Phase separation as an instability of the Tomonaga-Luttinger liquid

Asymptotic behavior of the Tomonaga-Luttinger liquid in the vicinity of the phase-separated region is investigated in the one-dimensional $t-J$ model to study the universal property of the $c=1\mathrm{}$ conformal field theory with U(1) symmetry near the $K\to \infty$ instability. By analogy to the spinless fermion, we argue that the compressibility behaves as $\kappa \propto {(J}_c-{J)}^{-1},$ and that the Drude weight is constant and changes to zero discontinuously at the phase boundary. This speculation is confirmed by analyzing the finite-size effect from the result of the exact diagonalization.