Finite compressibility in the low-doping region of the two-dimensional $t{-}J$ model

We revisit the important issue of charge fluctuations in the two-dimensional $t{-}J$ model by using an improved variational method based on a wave function that contains both the antiferromagnetic and the d-wave superconducting order parameters. In particular, we generalize the wave function introduced some time ago by J.P. Bouchaud, A. Georges, and C. Lhuillier [J. de Physique {\bf 49}, 553 (1988)] by considering also a {\it long-range} spin-spin Jastrow factor, in order to correctly reproduce the small-$q$ behavior of the spin fluctuations. We mainly focus our attention on the physically relevant region $J/t \sim 0.4$ and find that, contrary to previous variational ansatz, this state is stable against phase separation for small hole doping. Moreover, by performing projection Monte Carlo methods based on the so-called fixed-node approach, we obtain a clear evidence that the $t{-}J$ model does not phase separate for $J/t \lesssim 0.7$ and that the compressibility remains finite close to the antiferromagnetic insulating state.