our result remedies some failures of the slave-boson theory as employed
to calculate superfluid density in cuprates and may be useful in the
understanding of the related physics in other strongly correlated systems, e.g.
Na$_{x}$CoO$_{2}$$\cdot$yH$_{2}$O and certain iron-based superconductors with
dominating local magnetic exchange interaction.
