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 $c=1$ conformal field theory with U(1) symmetry near the $K\to\infty$ instability. On the analogy of the spinless fermion, we discuss 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.